Paper Title: Mitigating calibration errors from mutual coupling with time-domain filtering of 21 cm cosmological radio observations.



firstpage–lastpage
Magnetic field diagnostics of prominences with the Mgii k line
 3D Stokes inversions versus traditional methods
    
Jiří Štěpán1
Tanausú del Pino Alemán2,3
Javier Trujillo Bueno2,3,4

    Received XXXX; accepted XXXX
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§ ABSTRACT 
The 21 cm transition from neutral Hydrogen promises to be the best observational probe of the Epoch of Reionisation (EoR). This has led to the construction of low-frequency radio interferometric arrays, such as the Hydrogen Epoch of Reionization Array (HERA), aimed at systematically mapping this emission for the first time. Precision calibration, however, is a requirement in 21 cm radio observations. Due to the spatial compactness of HERA, the array is prone to the effects of mutual coupling, which inevitably lead to non-smooth calibration errors that contaminate the data. When unsmooth gains are used in calibration, intrinsically spectrally-smooth foreground emission begins to contaminate the data in a way that can prohibit a clean detection of the cosmological EoR signal. In this paper, we show that the effects of mutual coupling on calibration quality can be reduced by applying custom time-domain filters to the data prior to calibration. We find that more robust calibration solutions are derived when filtering in this way, which reduces the observed foreground power leakage. Specifically, we find a reduction of foreground power leakage by 2 orders of magnitude at k_∥≈ 0.5 h Mpc^-1.  








cosmology: dark ages, reionization, first stars – cosmology: observations – methods: data analysis – instrumentation: interferometers








§ INTRODUCTION

The detection of the redshifted 21 cm emission line from neutral Hydrogen during the Epoch of Reionisation (EoR) is one of the main goals of (current and upcoming) low-frequency radio telescopes such as the Low-Frequency Array <cit.>, the Murchison Widefield Array <cit.>, the Giant Metrewave Radio Telescope EoR experiment <cit.>, the Hydrogen Epoch of Reionisation Array <cit.>, and the Square Kilometre Array <cit.>. The EoR is one of the least constrained areas of cosmology, with several cosmological models predicting different reionisation scenarios. Thus, more sensitive power spectrum limits are still required to discriminate between reionisation models. The achievement of sensitive power spectrum limits will be challenged by the presence of systematic errors, either caused by instrument response <cit.> or by calibration errors due to the use of incomplete sky models during calibration <cit.>.

Measurements of the 21 cm signal are challenged by the presence of foreground emission from the Galaxy and extra-galactic sources, which are orders of magnitude brighter <cit.>. Fortunately, foreground emission is spectrally smooth, unlike the 21 cm emission line, which fluctuates rapidly <cit.>. The delay spectrum approach is a method designed to separate the 21 cm signal from the foreground emission, making use of interferometric delays to isolate the power spectrum of the 21 cm emission <cit.>. Due to the unsmooth spectral nature of the 21 cm signal, its power spectrum appears at all k modes. In contrast, the foreground emission, due to its smooth spectral emission, is limited to a wedge-like region in k-space <cit.>. The region in k space where the 21 cm signal is isolated from the foreground emission is referred to as the EoR window. 

However, the clean separation of the 21 cm signal requires the precision calibration of interferometric data <cit.>. Closely packed radio interferometric arrays, like HERA, are prone to effects of mutual coupling, where the sky signal incident on an antenna is partially reflected and subsequently received by a nearby antenna. Mutual coupling in a multi-element array such as HERA is complex, and it is difficult to fully model and characterise its impact directly <cit.>. If the effects of mutual coupling are not accounted for during calibration, they ultimately lead to unsmooth calibration errors, and 

cause foreground emission to leak into the EoR window <cit.>.  

Custom time-domain filters, called fringe rate filters, have been widely used to partially mitigate these systematics in drift-scan radio observations as a post processing step <cit.>. Recent works have also explored how these filters can be used to mitigate calibration errors from inaccurate sky models <cit.>, and from antenna displacement errors <cit.>.
However, previous works have not explored how these filters could be used to mitigate mutual coupling systematics in the raw data prior to calibration in order to improve calibration quality.

In this work, we investigate the prospect of improving HERA's calibration pipeline by filtering some of the mutual coupling features in the data before running calibration, in order to mitigate their impact on the recovered antenna gain solutions. We aim to do this through the application of fringe rate filters <cit.>. We consider two specific kinds of filters: a simple baseline-independent high-pass filter, which we refer to as the notch filter, and a baseline-dependent filter, referred to as the main-lobe filter <cit.>. We assess the calibration improvement by computing the reduced chi-squared after calibration and by further examining the structure of the recovered gains in Fourier space to assess the amount of residual spectral structure. Lastly, we show how our technique helps mitigate the foreground leakage in the EoR window. Note that the exact form of the adopted main-lobe filter is specific to this work: other works may use similar kinds of "main-lobe filters" but may have slightly different parameterizations depending on their desired outcome.

This paper is divided into three sections, in Section <ref> we give an overview of calibration and the simulation set up, Section <ref>  we discuss the application of fringe rate filters before calibration and the relevant metrics that we use to measure the improvement in calibration, and we conclude in Section <ref>. 
 





§ CALIBRATION OVERVIEW





 §.§ Sky-model based calibration


When observing the sky with a radio interferometer, the electromagnetic waves from celestial sources are measured by two antennas that are correlated, forming interferometric visibilities.
The measured complex-valued visibilities are related back to the intrinsic sky emission and the instrumental response of the antennas via the radio interferometer measurement equation <cit.>.

    V^0_ij(ν)=∬ J(s,ν))  I(s, ν)   J^*(s,ν)   e^-2πν/cb_ij·s  dldm/n(s),

where V^0_ij is the visibility formed between antenna i and j, and ^* denotes the complex conjugate, J_i(s,ν) denotes Jones terms describing the propagation effects along the path from the source to the i-th antenna in the array. I(s,ν) is the brightness distribution of the sky, ν is the frequency of the incoming electromagnetic wave, b_ij is a baseline vector connecting antenna i and j, s = [l, m, n]^T is the position vector on the celestial sphere with an origin centred at the target field (phase centre), and (l, m, n) are the direction cosines, with n = √(1 - l^2 - m^2). Note that in this paper, we will only consider a single polarization. The Jones terms are classified into two main groups: direction-dependent and direction-independent. The direction-dependent Jones varies with observing direction, these effects include the primary beam E(s,ν) response of the antenna, which varies across the sky for a fixed pointing. Whilst the direction-independent Jones terms are independent of observing direction. These effects include (not limited to) the band-pass response of the antenna B(ν) and gain amplification term G(ν).  

Note that in this paper, Equation <ref> assumes a trivial primary beam response in that the signal from the celestial source is completely absorbed by array antennas. However, in real observations, a portion of the signal is reflected due to departure from the conjugate match at the terminals of antenna feeds. If the array is compact, the reflected signal will be absorbed by a neighbouring antenna. This effect, known as mutual coupling, is a known concern for interferometric 21 cm experiments. More analysis of the upper limits of the power spectrum cite mutual coupling as one of the unresolved possibilities for their excess power <cit.>. 

Recent works have developed semi-analytic approaches to modelling the complex nature of coupling effects within an interferometric array.
<cit.> developed a semi-analytic approach for single antenna-pair coupling, which was highly effective in suppressing observed coupling in HERA Phase I data <cit.>.
Future work expanded upon this approach for multi-antenna coupling, which showed new phenomenology that matched the observed systematics in HERA Phase II data <cit.>.

In this work, we will consider only first-order, multi-antenna mutual coupling effects as described in <cit.>, where each antenna in the array absorbs most of the incident signal from the source, and some of the signal is reflected and completely absorbed by other antennae in the array (i.e., the incident signal is only reflected once). If we consider unpolarised radiation from the sky and assume that all beams in the array have identical electromagnetic properties, that is they have the same reflection and absorption coefficients, then the first-order visibilities (visibilities with first-order mutual coupling effects) are given by: 


    V^1_ij =  V^0_ij + (∑_k ≠ i  V_ik^0 X^*_jk  + X_ik V_kj^0 ),
 
where

    X_ik= i η_0/4λΓ/R   |b_ij|  e^2π i ν/c |b_ij|   h_0^2   J(b_ik) J^*(b_ki)


where η_0 is the impedance of free space, Γ is the frequency-dependent reflection coefficient, R is the real part of the impedance term and J(b_ki) denotes direction-dependent effects (primary beam response) along the path travelled by the reflected signal from antenna k to antenna i and h_0 is the antennas so-called effective height <cit.>.


In real observations, antennas in an array also impart direction-independent corruptions to the visibilities, which can be modelled using antenna-based complex gain terms. The corrupted visibilities are related to intrinsic visibilities as

    V^c_ij=g_i  V_ij  g_j^*   + n_ij,

where V^c_ij are the corrupted (or measured) visibilities, g_i and g_j are the complex gain terms of antenna i and j, and n_ij is the complex thermal noise generated by the telescope.

Antenna gain calibration is the process of deriving the direction-independent gain terms (g_i) from the data and then correcting the data to remove their effects.
This is commonly done by constructing a model of the true visibilities and then minimising a χ^2 statistic:

    χ^2 =∑_i,j |V_ij^c-g_i  g_j^*V^m_ij|^2/σ_ij^2,

where V_ij^m are the constructed model visibilities, and σ_ij^2 is the variance of the visibility, and the sum runs over all unique antenna pairs.

The chi-square minimisation depends on the accurate modelling of the intrinsic sky visibilities V_ij. It is crucial, therefore, that the sky model matches the intrinsic sky brightness as closely as possible. Unmodelled mutual coupling effects can cause significant deviation between intrinsic sky visibilities and the model visibilities in calibration, leading to calibration errors that can impart spectral structure on smooth foregrounds <cit.>.




 §.§ Redundant calibration


Redundant calibration is less reliant on a sky model compared to sky-base calibration, to solve for antenna gains, the gains are obtained by exploiting the internal redundancy of the interferometric array. Redundant baselines have the same length and orientation and, therefore, measure the same Fourier mode of the sky brightness distribution.
For example, assume we have a single baseline type, A, uniquely identified via its baseline vector b_A. For all antenna pairs i,j that share this baseline vector, the calibration equation, Equation <ref> from before now becomes

    V^c_ij = g_i   V_A  g_j^* + n_ij,

where V_A is now a parameter of the model, called the “redundant model visibility”.
This is repeated for all unique baseline types, eventually building up an overconstrained system of equations, which can be solved via a χ^2 minimization <cit.>:


    χ^2 =∑_i,j |V_ij^c-g_i  g_j^*V_A|^2/σ_ij^2,

where V_A is the corresponding redundant model visibility for the b_ij baseline vector. Note that in this paper, the χ^2 is both a function of frequency and time. The gain solutions obtained from the system of linear equations set up in the redundant calibration are not unique; there are at least four degenerate parameters that need to be solved for after redundant calibration: the overall gain amplitude, the model visibility amplitude and the phase gradient across the array in the east-west and north-south orientations <cit.>. These parameters are, in principle, a function of polarization, frequency and time and require a sky model to be constrained.  
It is in this step that, redundant calibration itself makes use of a sky model. In this study, we will refer to this step in calibration as absolute calibration to differentiate this from sky calibration or the typical sky-based calibration discussed previously.

In the case of perfect calibration, where the sky model is complete, model visibilities only differ from the data by noise. The expectation value of chi-square ⟨χ^2 ⟩ is thus two times the degrees of freedom (DoF) for complex data <cit.>. The number of DoF, in general, is given by the difference between the number of data points and the number of fitted parameters. If we consider a single polarization and the case of sky-based calibration, the DoF are given by the difference between the number of visibilities and the number of antennas:

    DoF = N(N-1)/2 - N = N_ bl - N,

where N is the number of antennas and N_ bl is the number of baselines given N antennas. If we histogram the χ^2/ DoF quantity, it is expected to follow a theoretical chi-square distribution <cit.>. 
In the case of redundant calibration, the DoF is given by:

    DoF = N_ bl-N_ ubl-N+2,

where N_ ubl is the number of unique baselines that have a single sky model visibility V_A, i.e. the number of redundant baseline groups <cit.>.





 §.§ Simulation set up


In this paper, we simulate sky models composed of diffuse emission from GSM <cit.> and compact source flux density consistent with the GLEAM catalogue <cit.>, but with no spatial clustering of sources (uniformly distributed). We also include bright radio galaxies <cit.>, modelled as compact sources. To simulate mutual coupling effects, we make use of the mutual coupling simulation formalism from <cit.> to produce visibility products with first order mutual coupling effects. Their formalism included modelling the first-order antenna-antenna coupling in radio interferometers.  Specifically, they derive a semi-analytic model of the interferometric visibility Equation <ref>, in which first-order coupling effects are considered. In their model, incident radiation enters all interferometric elements in the array, and then each element absorbs most of the incident radiation, but some of the power is then reflected due to departure from the conjugate match at the terminals of the antenna feeds. Thus, each element in the array not only absorbs the incident array from the sky but re-radiates it across the array, which is subsequently absorbed by other elements. This simulation only considers the first-order formalism, and thus, it only considers the effects of re-radiated radiation being absorbed only once by all the array elements. In this paper, we refer to simulated visibilities containing effects of mutual coupling as first order visibilities. And simulated visibilities without effects of mutual coupling are to as zeroth order visibilities.  

The simulated visibilities span a frequency range of 140-190 MHz, with a frequency resolution of 122.07 kHz. We consider observations in the Local Sidereal Time (LST) range 0^h-4^h, with a time resolution of dt=58 s. The simulation is done for a HERA array consisting of 174 antennas shown in Figure <ref>. 
Figure <ref> shows the visibility spectra from 14 m East-West redundant baselines with (first order visibilities) and without mutual coupling (zeroth order visibilities). The visibility spectra from redundant baselines have a higher amplitude at low frequency, as expected, due to the brightness of diffuse emission at low frequencies <cit.> for both zeroth and first order visibility. Primary beam models containing the effects of mutual coupling differ from the primary beam model that is without mutual coupling <cit.>. As a result, the visibility spectra from the 14 m baselines without mutual coupling (solid red line) differ significantly from the visibility spectra of the same baseline with mutual coupling. Also, notably, the visibility spectra from 14 m redundant baselines with mutual coupling deviate from each other, and these deviations are as high as 6% across the frequency band. 


We make use of gains that are similar to ones simulated by <cit.> to create mock raw data.  The gain g for the jth antenna is given by:

    g_j(ν)=A_j (ν) e^i ϕ_j(ν),

 where the amplitude A follows a frequency power law 

    A_j(ν)=A_j ( ν/150 MHz)^b_j,

and A_j and b_j are drawn for each antenna from a Gaussian distribution 𝒩∼ (x̅_A=0.30, σ_A=0.001) and 𝒩∼ (x̅_b=-2.6,σ_b=0.2), respectively. The phase ϕ_i(ν) is modeled as:


    ϕ_j(ν)=sin(w_a  ν) + cos(w_b  ν),

where w_a and w_b are drawn for each antenna from a Gaussian distribution 𝒩∼ (x̅_w=0.005, σ_w=0.0005). The mean values of the parameters A and b, as well as their standard deviation σ_A and σ_b, are informed by the gain solutions from actual HERA observations <cit.>. The mean phase variation w̅ is based on single antenna phase dependency of actual HERA gains where the cable delay and geometric phase offset have been removed, and the variation of the mean phase between antenna stations, i.e. σ_w, is chosen to be within 10%. We then add noise to the data, such that signal-to-noise ratio (SNR) of the simulated bandwidth ∼ 10^4, mutual coupling effects are known to be prominent at this dynamic range <cit.>. Note that this SNR is far higher than in real observations <cit.>.  We simulate two sets of raw data products; one includes the effects of mutual coupling, and the other does not. The latter raw data product is used to verify our pipeline and serves as a benchmark to quantify the effects of mutual coupling. 

We subdivide the band into two sub-bands, which we name low-band and high-band, with a frequency range of 140-165 MHz and 165-190 MHz, respectively. We make use of a sky model that contains no mutual coupling effects, i.e., our sky model is composed of zero-order visibilities only. First, we make use of redundant calibration to calibrate raw data. The gains from the redundant calibration step are then applied to the raw data, i.e., the raw data is partially calibrated. The calibrated visibilities are then further calibrated using the absolute calibration step. Here, the data is calibrated using the sky model (zeroth order visibilities), and the gains recovered from this step are combined with those recovered from the redundant calibration step to make a full set of gains. We then apply the full set of gains to calibrate the raw data. To quantify the robustness of the calibration,  we compute the reduced chi-square from redundant calibration and absolute calibration steps. 



 §.§ Calibration with mutual coupling effects


Figure <ref> shows the reduced chi-square obtained from the redundant calibration step for an ideal calibration scenario, i.e., where the raw data does not contain the effects of mutual coupling. The histogrammed reduced chi-square from the data set without mutual coupling, as expected, follows the theoretical χ^2 distribution, indicating that model and raw data visibilities only differ by noise, and the recovered gains from calibration are perfect (match the true gains), thus also verifying the calibration pipeline works as expected. Figure <ref> shows the recovered chi-square from the data set with mutual coupling for both low and high sub-bands. The chi-square is notably biased to higher values for both sub-bands, but notably significantly higher at the low sub-band. The mean value of the reduced chi-square χ^2 /DoF is 3.3× 10^3 and 1.1× 10^3 for low and high sub-band, respectively. This bias is due to differences in visibility products from redundant baselines (see Figure <ref>) caused by the effects of mutual coupling, since redundant calibration makes no use of the sky model. Given the mean value of the reduced chi-square, we can also infer that the redundant calibration errors are larger at lower frequencies (low band) than at high frequencies. The reason for this is that the increased brightness at lower frequencies of the diffuse emission, which couples to the primary beam response, thereby effectively enhancing differences in the apparent sky observed by two antennas in the array. Thus, when redundantly calibrating the data, larger differences are found between the visibility spectra of supposedly redundant baselines. Figure <ref> further shows that the reduced chi-square depends upon the looking direction, thus indicating that the chi-square depends on the brightness of the diffuse emission. 

The right panel of Figure <ref> show the chi-square obtained after the absolute calibration step. The mean value of the recovered chi-square χ^2 /DoF from the absolute calibration step is 1.6× 10^6 and 1.1× 10^4 for both low and high sub-band, respectively. At the low sub-band, the mean value is at least 10^2 times greater than that obtained from the redundant calibration step. The increase in chi-square is caused by the fact in our sky model we do not incorporate the effects of mutual coupling, which are found in the data, thus, the model and raw visibility products differ significantly (see Figure <ref>). In addition, calibration errors in the redundant calibration step are carried through to the absolute calibration step, and hence, thus calibration errors accumulate, and the obtained chi-square is significantly larger in the absolute calibration step. We note the recovered chi-square values are large compared to the values (χ^2 /DoF around 1-3) one would find in typical HERA observations <cit.>, this is just a consequence of the low noise (high SNR) in our simulation compared to actual observations. Small differences between model and raw data visibilities are effectively 'amplified' by the small variance σ^2 ∼ 10^-8 in our simulation, whereas in actual observation the noise variance is typically σ^2 > 10^-4.















§ FOURIER REPRESENTATION OF INTERFEROMETRIC DATA


We will now define relevant quantities that will be used to quantify the impact of mutual coupling in calibration. Interferometric visibilities are a function of frequency (ν) and time (t). We refer to the Fourier dual of frequency as the delay domain, formed by taking a Fourier transform of the visibilities across the frequency axis:


    Ṽ(τ) = ∫ V(ν)  ^-2 π i τν dν,


where τ is a delay (in seconds) and Ṽ(τ) is the Fourier pair of the frequency. 

We can also define delay-transformed antenna gains to be the Fourier transform of the antenna gains along the frequency axis:


    g̃(τ) = ∫ g(ν)  ^-2 π i τν dν.

We define the Fourier dual of time as the fringe rate and the fringe-rate visibility as the Fourier transform of the visibility along the time axis:

    Ṽ(f) = ∫ V(t)  ^-2 π i f t dt,

where f is the fringe rate (in units of Hz). For interferometric arrays that operate in drift-scan mode, the fringe rate basis also separates signals based on their relative motion through the fixed interferometric fringes, acting as another form of separation of signals on the sky <cit.>.

 


 §.§ Recovered gains from calibration

Figure <ref> shows the gains recovered from absolute calibration for both low and high sub-band. Let's first consider the gains recovered from the low-band data set. The recovered gains show a pronounced frequency structure in both amplitude and phase, caused by calibration errors resulting from unmodelled mutual coupling effects. The calibration errors are unsmooth, resulting in excess power at delays τ>250 ns (Figure <ref>). The dynamic range, i.e., the ratio of the gain amplitude at delay τ=0 to that of a specific delay τ, of the gains without mutual coupling at τ = 250 ns, is ∼ 10^5. Due to the effects of mutual coupling, this is reduced to ∼ 10^2.  Notably, the mutual coupling effects lead to excess power in the gains at a broader range of delays (τ>250 ns) compared to calibration errors from incomplete sky modelling (200 <τ< 600 ns) <cit.>. Indeed <cit.> also found that mutual coupling effects tend to contaminate a wider area of the EoR window. The recovered gains with mutual coupling at the higher band, however, have comparatively less excess power, and the excess power spans a narrower delay range 250< τ <2000 ns. This is due to the reduced brightness of the diffuse emission, within the high sub-band.









 §.§ Application of Fringe Rate Filters



We propose the use of a fringe rate filter as a way to reduce the impact of mutual coupling in calibration. The idea is to filter out some mutual coupling effects in the raw data, thus bringing the raw data closer to the sky model. We employ a similar approach to that used by <cit.> to improve calibration. We also apply fringe rate filtering to both the raw data and the sky model visibilities in the same manner for each of the different filters that we use (described below). We then apply the derived gains to the unfiltered visibilities, thus mitigating concerns of a possible cosmological signal loss.


Note that, for the two filters described below, we apply them to the data using the DAYENU filtering formalism described in <cit.>, which relies on the Discrete Prolate Spheroidal Sequences <cit.>.



  §.§.§ Notch Filters


We first consider a symmetric baseline-independent notch filter F(f) centred at f = 0 mHz fringe rate,  i.e. a high-pass filter, defined as 


    F(f) = 
    {[       10^-8, |f| ≤ f_ max;           1, |f| > f_ max ]},


with 

f_ max = 0.25 mHz,  we refer to the filter as f_25. See Figure <ref> for a visual representation of the notch filter.




  §.§.§ Main Lobe Filters



We also considered a second type of filter, which we refer to as a “main lobe” filter, because it aims to suppress emission from outside the primary beam field-of-view.
In contrast to the baseline-independent notch filter, which only filters out emission in a region near f∼0 mHz, the baseline-dependent main-lobe filter suppresses the signal everywhere except near the peak emission of the sky model in fringe rate.

Technically, the main-lobe filter is a frequency-dependent filter, but because we operate over a relatively small bandwidth (25 MHz within each band), we approximate it as frequency-independent. The response of the filter within the pre-defined fringe rate bounds is uniform, such that it can be thought of as a top-hat filter.

The bounds of the baseline-dependent main-lobe filter are determined by its centre f_0  and its half-width f_w. These parameters are determined for each baseline by fitting a Gaussian profile G(f) to the sky model visibilities in fringe rate space:


    G(f) = A   e^-(f - f_0)^2/2 σ^2,


where A is the amplitude of the Gaussian, σ is its standard deviation, and f_0 is its mean.
After the fit, we set the main lobe filter centre to be f_0 and its half-width to be f_w = 2σ, such that the full width of the main lobe filter is 4× the fitted Gaussian's standard deviation. We denote the baseline dependent main lobe filter as M_de (Figure <ref>).







 §.§ Using the Notch Filter to Improve Calibration

In Figure <ref>, we have noted that the recovered chi-square is dependent on the looking direction (LST), we have shown that the chi-square is coupled to the diffuse emission, that is, it is higher when the diffuse emission is bright. To show the effectiveness of the fringe rate filters we consider a field where the chi-square is largest (bright diffuse emission), i.e., we are considering the 'worst case-scenario'. We consider a field centred at an LST range 0^h-1^h. We first examine the χ^2 obtained after the absolute sky calibration step. The sky emission can be separated in fringe space, as different parts of the sky occupy different fringe rate values, for an array observing in tracking mode i.e., with the sky emission drifting across the array <cit.>. For a baseline in the HERA array, we note, especially, that emission originating from the celestial South Pole will occupy fringe rate value zero, as sources at the celestial South Pole do not drift across the array <cit.>. Thus, we expect these sources to fall on primary beam side lobes. Due to the unsmooth spectra of the primary beam with mutual coupling <cit.>, when this emission couples with primary beam response, we expect the resulting calibration errors to be unsmooth, as shown in Figure <ref>. Note, however, that this is only true for a baseline with a non-zero East-West projection <cit.>. <cit.> also showed that mutual coupling effects occupy in fringe rate space occupy fringe rate values near zero. Thus, in this paper, we design fringe rate filters to filter out mutual coupling effects at zero fringe rate. We employ a baseline cut and downweight all baselines during the calibration process that have an East-West projection of less than 30 m, to ensure that all baselines have a peak of the main lobe emission (in fringe rate space) at fringe rate values greater than 0.5 mHz.  

Figure <ref> shows the chi-square obtained for the low sub-band when a notch fringe filter is applied to the data set with mutual coupling before calibration. The chi-square has reduced significantly, with a mean value of 1.7× 10^3. This represents a reduction in χ^2 of factor ∼ 10^3. This occurs because the filter suppresses some of the mutual coupling features in the raw data, thus bringing the raw data closer to the model, leading to a smaller chi-square value. A smaller χ^2 value is indicative of robust calibration. Thus, we expect the recovered gains to improve as well. Figure <ref> shows the recovered gains after applying fringe rate filters. The recovered gains have comparatively less frequency structure in both amplitude and phase, and as a result, the delay transform of the gains has comparatively less excess power at delays τ >250 ns for both the main lobe and notch filter. The dynamic range is improved by a factor ∼ 10 at delays τ≈ 250 ns for both high and low sub-band, when the notch filter f_25 is applied. However, the gains recovered after applying the main lobe filter show comparatively less improvement (compared to the notch filter), in terms of dynamic range at delays 250<τ<1000 ns at low sub-band. Similarly, at the high sub-band, the notch filter performs comparatively better than the main lobe filter over the delay range  250<τ<500 ns. Notably, the gains recovered from the notch-filtered data have a higher noise floor compared to unfiltered data, and this is due to lower SNR as some of the sky signal is suppressed by the filter, thus reducing the SNR. However, the gains recovered when the main lobe filter is used have a lower noise floor compared to the notch filter, and this is because the main lobe filter retains most of the sky signal, and in addition, it also significantly attenuates some of the noise. Note that the current HERA calibration pipeline includes smoothing the derived gains, i.e., a prior on the spectral structure of gains is imposed, and this mitigates the some of effects of mutual coupling <cit.>. However, the advantage of using fringe rate filters is that, if there is real spectral structure in the intrinsic antenna gains, say above ∼ 100 ns, then the use of fringe rate filters has a better chance of finding the true gain response.

We can see that the notch filter performs better than the main-lobe filter at lower delays τ<500 ns, given the difference in the dynamic range of the recovered gains. The main key to understanding the improved performance of notch filters is to examine closely the filtering and calibration process. The main key to getting robust gains after calibration is to minimise the differences between the sky model and raw data visibility products. The main lobe filter is designed to retain the main lobe emission in fringe rate, and for a baseline with an East-West baseline projection greater than 30 m, this emission also falls within the main lobe of the primary beam, with sidelobe emission occupying fringe rate near zero. Although the effects of mutual coupling are smaller in the main lobe of the primary beam compared to the sidelobes, there are still significant differences in the antenna primary beam main lobe, between mutual coupling and non-mutual coupling beam models <cit.>, and these primary beam differences would contribute significantly to calibration errors, even more so when bright emission falls on the main lobe of the beam (see Figure <ref>). Unlike the sky emission falling on the sidelobe, the sky emission falling on the main lobe (with some mutual coupling effects) is not significantly attenuated. Thus, it dominates calibration. Mutual coupling effects in the main lobe of the beam have a much larger impact on the calibration compared to mutual coupling from the sidelobe of the primary beam, and are typically relatively smooth in frequency compared to side lobe mutual coupling effects. Thus, we would expect calibration errors due to mutual coupling effects to be relatively smooth in frequency as compared to mutual coupling effects from the sidelobes. The key goal of the filters is to suppress a large portion of emission where mutual coupling effects are pronounced, whilst retaining a good portion of the sky signal that can be used for calibration. On baselines with short East-West projection (main lobe emission is close to  f∼ 0.5 mHz), the baseline independent notch filter, filters a portion of the emission that lies in the main lobe emission,  whereas the main lobe filter retains this emission, which may contain some mutual coupling effects. 

An example of such baseline emission is shown in Figure <ref>. The shoulder emission (emission at fringe rate value f≈ 0.25 mHz) in the main lobe is filtered out by the notch filter (green), minimising the impact of mutual coupling effects in the main lobe emission, thus bringing the raw data closer to the sky model data, reducing errors in calibration. However, the baseline-dependent main lobe filter (black) retains this emission. Figure <ref> shows the visibility spectra of the main lobe and notch filtered visibility of both zeroth order (without mutual coupling, solid line) and first order (with mutual coupling, dotted line) visibilities. The main lobe filtered visibilities (with and without mutual coupling) deviate significantly from each other as the main lobe filter essentially fails to attenuate the relatively smooth frequency varying unmodelled mutual coupling terms, which can be seen as the ripples in spectra (black dotted line). Thus, model and raw data visibilities differ significantly, leading to larger calibration errors. However, the notch-filtered visibility closely match with each other, as ripples in visibility spectra caused by mutual coupling effects are attenuated significantly, leaving an overall smooth visibility spectra for both filtered and unfiltered. The difference between filtered and unfiltered visibility spectra is notably small for the notch filter compared to the main lobe filter,  thus, the notch filter would yield smaller calibration errors, leading to more robust recovered gains that can be used to calibrate the data. In order to achieve the same improvement as the notch filter in the calibration, particularly at the low delays τ <500 ns, we need to implement a E-W baseline cut larger than 50 m. In Figure <ref>, we show that the recovered gains from calibration using the notch filter have the same dynamic range as gains recovered from the main lobe filter, but with an E-W baseline cut of 50 m. 



















 §.§ Foreground Power Spectrum


In this section, we will introduce the power spectrum as a metric to assess the impact of fringe rate filters on calibration. We compute the per-baseline power spectrum of the visibilities, P(τ, b), following the delay approximation <cit.>.



    P(τ, b)=|V_b(τ)|^2  (λ^2/2k_B)^2(D_c^2Δ D_c/B_ eff)(1/Ω B_ eff),



where λ is the centre wavelength of observing bandwidth, k_B is the Boltzmann constant, B_ eff is the effective bandwidth, D_c is the comoving distance at the redshift of our measurement, Δ D_c is comoving distance parallel to the line of sight, Ω is the field of view solid angle, Ṽ_b is the Fourier transformed visibility, and b is the visibility baseline length.
We can map τ to the line-of-sight cosmological Fourier wavevector k_∥ using the relation <cit.>:

    k_|| = 2πν_21 H_0 E(z)/c(1+z)^2 τ,

where ν_21 = 1420 MHz, H_0 is the Hubble constant, E(z)=[Ω_m(1+z)^3+Ω_Λ]^1/2, Ω_m is the normalized matter content and Ω_Λ is the normalized dark energy content.

In Figure <ref>, we show the residual visibilities obtained after calibrating raw data with gains recovered from the data set with and without mutual coupling. The residual visibilities are obtained by taking the difference between calibrated visibilities and the sky model visibilities with mutual coupling effects. Recall that simulated true sky model visibilities contain effects of mutual coupling, thus a perfect calibration needs to yield calibrated visibilities that match closely with the sky model visibilities that contain effects of mutual coupling. As expected, when the raw data visibility products are without mutual coupling, we get a perfect calibration, yielding robust gains, and thus, the calibrated visibilities differ from the sky model visibilities by noise. Hence, the residual visibilities obtained are just noise (left panel). However, when our simulated raw data includes the effects of mutual coupling, our calibration is not perfect, and the recovered gains (Figure <ref>), which are subsequently used to calibrate the raw data are erroneous. Thus calibrated visibilities differ significantly from the true sky model visibilities. Additionally, when the recovered gains are used to calibrate the raw data because of their non-smooth spectral structure (due to calibration errors) they impart spectral structure on the relatively smooth foreground spectra, some of the foreground emission is pushed to higher delays beyond the horizon limit (white line). This is most prominent on the shorter baselines. In the panel second panel from the left, the residual visibilities have foreground emission at short baselines beyond the horizon limit, and this residual foreground emission extends up to delays of τ∼ 500 ns.  When the notch filter f_25 is applied (third panel from left) before calibration, more robust gains are used to calibrate the unfiltered raw data. We find that the calibrated visibilities closely match the sky model visibilities. As a result, the foreground residual emission beyond the horizon limit is significantly reduced by a factor of ∼ 10^2. The excess residual foreground within the horizon line poses no issue, as this region in delay space is expected to be occupied by foreground emission. However, when the main lobe filter M_be is used (right panel), there is comparatively less reduction in the foreground emission in residual emission compared to the notch filter. This is expected as the gains recovered when the main lobe filter is used are not as robust as gains obtained through the use of the notch filter, and can therefore cause foreground leakage when they are used to calibrate the data.  

Figure <ref> further shows the residual power spectrum for 14 m redundant baselines for both low and high sub-bands. The foreground residual power is significantly less in the high sub-band than in the low sub-band as the foreground residual emission is reduced to the noise level at k_∥∼ 0.75 h Mpc^-1 in high sub-band, but at low sub-band, there is still residual power at k_∥ > 1.5 h Mpc^-1 (cyan line). This is again due to the increased mutual coupling effects at the low sub-band.  We also note the main-lobe filter also suppresses foreground residual emission (black line). However, the suppression is not as significant as the notch filter (green line) in both the low and high sub-band. In the low band, the notch filter suppresses residual foreground by a factor ∼ 10 more than the main lobe filter at k_∥ < 0.75 h Mpc^-1. Additionally in the low sub-band, with the notch filter, the residual emission is reduced to levels below the noise floor at k_∥≈ 0.60 h Mpc^-1, whereas for the main lobe filter, this only occurs at k_∥≈ 0.75 h Mpc^-1. In the high sub-band, the reduction of residual power by the use of the notch filter is by a factor of ∼ 10^3 larger than that of the main lobe filter at k_∥≈ 0.5 h Mpc^-1. Thus, the use of the f_25 notch filter produces robust gains that significantly reduce the foreground power leakage.


Lastly, Figure <ref> shows the residual power from 14 m redundant baselines at an LST centred at 2^h, typically used for calibration of HERA data <cit.>. The sky at this LST has strong compact sources and the diffuse component of the sky is colder compared to the LST of 0^h. As a result, the error in the calibration is reduced significantly, as evident by the low recovered chi-square (Figure <ref>). We again note that in this field the use of fringe rate filters improves the calibration as the foreground residual power is reduced by 10^2 at k_∥≈ 0.5 h Mpc^-1. We, however, note that there is no significant difference in the improvement in calibration when either the main lobe or notch filter is used, particularly at the high sub-band. There is slightly better performance of the main lobe filter at low sub-band at k_∥≈ 0.25 h Mpc^-1. In this field, the diffuse emission component falling on the main lobe is cold, compared to bright compact sources, however, the hot diffuse component still resides in the side lobes of the primary beam and thus would introduce calibration errors if the emission is not filtered out (cyan line). When this emission is filtered through with a notch filter (green line), the calibration is improved, and even more so when the main lobe is used, due to more of the emission being filtered. In summary, both filters have shown effectiveness at reducing foreground power leakage due to mutual coupling effects, with the notch filter showing better performance when the diffuse emission dominates the sky. Both filters were also shown to be effective at mitigating calibration due to the use of an incomplete sky model during calibration <cit.>, especially the main lobe filter at LST of 2^h. Thus, the main lobe filter can be used to mitigate both calibration errors due to incomplete sky modelling and the effects of mutual coupling.  











§ CONCLUSION



In this paper, we investigate the prospect of mitigating the effects of mutual coupling in calibration through the application of fringe rate filters prior to calibration. We make use of two types of filters; baseline dependent main lobe filter and baseline independent notch filter. The main lobe filter is designed to attenuate all emission that in not within the main lobe of the primary beam, whilst the notch filter is designed to attenuate emission that is near zero fringe rates. For a baseline in the HERA array with an East-West projection greater than 30 m, this emission will typically resides on the sidelobes of the primary beam. 

We show that mutual coupling effects lead to non-smooth calibration errors, which impart spectral structure in gains derived from calibration in the delay τ >250 ns. When these gains are used to calibrate data, we find significant contamination of the EoR window by foreground emission, which is about 4 orders of magnitude at k_∥≈ 0.5 h Mpc^-1 at z=8.3 for a 14 m redundant baselines. By applying the filters before calibration we find in this paper that the calibration significantly improves, as the recovered gains are much more robust i.e. much closer to the true gains, and when these gains are used to calibrate data we observe a significant decline in foreground contamination of EoR window. The foreground contamination is reduced by at least 2 orders of magnitude at  k_∥≈ 0.5 h Mpc^-1. In addition, we find that the application of the notch filter leads to reduced calibration errors compared to the main lobe filter. We find that the foreground contamination is an order of magnitude below for notch filter compared to the main lobe filter. This is mainly due to mutual coupling effects that are within the main lobe of the primary beam, to which the main lobe filter retains, thus resulting in erroneous gains compared to the notch filter, where a portion of this emission is filtered.

Although the use of fringe rate filters has shown significant improvement in the calibration, we, however, note that more investigation is still needed, because real observations normally include flags, that can potentially affect the filtering of data. When data containing flags is Fourier transformed along the time axis, we expect there to be “spectral leakage” resulting in the mixing of fringe rate modes of the sky due to the sidelobe structure introduced by the null/flags in the data. Thus, flagging on the time axis can reduce the effectiveness of the filter. To reduce the impact of missing data points, the data can be inpainted prior to applying the filters thus preventing the mixing of fringe rate modes of sky prior to filtering the data. A variety of inpainting techniques for radio data could be leveraged in this effort <cit.>.



§ DATA AVAILABILITY

Data used in this work may be made available upon reasonable request to the corresponding author. The code used for the analysis of this work is publicly available at https://github.com/Ntsikelelo-Charles/Fringe_rate_filtershttps://github.com/Ntsikelelo-Charles/Fringe_rate_filters. This work relied on publicly available and open-sourced community Python software, including numpy<cit.>. Visibility simulations were made using matvis <cit.>, and simulation of mutual coupling effects hera_sim (https://github.com/HERA-Team/hera_simhttps://github.com/HERA-Team/hera_sim) and pyuvdata <cit.>. 



§ ACKNOWLEDGEMENTS


This work was supported in part by the Italian Ministry of Foreign Affairs and International Cooperation”, grant number ZA23GR03. NK acknowledges support from NASA through the NASA Hubble Fellowship grant #HST-HF2-51533.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. MGS acknowledges support from the South African Radio Astronomy Observatory and National Research Foundation (Grant No. 84156). OMS's research is supported by the South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation (grant No. 81737). AL and RP acknowledge support from the Trottier Space Institute, the Canadian Institute for Advanced Research (CIFAR) Azrieli Global Scholars program, a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant, a NSERC/Fonds de recherche du Québec-Nature et Technologies NOVA grant, the Canada 150 Research Chairs Program, the Sloan Research Fellowship, and the William Dawson Scholarship at McGill.






mnras
 















 Paper Title: gwforge: A user-friendly package to generate gravitational-wave mock data.





kbc595@psu.edu


0000-0003-4750-5551]Koustav Chandra
Institute for Gravitation & the Cosmos and Department of Physics, The Pennsylvania State University, University Park PA 16802, USA
















§ ABSTRACT 

The xg gw detectors, with their improved sensitivity and wider frequency bandwidth, will be capable of observing almost every cbc signal from epochs before the first stars began to form, increasing the number of detectable binaries to hundreds of thousands annually. This will enable us to observe compact objects through cosmic time, probe extreme matter phenomena, do precision cosmology, study gravity in strong field dynamical regimes and potentially allow observation of fundamental physics beyond the standard model. However, the richer data sets produced by these detectors will pose new computational, physical and astrophysical challenges, necessitating the development of novel algorithms and data analysis strategies. To aid in these efforts, this paper introduces , a user-friendly, lightweight Python package, to generate mock data for xg detectors. We demonstrate the package's capabilities through data simulation examples and highlight a few potential applications: performance loss due to foreground noise, bright-siren cosmology and impact of waveform systematics on binary pe.






§ INTRODUCTION
 




    
    
    



The xg ground-based gw detectors, such as the ce and the et, promise to significantly improve both the detection horizon and the frequency bandwidth of observable cbc signals <cit.>. This expanded detection horizon will allow us to observe cbc signals from greater distances, thereby increasing the number of detectable binaries by several orders of magnitude. Additionally, the increased bandwidth of these detectors could allow for the observation of a cbc signal over durations spanning minutes to even hours <cit.>. 

Given this, the xg gw detectors will have far-reaching scientific potential. However, the “richer” gw detector data will present computational, physical and astrophysical challenges. Existing gw analysis tools, waveform models and detector calibration techniques will be insufficient in xg era <cit.>. Also, the sheer volume of observed signals will necessitate the development of novel algorithms for detecting the signals and extracting the science out of them. It is, therefore, imperative to quantify the limitations of current tools when employed in xg data analysis. 


To support this effort and ensure that xg detectors will achieve their scientific potential, it is crucial to generate playground datasets to allow the gw community to develop and tune their algorithms. Such an effort will drive the development of new data analysis strategies, such as faster search and pe techniques, improved waveform modelling, dealing with non-stationary noise and methods to identify subtle physical effects, such as finite size effects in ns. 

This paper introduces , a user-friendly Python package to simulate the mock data for the proposed xg detectors based on our understanding of our universe. 

One primary reason for developing is the anticipated abundance of signals in xg data, necessitating systematic data generation that accounts for realistic population models and the complexities of overlapping signals. This code is built on existing gw packages, such as , ,  and .



The subsequent sections of this paper are structured to highlight the user-friendly nature of the code. Sec <ref> provides an overview of the code, including installation instructions in Sec. <ref>. Sec. <ref> provides the necessary theoretical background and an example to generate a day's worth of coloured Gaussian xg network data with cbc signals. Sec. <ref> is devoted to a few studies one can do using the prepared dataset in the previous section, and we conclude in Sec. <ref>.




    



§ CODE OVERVIEW






 §.§ Installation


The source code is available on the repository <cit.>, which includes instructions for contributing to code development. Comprehensive documentation regarding installation, functionality, and user syntax is provided on the project’s website <cit.>. Example setup scripts discussed in this work are available in the repository. 




 §.§ Packages



At the top level, comprises three main packages: , and . The package provides functions essential to generate individual compact binary properties θ based on hyper-parameters Λ that describe the population characteristics. These characteristics are defined by phenomenological population models or detailed physical simulations, such as population synthesis or N-body dynamical simulations generated using population synthesis codes such as Compas <cit.>. integrates all phenomenological population models from  and includes additional distribution functions such as a  and . Users can also utilise outputs from population synthesis simulations to generate their own populations and define cosmological parameters, such as the Hubble constant H_0, the CMB temperature T_CMB, etc., to specify their Universe.

The package provides core functionality for generating either zero-noise or coloured Gaussian noise given a noise psd. It implements current and future ground-based gw detectors, detailing their locations, orientations, and various noise PSDs of existing and future instruments. Users can also implement custom gw detectors akin to  <cit.>. Additionally, this package allows users to introduce data gaps, enabling the simulation of detector downtimes. This is important, as it creates a more realistic scenario, reflecting the potential for missing signals or having signals that are only partially present. By accounting for detector downtimes and data gaps, one can better understand the impact of these factors on gravitational wave detection and analysis, thereby improving the robustness of signal detection and parameter estimation methodologies. 



Finally, the  module provides essential functionalities for injecting or adding simulated gw signals into the data. It supports all standard waveform approximants available via the  package <cit.> and/or those made available via 's waveform plugin.

Future updates will expand this module to include the capability to add non-Gaussian transients, SNe signals, and other gw transients, thereby making the network output more closely “mock” the real data.




 §.§ Executables



To facilitate usability, we developed three key executables using the aforementioned packages: , , and . Each executable operates based on a configuration file tailored to its specific function. Additionally, the  executable integrates all three tools and utilises the high-throughput computing framework HTCondor. This integration significantly accelerates the data generation process and simplifies user interaction by managing all ancillary tasks automatically. 


The following section provides an explicit example of using these executables.





§ SIMULATING A DAY'S WORTH OF MOCK DATA








 §.§ Background






The observed gw strain s(t |θ, I) at a detector I can be expressed as:

    s(t |θ, I) = 1/D_Lℝ[F^I h]

where θ is a vector representing the source's properties,  D_L is the luminosity distance to the source, ℝ denotes the real part, and

    F = F_+^I + iF_×^I

is the antenna response function that depends on the sky location (α, δ) and the polarization angle (ψ). The term h represents the complex binary waveform on a sphere. This waveform can be decomposed as a sum of spin-weighted spherical harmonic modes h_ℓ m, so that the waveform along any direction (ι, φ) in the binary's source frame is given by:

    h = ∑_ℓ≥ 2∑_m=-ℓ^ℓ^-2Y_ℓ,m (ι, φ) h_ℓ m(t |Ξ)

Here, ^-2Y_ℓ,m (ι, φ) are the spin -2 weighted spherical harmonics, and (ι, φ) represent the binary's inclination and azimuth, respectively. The vector Ξ denotes the binary's intrinsic parameters whose dimensionality depends on the binary type. For example, in quasi-spherical bbh, Ξ includes the two component masses m_i and the six spin degrees of freedom χ_i. In contrast, for quasi-circular or non-precessing bns, Ξ represents the component masses m_i,  the tidal parameters Λ_i and the aligned spin components χ_i ·L̂, with L being the binary's orbital angular momentum.

Since the source's total merger rate and redshift distribution determine the number of signals in a given simulated dataset, begins by first estimating the average time interval ⟨Δ t ⟩ between two adjacent signals of the same type given the source population and the cosmological parameters. It assumes that all cbc systems are formed through isolated formation channels involving a common envelope phase. This allows it to derive the redshift distribution of cbc in the source frame as follows <cit.>:

    ℛ(z)    ∝∫_τ_min^∞ℛ_∗ [z_f(z, τ)] P(τ)dτ
       ∝∫_z^∞ℛ_∗ (z_f) P[τ(z, z_f)] dt(z_f)/dz_f  dz_f .

Here, ℛ_∗ is the cosmological star formation rate density, P(τ) is the probability density function of the delay time, z_f is the redshift at which the stellar binary forms, z is the redshift at which the compact binary merges, τ=t(z)-t(z_f) is the delay time and t(z) is the lookback time at redshift z. The proportionality constant is defined such that ℛ(z=0) = ℛ_0 where ℛ_0 is the local merger rate density of the cbc source.

The lookback time,

    t(z) = 1/H_0∫_z^∞dz/(1+z) √(Ω_Λ + Ω_m(1+z)^3),

depends on the Hubble constant, H_0, the dark matter, Ω_Λ and matter density, Ω_m. All these cosmological parameters, along with the star formation rate, are adjustable by the user, allowing them to simulate their own Universe. However, currently only implements P(τ) ∝ 1/τ.

Given this, the merger rate per redshift bin in the detector frame is then given by <cit.>:

    dR/dz = ℛ/1+zdV(z)/dz,

where the comoving volume element is given as:


    dV(z)/dz = c/H_04 π D_L^2/(1+z)^2 √(Ω_Λ + Ω_m(1+z)^3) .



Here, c is the light speed, and D_L is the luminosity distance to the source defined as:

    D_L =  c(1+z)/H_0∫_0^z dz'/√(Ω_Λ + Ω_m(1+z')^3)

The 1/(1+z) factor in Eq. (<ref>) accounts for time dilation.

obtains ⟨Δ t ⟩ as follows:

    ⟨Δ t ⟩ = 1 / ( ∫^z_max_0 dR/dz dz ) .

where z_max is the maximum redshift used in the simulation. Based on ⟨Δ t ⟩ and the total analysis time, T_a of the network, determines the expected number of signals, N = T_a / ⟨Δ t ⟩, and correspondingly draws the binary parameters from the provided population models.


Below, we discuss an example to build a mock catalogue based on our current understanding of bbh binary population distribution. 




 §.§ Simulating the mock GW catalogue


To create a mock bbh catalogue using , one must first define a configuration file like the one provided below:

    [style=bash]
[Redshift]
redshift-model = MadauDickinson
redshift-parameters = `gamma': 2.7, `kappa': 5.6, `z_peak': 1.9
local-merger-rate-density = 22
maximum-redshift = 30
; custom cosmology
cosmology = custom 
H0 = 67.7
Om0 = 0.31
Ode0 = 0.69
Tcmb0 = 2.735
; analysis start time
gps-start-time = 1893024018

[Mass]
mass-model = PowerLaw+Peak
mass-parameters = `alpha':3.37, `beta': 0.76, `delta_m':5.23,  `mmin':4.89, `mmax':88.81, `lam':0.04, `mpp': 33.60, `sigpp':4.59

[Spin]
spin-model = default
spin-parameters = `mu_chi':0.26, `sigma_squared_chi':0.02, `sigma_t':0.87, `xi_spin':0.76

[Extrinsic]
    



Within the  section, users must specify the redshift evolution model, local merger rate density in , analysis start time (in GPS time), and the maximum redshift. Currently, supports two different redshift evolution models: the power-law redshift evolution model and the other that convolves the Madau-Dickinson star-formation rate with an inverse time-delay model to determine the cbc's redshift evolution model <cit.>. Users can define custom cosmological and redshift parameters; otherwise, default values from <cit.> and Madau-Dickinson parameters are used <cit.>.

Similarly, the  and  sections provide options for various phenomenological mass and spin distribution models, as detailed in <cit.>. Besides predefined models, can utilize outputs from population-synthesis codes, creating interpolated distribution functions from variable and associated probability arrays, enabling sampling from simulated distributions.

Similarly, the  and  sections offer a selection of phenomenological mass and spin distribution models, as documented in <cit.>. 

The  section deals with the other extrinsic parameters. Unless otherwise specified, employs the following default distribution choices for these parameters (See Table <ref>) for definitions of the symbols):





Finally, if the binary contains a ns or a pair of ns, can set the tidal parameters Λ_i based on an eos. This necessitates a pre-shipped  table, examples of which can be found at <cit.>. Additional configuration file examples are also provided at the same location.

Once the configuration file is defined, the next step is to execute:

[style=bash]
    gwforge_population –config-file bbh.ini –output-file bbh.h5


to create the mock bbh catalogue and store as in an  file.





 §.§ Mock noise generation



After generating the mock gw catalogue, the next step is to define the detector network and simulate their noise. This can be achieved by creating a configuration file, for example, , as shown below:

    [style=bash]
[IFOS]
detectors = [`CE20', `CE40', `ET']
sampling-frequency = 8192
noise = gaussian
    

and executing:


[style=bash]
    gwforge_noise –config-file xg.ini –output-directory output/data –gps-start-time 1895616018 –gps-end-time 1895656978


This initialises a three-detector network consisting of two L-shaped ce observatories at two fiducial locations: ce40 off the coast of Washington State and ce20 off the coast of Texas, along with an underground triangular et located in Sardinia, Italy <cit.>. The simulated noise thus produced will be Gaussian coloured by the detector noise psd, with data sampled at 8192 Hz over approximately one day. The data will be stored as  files in the  directory, named in the format , with channel names . To simulate zero-noise realisations, the user has to set the flag .

also includes the two LIGO detectors in the US <cit.>, Virgo in PISA <cit.>, LIGO-Aundha in India <cit.> and the KAGRA detector in Japan <cit.>. It is also possible to define new detectors characterised by their geometry, location and frequency response. This feature is particularly useful for developing the science case for new proposals and optimising the design and placement of new detectors, such as placing the et observatory in the Netherlands or building a pair of et observatories at two different locations <cit.>.




 §.§ Mock data generation






The final step involves injecting the simulated gw signals into the network data. Similar to the previous steps, this requires a configuration file, for example, , as shown below: 


[style=bash]
[IFOS]
detectors = [`CE20', `CE40', `ET']
channel-dict = `CE20':`CE20:INJ', `CE40':`CE40:INJ', `ET':`ET:INJ'
sampling-frequency = 8192
minimum-frequency = 6

[Injections]
injection-file = bbh.h5
injection-type = bbh
waveform-approximant = IMRPhenomXO4a



As with the noise generation step, the  section requires specification of the detector network and sampling frequency. Additionally, this section must include the minimum frequency cut-off for signal generation and the channel names for the frame files. If not specified, assumes that the reference frequency for measuring the spin angles is the same as the minimum frequency used for signal generation.

The  section should contain the following keys: , which points to the file containing the mock bbh catalogue (e.g., ); , which specifies the type of injection being made; and , which indicates the waveform model to use for signal generation, with  being an example <cit.>. Currently, supports all waveform approximants available via the  package <cit.>. To use waveforms beyond those available in , a new plugin package must be created to advertise the new waveform model to for waveform generation <cit.>.




To inject the signal waveforms into the generated data over the specified GPS time interval, the user needs to execute the following command:

[style=bash]
gwforge_inject –config-file forged-data/inject-bbh.ini –data-directory /output/data –gps-start-time 1895616018 –gps-end-time 1895656978        



This command injects the signal waveforms into the generated noise over the specified GPS time interval, creating the mock data for further study.




 §.§ Using 




Since the data generation process can be computationally expensive and time-consuming, users are encouraged to take advantage of , which leverages the high-throughput computing framework HTCondor. This significantly accelerates the data generation process by abstracting all other housekeeping logic away from the user. Once the mock catalogues are generated and the different configuration files are created, the following commands can be executed:

[style=bash]
# Activate Conda environment
conda activate gwforge-test ||  echo "Failed to activate Conda environment." >   2; exit 1; 

# Set output directory
output_directory=forged-data

# Workflow submission script
gwforge_workflow     –gps-start-time 1895616018     –gps-end-time  1895702418     –output-directory "output_directory/output" –noise-configuration-file "output_directory/xg.ini"     –bbh-configuration-file "output_directory/bbh-inject.ini" –bns-configuration-file "output_directory/bns.ini"     –nsbh-configuration-file "output_directory/nsbh.ini"  –workflow-name work –submit-now


This will create an HTCondor meta-scheduler to manage dependencies between jobs and systematically parallelize the tasks, accelerating the mock data generation process. Fig <ref> shows 1024s of simulated data when assuming Gaussian noise, including signals from Population-A (See Sec. <ref> for details). As can be seen, the data will never be “signal-free”. 





§ EXAMPLE USAGE OF THE MOCK DATA




 §.§ Foreground Noise


The increased bandwidth of xg detectors will allow low-mass cbc signals to remain within the detector's sensitive bandwidth for hours. Consequently, multiple signals will simultaneously appear in the data stream, the number of which will depend on the sources' astrophysical merger rate density. Characterised by their rapid frequency evolution, these signals will predominantly overlap at low frequencies and “separate” as they approach merger. Therefore, current modelled detection techniques can identify and measure their merger time to𝒪(10ms)accuracy <cit.>. As a result, unlike in LISA, where overlapping signals lead to source confusion, these signals will be resolvable and only contribute to foreground noise.



However, as the number of overlapping signals per time-frequency bin is small, their combined contribution won't obey the central limit theorem. Therefore, the data cannot be assumed to follow a wide-sense stationary Gaussian process — an assumption critical to the currently used noise psd estimation technique such as the Welch averaging <cit.> and the BayesLine method <cit.>. As a result, using these techniques without modification can result in biased psd estimates. 
Such a deviation can introduce systematic biases in parameter inference and reduce cbc detectability.

To demonstrate this, we simulate multiple realisations of 512s mock ce40 data using the  package and estimate the noise psd using the Welch method. Mathematically, this method assumes the data to be wide-sense stationary and divides the discrete time domain data d[j] into K overlapping segments d_k[j] of length L, each windowed by w[j] which for our purpose, is the Hann truncation function. The discrete Fourier transform d̃_k[i] of each windowed segment is computed using an FFT routine, and the periodogram P_k[i] is obtained as

    P_k[i] = 1/U |d̃_k[i]|^2,

where U = ∑_n=0^L-1 w[j]^2. We, following <cit.>, use mean-median averaging of the periodograms over all segments to obtain the final psd estimate. This approach significantly reduces the variance of the spectral estimate due to glitches, thereby providing a more accurate representation of the data's power distribution across frequencies.

We assume two different source populations — population-A quantified by bns and bhns merger rate density ofℛ_BNS^A=320/Gpc^3/yr,  ℛ_BHNS^A=90/Gpc^3/yrrespectively and population-B with merger-rate density ofℛ_BNS^B=1700/Gpc^3/yr,  ℛ_BHNS^B=200/Gpc^3/yr<cit.>. While overlapping bbh signals will lead to larger foreground noise, their lower local merger rate and shorter duration in the detector band suggest that they will rarely contribute to the foreground noise. We therefore useℛ_BBH=30/Gpc^3/yrin both the cases.

Fig. <ref> shows the simulated and the estimated ce40 asd, with the latter obtained using the Welch method. It is estimated by dividing multiple-realisation of 512s of data into 16s subsegments and 50% overlap, akin to that used in a PyCBC search. Consistent with previous works <cit.>, we find that the deviation is mainly concentrated between 6-30Hz, reaching≳10%at 15Hz for the upper merger rate. This can reduce the detectability of bbh with a high detector frame (redshifted) total mass, as shown in Fig. <ref>. In particular, depending on the binary's detector frame mass, the horizon distance [The horizon distance, D_H, for our case, is the farthest luminosity distance to which we can confidently detect an optimally oriented, located, non-spinning symmetric mass bbh assuming an optimal snr threshold of 8.] loss to systems withM_T(1+z) ≳1000M_⊙can be≳15%. Similar to the Welch estimate, the median BayesLine output, which is widely used in gwpe, will also show a deviation, unless a joint spectral and signal estimate is performed with precision (see Appendix <ref> for a discussion).



    

 §.§ Bright-siren cosmology
gw signals from cbc sources can be used as probes for cosmic expansion <cit.>. The luminosity distance,D_Lto a cbc can be directly estimated using the gw waveforms(t|θ) ∝1/D_Lwithout needing external distance calibrators <cit.>. If the cbc's cosmological redshift,z, can be estimated by other means, then one can infer the local expansion rate,H_0and other cosmological parameters <cit.>, such as the fractional energy density of dark energy,Ω_Λand dark matter,Ω_m. For example, observing the em counterparts from GW170817 allowed us to identify the source's host galaxy, NGC 4993, enabling the first direct measurement ofH_0using gw <cit.>.

Another promising em-bright cbc source is bbh mergers occurring in gas-rich environments such as agn disks <cit.>. bbh in such environments can efficiently merge due to gas torques and dynamical encounters, potentially producing detectable em radiation during or after the merger. Such detection is feasible even against bright agn disks, provided the disks are thin and relatively low in luminosity or if the merger remnant is ejected from the optically thick midplane and undergoes super-Eddington accretion.

xg detectors promise to play a pivotal role in inferring cosmological parameters <cit.>. To demonstrate how the mock data can be used to perform bright-siren cosmology, we analyse 8s of xg detector data segment containing a GW190521-like binary that coincides with an em counterpart with{α_em, δ_em, z_em}same as the simulated signal. The data includes Gaussian noise and Population-A binaries, as introduced in the previous subsection. The binary and cosmological parameters used for the analysis are tabulated in Table <ref>. We use the Bayesian pe library  <cit.>, waveform model  <cit.> and off-the-shelf dynamic nested sampler  <cit.>. We use the same waveform model for injection and recovery to prevent waveform systematics. Consistent with the mock data generation set-up, we use a minimum-frequency cutoff off_min=6Hz and perform a flare-unrestricted pe with standard priors <cit.> and psd calculated using the Welch method. 


Fig. <ref> shows the posterior probability distributions for the sky-location of simulated binary. As can be seen, the sky location of the flare and the gw signal are consistent. In fact, the odds of a common source hypothesisC, against a random coincidence,𝒪_C/R > 2 ×10^7, if we assume the prior odds,π_C/R,  to be unity.



Having confirmed such an association, we use the obtained posterior samples to inferH_0<cit.> given the gw and em datasets,d_gwandd_em:

    p(H_0     | d_em, d_gw) ∝ π(H_0) 
       ×ℒ( d_gw|α_em, δ_em, D_L (z_em| H_0) ) 
       ×π_pop(z_em| H_0)/β(H_0) .

Hereπ(H_0)is the prior onH_0,ℒis the marginalised gw likelihood function evaluated at the distance and sky location of the electromagnetic source,π_pop(z_em |H_0)is the assumed redshift distribution of the binary population andβ(H_0)is the fraction of binary population that will be simultaneously detected in gw and em window. 




Fig <ref> shows our results, assuming a flatΛCDM cosmology and matter constraints from <cit.>. We assumeπ(H_0)=𝒰(35, 140)km/s/Mpcand consider two redshift distribution models: one that assumes that bbh merger rate is constant up to redshiftz=10and the other that follows the Madau-Dickinson star-formation rate used to simulate the binary population. For both cases, we first estimateβ(H_0)following <cit.> and then inferH_0to be67.7^+0.36_-0.36km/s/Mpcat 68% CI, thereby being consistent with the value used in simulating the binary population. It shows that if the uncertainty in luminosity distance and the counterpart redshift measurement is small, xgem-bright sources can precisely measureH_0, irrespective of the underlying population distribution assumption.


In Fig <ref>, we present the measurements ofΩ_mandH_0, when using flat priors in the rangeH_0=[35,140]km/s/Mpc,Ω_m=[0,1]and assumingΛCDM cosmology. For this, we obtain the joint posterior on(H_0, Ω_m)by modifying Eq. (<ref>) to the following:

    p(H_0, Ω_m     | d_em, d_gw) ∝ π(H_0, Ω_m) 
       ×ℒ( d_gw|α_em, δ_em, D_L (z_em| H_0, Ω_m) ) 
       ×π_pop(z_em| H_0, Ω_m)/β(H_0, Ω_m) .

Consistent with Fig. <ref>, the maximum likelihood point is close to the true value. However, as expected, the posterior forH_0is broader.


    

 §.§ PE biases due to waveform systematics

gw from cbc carry characteristic information about the astrophysical properties of compact objects, such as their masses and spins. A significant challenge in inferring these parameters is the systematic error introduced by the waveform approximants. If two waveform models,𝒲_1and𝒲_2do not agree for a particular choice ofθ, i.e.δh = h_1(t |θ, 𝒲_1) - h_2(t |θ, 𝒲_2) ↛0, then it can significantly skew pe outcomes. This is especially problematic if the difference exceeds the detector noise level.


More rigorously, if the two waveform models are not faithful up to a particular level determined by the signal's snr, then the  disagreement between the waveform models will lead to differences in pe outputs.  This issue is particularly critical in xg era, where we anticipate observing over10^5compact binary signals per year, few of which can have a snr greater than 100. Since the statistical uncertainty of parameter estimates decreases with an increase in snr, the measurability of signal parameters can be significantly affected if the current waveform models are used, particularly when analysing high snr events <cit.>. Consequently, it is crucial to understand how waveform systematics can lead to incongruous results.

While a detailed study is deferred to future work, we provide an illustrative example of using the simulated mock data to identify biases in pe due to waveform systematics. We note that these studies must be performed in zero-noise so that no systematics are introduced due to noise. Here, we give an example where the noise is Gaussian and includes population-A signals.

To this end, we re-analyse the mock data segment containing the GW190521-like  signal with parameters tabulated in Table <ref> with three different state-of-the-art waveform models, namely ,  and  <cit.>. For each of these analyses, we use , employ standard priors and the standard likelihood function for gw transients <cit.> and evaluate it from a fixed minimum frequency off_min=6Hz identical to the reference frequency choice.  

Fig. <ref> shows a corner plot comparing a few binary parameters. As can be seen, depending on the waveform approximant, the recovered parameters can be significantly different from the injected value. This is because these waveform models differ in their construction; therefore, measurements relying on them can have systematic uncertainties. This has not been an issue for almost all second-generation gw candidates, as differences caused by different approximant choices have been lower than those induced by noise fluctuations. But for loud xg events, this can introduce significant biases as the statistical uncertainty of parameter estimates decreases with an increase in snr <cit.>. Standard approaches to account for modelling errors, such as combining posterior for analysis performed using different approximants to yield a joint posterior that is effectively marginalised over the uncertainty inherent in the waveform models, won't alone be able to account for the errors and biases that occur during the recovery of source parameters due to waveform modelling choices.




 §.§ Other possible challenges


The data generated using  can be used to address several other challenges, some of which are summarised below:

  * Testing faster and more robust search and pe algorithms: The xg detectors are expected to observe hundreds of thousands of binary signals annually. Developing optimal signal identification methods and rapid parameter estimation algorithms tailored to compact binary analysis will be essential for extracting science from the signal-rich dataset. In this context, the traditional requirement of achieving a ≳ 5σ confidence level may no longer be necessary. Instead, search design requirements can be adapted to enable faster detection and pe algorithms can be developed and/or tuned for faster and more precise estimation of signal properties.

    Moreover, non-stationary noise due to glitches and foreground noise will necessitate a joint analysis of the signal of interest and the remaining data. In fact, current search techniques can be replaced by coherent Bayesian search algorithms that will allow for the identification of potential candidates, give rapid, fully coherent parameter estimates of the signal source, and assign astrophysical probabilities of the candidate events <cit.>. The simulated datasets will allow the gw community to develop, test, and refine algorithms to meet these scientific objectives.


    
  * Measurement of nseos:gw from binaries consisting of a pair of ns, or a ns and a bh can be used to measure the tidal deformability of ns and constrain the nature of matter at supernuclear densities <cit.>. Given that these effects appear in the fifth pn order, this will be best measured for signals with snr of order thousands, akin to those expected to be observed in xg era. The  dataset will contain such signals depending on the initial conditions and can be used to verify how precisely and accurately one can measure the nseos. Further, it can be used to quantify any observed bias due to the foreground noise. Also, depending on the waveform model used, the post-inspiral portion of the signals may also be present in the simulated dataset which can be used to study how well we can constrain the presence of phase transitions at several times the nuclear densities <cit.>.        

    
  * Testing rapid pre-merger sky localisation algorithms: Low latency detection and pre-merger localisation of potential bns signals, especially those within the reach of electromagnetic observatories, will be an important scientific goal for xgcbc algorithms as they can allow for a multiband understanding of bns. The  dataset can be used to mimic online analysis. It will allow the community to combine multi-messenger observation of future mergers to more precisely determine the nseos and learn about the engine powering gamma-ray bursts, kilonovae and the formation of heavy elements.     
    
    
  * Estimate the strength of the stochastic background due to compact binary population: The aggregate of all sources, particularly binary neutron stars will contribute to foreground noise which could interfere with the study or the establishment of upper limits on the cosmological gw background. Therefore, it is crucial to either remove or account for this foreground to measure the properties of a primordial background accurately.  datasets will facilitate extensive testing of xg algorithms on large and realistic samples.

    
  * Investigating Deviations from gr and False Violations: allows injecting non-GR signals through a  plugin, enabling users to introduce and analyze these signals. This feature facilitates studies on the efficacy of conventional tests in detecting deviations from GR. Additionally, users can inject GR signals to investigate scenarios where tests of GR may fail. Such failures could arise due to waveform systematics, sub-optimal detector noise modelling, and various astrophysical aspects. By examining both non-GR and GR signals,  dataset will allow us to perform mock tests of GR in xg era.

    
  * Determining Merger Rates and Reconstructing the Underlying Distribution of Source Populations: Given the anticipated high number of detections, utilizing current hierarchical Bayesian inference techniques to determine the population properties of a cbc type may become impractical. However, the sheer volume of signals allows for an alternative approach: multiplying the posteriors of individual events of a particular source type to infer population properties. The  dataset will enable the validation of this hypothesis and facilitate the testing of faster and more accurate Bayesian inference techniques.

    



    




§ CONCLUSION


In the upcoming decades, xg detectors are expected to replace the current gw detection facilities. With longer baselines and new infrastructures, these detectors can make transformative discoveries across astronomy, physics and cosmology. However, the “richer”gw detector data will pose new computational, physical and astrophysical challenges, which may not be straightforward to mitigate. 

For example, the sheer volume of signals will require developing novel signal identification and extraction algorithms. Non-stationarity due to foreground noise will pose additional challenges, especially in noise modelling, complicating subsequent analysis.  Furthermore, detecting loud signals will necessitate the development of improved waveform models to achieve unprecedented precision in parameter estimations and avoid false indications of deviations from gr. 

To simulate these challenges, we introduce , a user-friendly and lightweight Python package designed to generate mock data for xg detectors. Through various data simulation examples, we demonstrate the capabilities of . These examples include examining foreground noise from overlapping signals with intersecting time-frequency tracks, demonstrating the potential of bright sirens forH_0measurement, and discussing parameter estimation biases due to waveform systematics. We also highlight a few potential challenges that can be addressed using the simulated dataset. These demonstrations underscore the potential applications of  in supporting the development of new analysis methods and preparing the scientific community for the next era of gravitational wave astronomy.




§ ACKNOWLEDGEMENTS

The author thanks B.S. Sathyaprakash, Ian Harry, Rossella Gamba and  Ish Gupta for their insightful comments and valuable suggestions. He extends special thanks to Sophie Hourihane for her assistance with BayesLine. Heartfelt appreciation is also given to Lorine Talhaoui, Tobias Forge, Papa Nihil, and N. Ghouls for their continuous inspiration throughout the development of this work. 

The author acknowledges the generous support of the National Science Foundation (NSF) through grant PHY-2207638. He is also grateful for the computational resources provided by the Gwave (PSU) cluster, which were essential for the numerical work conducted in this research.

The  package extensively utilises the following  packages:  <cit.>,  <cit.>,  <cit.>, and  <cit.>. Furthermore, various Python packages such as  <cit.>,  <cit.>,  <cit.>, and  <cit.> are used in this work.




§ BAYESLINE PSD ESTIMATE



The Welch method for estimating noise power spectral density (PSD) generates an “off-source” noise PSD by median-averaging the spectral estimates of neighbouring data segments. However, it is not widely used in gwpe due to its sensitivity to non-Gaussian transients or “glitches”. Moreover, even for wide-sense stationary Gaussian noise, the data whitened by the Welch method doesn't produce a𝒩(0,1)distribution, leading to biased pe estimates <cit.>.

Hence, current lvk analyses use the “on-source” Bayesian spectral estimates, such as those provided by the BayesLine algorithm <cit.>. This method approximates the noise psd by generating a parameterised fit involving a combination of splines and Lorentzians. This has yielded unbiased and statistically consistent results, with whitened residuals that follow a𝒩(0,1)distribution. 

Furthermore, this methodology, originally developed for joint spectral and signal inference, will be indispensable in the upcoming xg era, where no signal-free off-source data will be available. However, current lvk analysis uses sequential modelling, which may lead to notable discrepancies from joint analyses as detector sensitivities improve. This discrepancy becomes more pronounced, particularly for ground-based xg detectors, as the sum of foreground noise won't follow a Gaussian distribution.


To illustrate that, we re-compute the noise psd of a 512s segment of data as discussed in Sec. <ref> using the BayesLine algorithm. Fig. <ref> shows our result. Like the Welch estimate, the foreground noise significantly affects noise characterisation at lower frequencies, with a more pronounced impact observed for population-B data, which assumes upper merger rates for bns and bhns systems. This can affect pe results, potentially broadening constraints on parameters like chirp mass (ℳ_c) and/or symmetric mass ratio (η) for longer signals while not significantly affecting the measurability of parameters appearing at higher pn orders (see <cit.> for a possible way to mitigate this). However, this may substantially impact binary parameter measurements for short-lived signals, such as imbh binaries. A detailed investigation is deferred to future work.







aasjournal

 Paper Title: Developing and implementing a CubeSat's equations of motion.



Fast variational Bayesian inference for correlated survival data: an application to invasive mechanical ventilation duration analysis
    [
    August 5, 2024
=====================================================================================================================================



§ ABSTRACT 
As part of the Bristol PROVE mission, a nano satellite in low Earth orbit will
be required to track a ground based target during a 400 second flyover. This
requires agile attitude control that will be achieved using a system of
flywheels. To calculate the necessary torque from these flywheels, 
a controller was designed. Using newly derived equations of motion for
the system, an expression to optimise the gains was produced. With this
controller, simulations were run to evaluate the largest causes of
error in target pointing. Disturbance torques were safely handled by the
controller, but led to a 12% increase in wheel speeds, reaching 8325 rpm. 
This higher speed led to an increased gyroscopic torque, reaching 10^-7 Nm
in the worst case. However since the flywheels can deliver
10^-5 Nm of torque, the controller could also correct for this.
Hardware performance was then varied to assess the effect of each component
on pointing accuracy. Attitude sensor noise was found to increase pointing error by
1.9^∘ in the worst case. Minimum performance requirements were then 
determined for each component in order to maintain an acceptable pointing
accuracy.



gobble


arabic


§ INTRODUCTION




 §.§ Project Outline

This project aims to extend the previous work done on attitude control within
the Pointable Radiometer for Observation of Volcanic Emissions (PROVE)
mission run within the University of Bristol. This mission involves sending a small
satellite, known as a CubeSat, into low Earth orbit. The CubesSat will have an
infrared camera fixed to its base that must image a ground target whilst
passing above. This requires the satellite to rotate in a sharp arc about the
ground target, meaning that attitude control must be particularly accurate for
this mission. The attitude will predominantly be controlled by a system of
flywheels which are accelerated to generate torque within the CubeSat.

The key elements that are focussed on in this project are

  
  * Accurately representing the full dynamics of the rotational system
  
  * Understanding the gyroscopic effects of the rotating flywheels
  
  * The potential advantage of adding a fourth flywheel
  
  * A control law that improves on previous work
  
  * An analytical approach to optimise controller gains
  
  * A robust method of testing the controller with simulated noise


To accurately represent the full dynamics of the satellite during attitude
control, a more comprehensive derivation of its equations of motion will be
necessary. So far within the PROVE mission, the motion of the satellite has
been described using Euler's equation of motion for a rigid body;

    𝐈ω̇ = 
      τ_Tot -
      ω×𝐈ω,

where 𝐈 is the satellite's moment of inertia tensor,
ω is its angular velocity, and
τ_Tot is the net torque acting on it. This would
include the torque generated by flywheels within the satellite.

This equation treats the satellite as a single rigid body.
This is not accurate as there will be independently spinning flywheels
within the body contributing to the angular momentum of the system. The
omission of these terms thus far has meant that gyroscopic effects within the
satellite could not be quantified. Hence (<ref>) will be
re-derived, now with consideration for the independently rotating flywheels,
allowing for the gyroscopic interactions to be analysed. This will give an
approximate magnitude of the unintentional torques that will arise from non
parallel axes of rotation in the flywheels and the satellite as a whole.

The number of flywheels to use for the PROVE mission CubeSat has not yet
been settled upon. Hence the potential benefit of a four wheel configuration
will be assessed. The orientation and positions of the flywheels within the
satellite are also subject to change. For this reason the derivation outlined
in Section <ref> remains as general as possible to remain
applicable for any outcome.

Controllers in previous projects used error vectors for
orientation and angular velocity to calculate a corrective torque
vector that would be generated by the flywheels. This is known as a
proportional-derivative controller as it responds to both the attitude error and
the rate that it is changing. The addition of an integrated term is considered in this
project, as the increase in data given to the controller, allowing it to
remember previous errors, could improve its performance.

With this new controller, a method by which to automatically tune the gains
based on system parameters is explored. Because the inertia values and
computational capabilities of the controller are not yet known, these must be
included in the expression for controller gains. This could allow the
work done in this project to remain relevant despite changes to the satellite's
properties.

The controller will then be assessed with simulated noise added to the input
sensors. This will provide safe margins of operation for the CubeSat whilst
gaining an understanding of the most significant factors that will affect its
pointing accuracy.



 §.§ Bristol PROVE Mission

The Eyjafjallajökull volcano erupted in 2010, resulting in an ash cloud that
covered most of Northern Europe, causing a complete air-traffic 
shutdown for over a week. This grounded over 10 million people
and cost an estimated 130 million per day to the airline industry
<cit.>. 

Understanding the properties of these ash clouds and where they are safe to
pass through will benefit both airlines and consumers in the event of another
eruption. Bristol has been working on the PROVE mission since 2015. The mission
aims to attain infrared images of the ash cloud from different angles as the
CubeSat passes over it.

A CubeSat is formed of cubic units that are 10cm× 
10cm×10cm in size as shown in Figure <ref>.
This satellite will use a thermal infrared camera, sending the images back to
the PROVE mission lab for processing. With this data, a tomographic 
view of a volcano's ash cloud can be generated. This is a series of cross sectional images
through the vertical axis of the cloud, similar to those attained from an MRI
scan. During its flyover of the volcano, the satellite will pass approximately
300 km above the ash cloud. By rotating itself during this flyover, it can maintain a
constant alignment with the ash cloud, providing a range of angles from which
to capture it. A diagram of this manoeuvre
is shown in Figure <ref>. Due to the low altitude of the
satellite, it will be able to see the ash cloud from a side view as well as
above. Adding this third dimension to the images would provide a significant
improvement to images taken from directly above by larger satellites. 
This could allow planes to be rerouted through areas of low ash concentration,
avoiding the complete shutdown of a large airspace. 



The subsystems that are required for the PROVE mission are outlined in Figure 
<ref>. This project is focussed on the attitude
control component as highlighted.

Thus far, work has been done on each one of these subsystems, as well as preliminary
work on implementing multiple subsystems together through the use of a
`FlatSat'. The FlatSat is a method for trialling the core components of
the satellite within a lab environment. Components such as communication, computer and data
handling, and power supply are assembled on a workbench to assess how they will
function cooperatively in different configurations. Understanding
and improving the system at this fundamental level has been shown to yield a
higher rate of success in similar projects <cit.>. This work is
primarily focussed on the electronic hardware within the satellite. An equally
important aspect of the PROVE mission is the set of external systems that have been
developed. This includes the CubeSat lab, fitted with a clean room for component
assembly, that has housed various projects working on CubeSat hardware. The lab
is also linked to the CubeSat communications system that uses a pair of
antennas, capable of tracking the satellite, to send orbital data
as well as receive the images taken by the on-board infrared camera. 

It is through this communication system that the satellite will be sent
its position and orbital trajectory in the form of a `two-line
element' (TLE). This is a data format that encodes a satellite's orbit 
and can be used to estimate the position of the satellite as a function of
time. The data is attained through <www.space-track.org>, and periodically
broadcast to the CubeSat when its orbit aligns with the PROVE mission antennas
that are fixed to the roof of the Queen's building in Bristol. Although this is
one of the most accurate methods of orbit tracking, the cumulative effect 
of the forces discussed in Section <ref> will, over time,
result in the satellite diverging from its predicted path.
For a 3U CubeSat in low Earth orbit (below 350 km) one day after a TLE has been calculated,
the position estimates have a standard deviation between 10 km and
30 km, however this can be significantly reduced by factoring in previous TLEs <cit.>. The satellite's orbit will coincide with the antennas every
12 hours. Since there is no on-board GPS system, due to cost restrictions, this
is the only method through which to communicate to the satellite where it is in
its orbit. Accurate knowledge of the satellite's position is crucial for the Attitude
Determination and Control Systems (ADCS). 


 §.§ ADCS

The currently proposed measurement instruments for attitude determination in
the PROVE mission are a sun sensor and a magnetometer. To control the attitude,
a set of flywheels and a magnetorquer will be used.

The magnetorquer is a 3-axis system of electromagnets that can
interact with the Earth's magnetic field to produce an external torque on the
satellite. The magnetorquer used in this mission can create a maximum torque of 4.6×
10^-6 Nm <cit.>. This is also dependant on the satellite's
orientation within the Earth's magnetic field. Therefore, the magnetorquer
alone will not be sufficient to generate the
angular rate required for the satellite to remain pointing at the
ground target during its 6 minute flyover <cit.>.
Therefore in this project, the satellite's magnetorquer will not be included in
calculations that discuss torque produced from the satellite as it functions predominantly as a tool for
desaturating the flywheels and detumbling the satellite immediately after
deployment. By measuring the induced electromotive force through the three components of the
magnetorquer, it can also function as a magnetometer
<cit.>. With the measurements over three
dimensions, the direction of the field can be calculated, thus supporting the
sun sensor measurements to determine attitude.

The flywheel system uses at least three rotating wheels at an angle to each
other to give attitude control in each axis. The wheel configuration of this
project has not yet been confirmed. In response to this, the equations of motion
have been left in a general form to be applicable for both 3 and 4 wheel
configurations with the wheels in any orientation. 
For three axis orientation control, only three wheels are needed. However, a 
fourth wheel may be added
as a safety measure, allowing for continued control in the case that one wheel
fails. To demonstrate the operation of flywheels in a 2 dimensional case; a single flywheel
accelerating within a body will produce a torque at the point of rotation,
causing the body to
accelerate in the opposite direction resulting in zero net torque. This is due to
the rotational analogue of Newton's third law such that every torque exerted
results in an equal reaction torque in the opposite direction.
By combining the torque of multiple wheels, more complex rotations can be
performed. However, as flywheel speeds increase, the impact of gyroscopic effects between
the wheels and the body will increase. The significance of this is examined in Section
<ref>.

The PROVE mission satellite requires an attitude control system with higher
accuracy than most previous CubeSat projects. The satellite will be required
to rotate rapidly in order to maintain a line of sight with the ash cloud. In
researching the PROVE CubeSat's design, L. Hawkins developed a series of
constraints that the ADCS components must adhere to <cit.>. 
These include operational
requirements, such as having a minimum lifetime of 12 months, and physical
requirements, such as weighing no more than 0.67 kg and taking up a maximum of
half a unit of space in the CubeSat. These constraints, along with a financial
limit on components, mean that more work is required to achieve high
performance in ADCS. 

To increase the chances of safe operation over a long
period, most components must maintain a safe margin from their peak performance. This
was also addressed by Hawkins when assessing the performance of the flywheels.
The Faulhaber 2610 was the proposed flywheel motor at the time, which 
had a maximum angular rate of 7000 rpm. However, Hawkins found that the
proposed flywheels would become at risk of
tangential stress damage from rates beyond 6000 rpm <cit.>.
Since this paper, the choice of motor has changed to the Faulhaber
1509T006B as it can achieve angular rates of up to 10,000 rpm. This presents
the need for flywheels that can withstand a higher degree of tangential stress.
Flywheel performance was also investigated by M. Tisaev for the PROVE mission,
who showed there to be a 3% drop in efficiency for
flywheel speeds of 6395 rpm compared to 1223 rpm <cit.>.

Another constraint on the flywheels occurs due to saturation. This occurs once
the wheel has accelerated in one direction long enough for the maximum speed to
be reached, meaning no more torque can be produced in that axis and
desaturation is required. This is likely to occur due to external torques that
could continuously act in one direction on the CubeSat. This would require a
prolonged correcting torque from the flywheels meaning a constant acceleration
must occur. The magnetorquers can provide an external torque through
interacting with Earth's magnetic field, allowing the flywheels
to reduce their speeds without affecting satellite rotation <cit.>.

Initially, the attitude determination system consisted of a sun sensor, and
three motion references; an accelerometer, a gyroscope, and the magnetorquer.
The sun sensor approach, investigated by T. Kariniemi-Eldridge, uses sensors on
each face of the CubeSat, orientation is determined by which sensors are
illuminated <cit.>. This component, however, can only function when in direct
sunlight; passing through Earth's umbra renders it useless. Therefore, during
this period, the satellite would need to rely on dead reckoning using the
on-board motion sensors. This could be mitigated by using a star tracker to
determine orientation, although this is a more expensive option due to the
added computation required.

In response to this, R. Biggs assessed the use of an Extended Kalman Filter to
increase the accuracy of attitude estimations. Biggs also evaluated Earth
horizon sensors <cit.>. Kalman Filtering uses
probabilistic modelling to make predictions, based on previous measurements, and update
its model as new data is made available. This method has been implemented in other
satellites and is shown to largely improve attitude determination with minimal
extra cost <cit.>. Simulations of this approach reduced error
in the sun sensor to ± 0.25^∘ when used with an Earth horizon
sensor. The more expensive star tracker had an error of
just ± 0.1^∘. This is compared with unfiltered attitude 
determination, which produced an error of ± 0.5^∘ <cit.>.
An important result of Biggs' research was showing
that using an Extended Kalman Filter provides enough added accuracy that
a gyroscope may not be needed. This would reduce
computational resources and power within the satellite.



§ EXTERNAL TORQUES



There are four potentially significant external sources of torque that can act on a satellite in low
Earth orbit <cit.>. These are

  
  * Gravitational gradient
  
  * Aerodynamic drag
  
  * Environmental radiation 
  
  * Magnetic field interaction

The physical characteristics of the satellite, and the type of its orbit, will decide how strong each of these
torques are. 

Gravitational torque arises due to the difference in force felt across the
satellite. For a point mass, gravitational force acts at its centre, but for a
body with differently distributed mass, the net force may act at a point
away from the centre of mass. A second factor comes from the non-uniformity of
Earth's gravitational field. Earth is an oblate spheroid, i.e., it has a larger
radius at its equator than it does at its poles, this causes the gravitational
field of Earth to vary in strength during an orbit. Passing
through this field of varying strength can also produce a torque on the
satellite.

The air density at an altitude of 300 km
is approximately 11 orders of magnitude less than that of the density at sea level
<cit.>. However, this will still produce a small drag force on the
satellite, resulting in a disturbance torque.

Pressure is generated on the satellite's surface when sun light is reflected
and absorbed. This is due to the non-zero momentum of photons which is partially 
transferred on impact. The resultant force of this solar pressure will not
necessarily coincide with the centre of mass of the satellite, thus producing a
torque.

Finally, magnetic torque arises due to Earth’s magnetic field, behaving
like that of a dipole as the satellite passes through it. The magnetic moment of the satellite interacts with this
field and results in a torque as the two fields are drawn into alignment.
However, as discussed, this phenomenon can be exploited by using a magnetorquer
to control the satellite’s magnetic moment, producing an intentional external
torque that can be used for detumbling, or to desaturate the flywheels.

This external torque is given to be

    τ_Mag = μ_Sat×𝐁_Earth,

where τ_Mag is the generated torque, given as the cross product of the
satellite's magnetic moment, μ_Sat, with the
magnetic field vector of Earth, 𝐁_Earth. The magnetic
moment can be expressed as μ = nI𝐀, where n is the
number of turns in the magnetorquer solenoid, I is the current, and
𝐀 is the vector area of the solenoid. The cross product in
(<ref>) indicates that the torque produced from the
magnetorquer is dependent on the misalignment of the two magnetic fields. Using a
three-axis magnetorquer can ensure that the satellite's magnetic moment is not
parallel to the Earth's magnetic field, however the torque it produces will
always be perpendicular to this field. This limits the versatility and power
of the magnetorquer, hence why the more adept flywheels are used during the
flyover.

For a 3U CubeSat in low Earth orbit, the largest source of disturbance torque
comes from aerodynamic drag <cit.>. An upper limit on the
magnitude of this torque is 6× 10^-7 Nm <cit.>, whilst
the next largest torque comes from gravitational gradient and has an estimated
magnitude of 5.6× 10^-8 at an altitude of 300 km <cit.>.
Environmental radiation and magnetic field interaction are negligible in
comparison to these.
During its orbit, the satellite will register these disturbance torques with
the gyro and attitude sensors, and will correct for them with the controller. If this
disturbance torque is constant, the wheels will require frequent desaturation
from the magnetorquer. An upper limit on the frequency that this desaturation
could be required can be calculated.

The frequency that desaturation is required depends on both the moment of
inertia of the flywheels and the maximum speed at which they can spin.
The moment of inertia of the flywheels about their rotation axis is
approximately 1.1× 10^-6 kg m^2 <cit.>. Therefore, to produce 6× 10^-7 Nm 
of torque with one flywheel, it must accelerate at 0.55 rad/s^2,
meaning it would reach its maximum speed of 840 rad/s within 1540 seconds.
This means that desaturation could be needed every 25 minutes, in a worst-case
scenario. This only poses
a potential problem during the flyover, when the wheels may already be close to
their maximum speed after producing the necessary torque. 

The most strenuous flyover occurs when the satellite passes directly above the
ground target, needing to perform the sharpest turn. When simulating this
flyover in idealised conditions, the
wheel speeds reach a maximum of 7335 rpm. This assumes that the satellite
starts the flyover with zero flywheel speed. 
Even in this idealised condition, this is close to the
suggested speed limit of 8000 rpm, indicating that desaturation must occur
before the flyover begins in any case. The worst-case scenario of external
torque occurs when the satellite's rotation is directly opposed. Figure
<ref> shows this case, with a constant torque of magnitude
6× 10^-7 Nm acting along the body-fixed y-axis, y_B, whilst
rotation occurs in the opposite direction.



The dashed line plots in Figure <ref> show the wheel
speeds during the flyover if no external torque is present, whilst the solid
lines show their behaviour when the external torque, shown in Figure
<ref>, is present. In the case of no external torque, the wheels display 
symmetrical ramps up and down to match the changing severity of turn required.
However, in the presence of the opposing external torque, these wheels must
continuously accelerate at approximately 0.55 rad/s. As a result, the maximum
wheel speed increases to 8325 rpm, impinging on the proposed safety margin
from the actual maximum speed of 10,000 rpm.
Although this is acceptable for a brief period, higher speeds will
increase the gyroscopic effect produced by the flywheels. This is 
discussed in Section <ref>.



§ QUATERNIONS


For this project, quaternions will be the method used to describe orientations and
rotations.
A quaternion is a hyper complex number that consist of 4 elements; 1 scalar
term and 3 vector terms. The vector terms are named as such since their
basis elements; i, j, and k, can
be interpreted as unit vectors pointing along the three spatial axes.
Quaternions are generally expressed in the form

    q = q_1 + q_2 i + q_3j + q_4k,

however, in this application, they may also be written in the vector 
form q=[q_1,q_2,q_3,q_4]^⊤.

The products of the basis elements of a quaternion are defined by

    i^2 = j^2 = k^2 =
      ijk =  -1,

similar to the imaginary part of a complex number.
The multiplication of two quaternions is noncommutative, meaning
the order of the operands affects the result. The multiplication of
quaternions, q and p, is defined as

    q⊙ p = [ q_1p_1 - q_2p_2 - q_3p_3 - q_4p_4; q_1p_2 + q_2p_1 + q_3p_4 - q_4p_3; q_1p_3 - q_2p_4 + q_3p_1 + q_4p_2; q_1p_4 + q_2p_3 - q_3p_2 + q_4p_1 ].

The conjugate of a quaternion, much like with complex numbers, is defined as 

    q^* = q_1 - q_2i - q_3j - q_4k,

using the definition from (<ref>). The
inverse of a quaternion is then written as 

    q^-1 = q^*/|q|^2.

A unit quaternion, such that √(q_1^2+q_2^2+q_3^2+q_4^2) = 1, can be used
to describe rotations in 3D space.
Orientation can also be described using a rotation from a canonical form, as
shown in Figure <ref>. Rotations are achieved through
quaternion multiplication. To rotate a vector, v, it must first
be represented as a quaternion. This is achieved by setting the quaternion
scalar element to 0 and the vector elements to those of v.
Rotating by quaternion q is achieved by first multiplying from the
left by q, and then from the right by q^-1, which is the same as q^*
for unit quaternions. This is written as

    q⊙v⊙ q^-1 =
      [ q_1; q_2; q_3; q_4 ]⊙[   0; v_1; v_2; v_3 ]⊙[  q_1; -q_2; -q_3; -q_4 ] =
      [    0; v_1'; v_2'; v_3' ].

The polar decomposition of a unit quaternion is analogous to the polar form of
a complex number. It separates the quaternion into its real, and vector, parts
using trigonometric identities to relate the two components. This is expressed
as

    q = cosθ/2 + 𝐧̂sinθ/2.

This can be easily interpreted to understand the rotation it will perform on a given
object. 𝐧̂ from the vector part,
𝐧̂sinθ/2, is the unit vector of rotation
whilst the amount it will rotate is θ. This will
prove highly useful in designing a quaternion based controller, discussed in
Section <ref>.

The other commonly used method used to describe rotation and
orientation in 3D space is rotation matrices. These are orthogonal matrices
with a determinant of 1. An example of this is

    X(ϕ) = [     1     0     0;     0  cosϕ -sinϕ;     0  sinϕ  cosϕ ],

which will rotate a vector by ϕ about the x-axis.

By multiplying three, single axis, rotation matrices together, a composite
rotation matrix can be generated. Although this is an intuitive method 
for describing rotation, it comes with limitations.

Gimbal lock is a phenomenon by which two axes of rotation align, resulting in
the loss of a degree of freedom. This is a significant problem when dealing
with an orbiting body as the orientation is not constrained, meaning an
alignment of rotation axes is inevitable. This suggests the need for a more robust
method of expressing orientation.

And additional benefit of quaternions comes from their invariant constraint,
whereby they must maintain a norm of 1. This is computationally cheap to check
and, if necessary, correct for, by simply rescaling the four elements
proportionally. Comparing this to the
invariants of a rotation matrix, they must maintain a determinant of 1,
as well as remaining orthogonal. If this invariant constraint is not met due to
truncation errors, a single-value decomposition is required to return to
orthogonal, which is far less efficient. Thus, a quaternion based
approach is used for describing orientation and rotation, as well as
in the control law derived in Section <ref>.



§ EQUATIONS OF MOTION

The equations of motion that have been used in previous projects are written as

    𝐈ω̇   =
      τ_Ext - τ_W -
      ω×𝐈ω
    q̇   = 1/2q⊙ω
    τ_W   =f(𝐪_err,ω_err),

where (<ref>) and (<ref>) describe the satellite's angular
velocity and orientation respectively whilst (<ref>) describes
flywheel torque, as generated by the control law. The control law uses the
vector part of the pointing error quaternion, 𝐪_err, and
the angular velocity error, ω_err, making this a
proportional-derivative controller.

This will be improved upon by extending (<ref>) to treat the
flywheels as separate rotating bodies, giving a more accurate representation of
the system. This can then be used in simulating the performance of the newly
designed controller.

The motion equations that will be used for this project are given as

    𝐈_Totω̇_B   =
      τ_Ext -
      ω_B×𝐈_Totω_B -  τ_W -
      ω_B×𝐄_W𝐬
    q̇   =
      1/2q⊙ω_B
    τ_W   =
      f(𝐪_err,𝐠_err,ω_err),

where the derivation of (<ref>) is given in Section
<ref>, whilst the new error term given to the control law,
𝐠_err, will be explained in Section
<ref>.
The last term in (<ref>) is the cross product of body angular
velocity, ω_B, and the net
angular momentum of the flywheels, 𝐄_W𝐬. This
denotes the previously omitted gyroscopic effect that arises from the flywheels spinning.
The cross product here shows that the gyroscopic torque will act perpendicular to
the net angular momentum of the flywheels which could perturb the satellite
from its rotational path at a high enough magnitude.


 §.§ Definitions



There are four key reference frames that will be referred to throughout this
report, all of which are illustrated in Figure <ref>. The
first is the Earth-centred rotating frame. This is a non-inertial reference
frame since it will rotate about its z-axis, at an angular rate of 1
revolution per 24 hours.
The convention used is for the x-y plane to
intersect the equator, with the x-axis intersecting the prime meridian
(0^∘ longitude). The Earth-centred inertial frame will initially align
with the rotating frame at some arbitrary start time, however this frame will
remain fixed at this orientation for all time. The next reference frame is the
body-centred inertial frame. This is not technically
inertial as its origin is fixed to the centre of mass of the orbiting
satellite. However, this frame will only be used to describe rotations about
its origin, and the orientation of this frame will be fixed to align with the
Earth-centred inertial frame. This means, for all intents and purposes, it can
be considered inertial and will be referred to as such. The final frame that
will be used is the body-centred rotating frame. This frame has the same
angular velocity as the satellite, ω_B, 
keeping it's positive z-axis pointing along the sight line of the on-board camera.

Throughout the report, wheel acceleration can be taken to mean the angular
acceleration of the wheels relative to the body-centred rotating frame.

The orientation of the satellite at time t is expressed by the unit quaternion
q(t). This quaternion represents an active rotation of the satellite's
canonical form, as shown in Figure <ref>. The canonical form of
the satellite is attained by aligning the body-rotating axes with the body-centred
inertial axes. An active rotation then describes the transformation of the
canonical body to a new orientation. This is compared with a passive rotation, in which
the coordinate system itself is rotated
instead, which will not be used here. In this report the
quaternion multiplication operator, ⊙, will be used between a
quaternion and a regular Euclidean vector. This indicates that the vector is
first transformed into a quaternion following the procedure described in Section
<ref>.

In subsequent sections, the satellite will be considered as a composite of its wheels and body.
This is to isolate the separate rigid bodies that will be rotating during attitude
control. When referring to the satellite `wheels', this includes the flywheel
itself and the shaft connecting it to the motor since they rotate as
one body. The satellite's `body' refers to all that remains of the
satellite after excluding the wheels.



 §.§ Derivation


The system's angular momentum is first expressed as a sum of the angular momentum of the
satellite body and the angular momenta of the flywheels. 
This gives a top level description of the system's angular momentum to be

    𝐋_Tot = 𝐋_B + ∑_i
    𝐋_Wi,

where the subscript, B, denotes the satellite's body and
Wi corresponds with the i^th flywheel.

Angular momentum is an object's angular velocity around its centre of mass,
ω, multiplied by its inertia tensor, 𝐈. If the
centre of mass is then also rotating around a different point, this must be incorporated
into the expression for the object's angular momentum. This term is the cross
product of the vector from the centre of rotation to the centre of mass,
𝐫, and the linear momentum of the centre of mass,
𝐩=m𝐯, where m and 𝐯 are the object's mass and
velocity respectively. The vector,
𝐫_B, describes the displacement of the body's centre
of mass from the total centre of mass of the satellite with the mass of the
flywheels included. This gives the body's angular momentum as

    𝐋_B   =
      𝐈_Bω_B+
      𝐫_B×𝐩_B
       = 𝐈_Bω_B+m_B𝐫_B×𝐯_B
       = 𝐈_Bω_B+m_B𝐫_B×(ω_B×𝐫_B),

where the relation between linear and angular velocity,
𝐯=ω×𝐫, has been utilised to express
the full angular momentum as a function of angular velocity.
Similarly the angular momentum of the wheels becomes

    ∑_i 𝐋_Wi   = ∑_i 𝐈_Wiω_Wi +
      𝐫_Wi×𝐩_Wi
       = ∑_i 𝐈_Wiω_Wi +
      m_W𝐫_Wi×(ω_B×𝐫_Wi),

where the angular velocity of the body, ω_B,
also describes the angular velocity of the wheels about the satellite's total centre of
mass.
This derivation aims to produce an equation that will describe the
dynamics of the satellite as a function of varied flywheel speeds.
Hence, it is necessary to isolate the terms that describe the respective wheel
speeds, expressing them from the perspective of the flywheel motor, since this is the angular speed
that the controller must output. For this reason, the angular
velocity of the wheels about their axes of rotation is expressed as the sum
of the wheel's angular velocity from the perspective of the body, and so the
motor, and the angular velocity of the body itself.
This then gives

    ∑_i𝐋_Wi   = ∑_i𝐈_Wi[
    ω_B+(ω_Wi-ω_B)] + 
    m_W𝐫_Wi×
    (ω_B×𝐫_Wi)
       =∑_i𝐈_Wi( ω_B + s_i
      ê_i) + m_W𝐫_Wi×(ω_B×𝐫_Wi)
       = ∑_i𝐈_Wiω_B+s_i
      𝐈_Wiê_i + m_W𝐫_Wi×(ω_B×𝐫_Wi),

where the wheel's angular velocity, from the perspective of the body, is
expressed as a product of its angular speed and the unit vector ê_i
that describes the wheel's axis of rotation.

To prepare the equation for factoring, the
triple product 𝐫_Wi×(ω_B×𝐫_Wi) is written as a
matrix-vector product
𝐀_Wiω_B such that

    𝐀_Wi=[   r_y^2+r_z^2       -r_xr_y       -r_xr_z;       -r_xr_y   r_x^2+r_z^2       -r_yr_z;       -r_xr_z       -r_yr_z r_x^2 + r_y^2 ],

where 𝐫_Wi = (r_x, r_y, r_z)^⊤.

A full derivation of this matrix is provided in Appendix <ref>.
Using this new representation gives the angular momenta of the wheels to be

    ∑_i𝐋_Wi = ∑_i𝐈_Wiω_B+
      s_i𝐈_Wiê_i + m_W𝐀_Wiω_B.

Similarly for the angular momentum of the body, 

    𝐋_B   = 𝐈_Bω_B+m_B𝐫_B×(ω_B×𝐫_B)
       =
      𝐈_Bω_B+m_B𝐀_Bω_B.

Since the wheels will be rotating about axes that are independent of the body
frame axes, their inertia tensors, when expressed in the body frame, will not be diagonal.
However, each tensor can be expressed as the diagonal tensor for a hypothetical
wheel that spins about the x-axis in the body frame, which is then multiplied
with a rotation matrix. Using one wheel to demonstrate, the second term in
(<ref>) can be written as

    s_i𝐈_Wiê_i = s_i 𝐎_i𝐈_Wx𝐎^-1_iê_i,
 where 
    𝐈_Wx =
      [              1/2 m_W R_W^2                          0                          0;                          0 1/12m_W(h_W^2 + 3R_W^2
      )                          0;                          0                          0    1/12m_W(h_W^2 +
    3R_W^2) ],

with R_W and h_W representing the wheel radius and
height respectively. 𝐎_i is the rotation matrix from the x-axis to the
i^th wheel's axis of rotation, ê_i. Therefore the inverse rotation,
𝐎_i^-1, will rotate ê_i back to the x-axis unit
vector to give

    s_i 𝐎_i𝐈_Wx𝐎^-1_iê_i= s_i 𝐎_i𝐈_Wxê_x.

Since ê_x = [1,0,0]^⊤, the product, 𝐈_Wxê_x will become 1/2m_WR_W^2
ê_x, giving

    s_i 𝐎_i𝐈_Wxê_x    =
      1/2m_WR_W^2 s_i
    𝐎_i  ê_x
       =1/2m_WR_W^2 s_i ê_i,

where 𝐎_i  ê_x has become
ê_i from multiplying by the rotation matrix. Reducing this
term to just a scaled unit vector will improve the simplicity of the motion
equations later on. Substituting this term into the equation for the wheels'
angular velocity gives

    ∑_i𝐋_Wi   = ∑_i𝐈_Wiω_B+
      1/2m_WR_W^2 s_iê_i + m_W𝐀_Wiω_B
       = ∑_i( 𝐈_Wi+m_W𝐀_Wi)ω_B +
      1/2m_WR_W^2
      s_iê_i.


Substituting in the expressions for 𝐋_B and
∑_i𝐋_Wi from (<ref>) and (<ref>)
respectively gives the full equation of angular momentum to be 

    𝐋_Tot   = 𝐋_B +
    ∑_i𝐋_Wi
       =𝐈_Bω_B+m_B𝐀_Bω_B
      + ∑_i ( 𝐈_Wi+
      m_W𝐀_Wi)ω_B +  
      1/2m_WR_W^2 s_iê_i
       = (𝐈_B+m_B𝐀_B+
      ∑_i 𝐈_Wi+m_W𝐀_Wi) 
      ω_B + 1/2m_WR_W^2 ∑_i
      s_iê_i
       = 𝐈_Totω_B +
      1/2m_WR_W^2 ∑_i s_iê_i,

where many of the factored out terms have been absorbed into a single matrix
𝐈_Tot multiplying the body's angular velocity.

Using the satellite's total angular momentum, shown in (<ref>),
the equations of motion can be derived.

The relationship between the angular momentum of a system and the torque acting
on it comes from the rotational analogue of Newton’s second law, given as

    τ=
    (d𝐋dt)_in,

where τ is the net external torque, and
(d𝐋dt)_in
is the time derivative of the total angular momentum of the satellite
in an inertial reference frame. It is important to note that the internal
torques generated from the flywheels are not included in
τ as they have been in previous projects. The
torques that act from within the satellite must sum to zero as stated by
Newton's third law.

To keep 𝐈_Tot and ê_i constant over time, 
the time derivative is taken in the rotating body frame. The general
relationship, when changing the frame of reference for the time derivative of
vector 𝐯, is

    (d𝐯dt)_in =
      (d𝐯dt)_rot +
      ω_F×𝐯,

where ω_F is the angular velocity of the rotating
frame.
Applying the change of reference frame from (<ref>) to this case gives 

    τ_Ext = 
    (d𝐋_Totdt)_rot 
    + ω_B×𝐋_Tot.

Substituting the expression for 𝐋_Tot in (<ref>) gives


    τ_Ext =𝐈_Totω̇_B +
    1/2m_WR_W^2 ∑_i ṡ_iê_i+
    ω_B×𝐈_Totω_B +
    1/2m_WR_W^2 ∑_i
      s_iω_B×ê_i,

where the time dependent variables, ω_B and
s_i, have become accelerations ω̇_B and
ṡ_̇i̇ in the first two terms of the equation. Grouping together terms
that relate to either the body, or the wheels, gives 

    τ_Ext = 𝐈_Totω̇_B +
    ω_B×𝐈_Totω_B +
    1/2m_WR_W^2 ∑_iṡ_iê_i+
    s_iω_B×ê_i,

which resembles Euler's equations of motion for a rigid body, extended to take 
into account the separate movement of the flywheels.

The sum of flywheel terms can be expressed as a matrix-vector product. This format
will allow for easy calculation of the individual wheel accelerations from the
torque vector calculated in the control law. It will also prove necessary for
the analytical tuning of the control law gains, discussed in Section
<ref>. The
matrix is implicitly invertible for the three wheel case due to the required linear independence of 
the flywheels' rotation axes. More work is required for a four wheel system
to make the matrix invertible, this is discussed in Section <ref>.
For the three wheel case, the rotation axis vectors, ê_i, are
horizontally concatenated and multiplied by the reduced inertia term,
1/2m_WR_W^2, giving

    𝐄_W = 1/2m_WR_W^2
      [ ê_1 ê_2 ê_3 ] =
      1/2m_WR_W^2 [ e_11 e_21 e_31; e_12 e_22 e_32; e_13 e_23 e_33 ].

This matrix can be thought of as an inertia-like tensor for the wheels.
The values for each wheel's angular speed are then combined into a
vector such that for a three wheel case, 

    𝐬=[ s_1; s_2; s_3 ],
      𝐬̇ = [ ṡ_1; ṡ_2; ṡ_3 ].

This format allows for the net torque and angular momentum of the flywheels to
be treated each as a single term, whilst being able to access the specific wheel
accelerations and speeds of the wheels with multiplication by
𝐄_W^-1.
The terms relating to the wheel dynamics in (<ref>) can now be expressed as

    1/2m_WR_W^2 ∑_iṡ_iê_i+
      s_iω_B×ê_i =
      𝐄_W𝐬̇ +
      ω_B×𝐄_W𝐬,

giving the full equation as

    τ_Ext = 𝐈_Totω̇_B +
      ω_B×𝐈_Totω_B +
      𝐄_W𝐬̇ +
      ω_B×𝐄_W𝐬.

Finally, rearranging for ω̇_B gives 

    𝐈_Totω̇_B = -  𝐄_W𝐬̇ 
      +τ_Ext
      -ω_B×𝐈_Totω_B-
      ω_B×𝐄_W𝐬.


This equation describes the rotational dynamics of the satellite, now also
taking account of the gyroscopic effects due to the rotating flywheels. 
The 𝐄_W𝐬̇ term in (<ref>) is
the torque directly produced by the flywheels, which 
will be calculated from the control law discussed in Section
<ref>. The ω_B×𝐄_W𝐬 term is the gyroscopic
interaction that occurs between the wheels and the body they rotate within.
From Section <ref>, the magnitudes of
𝐈_Totω̇_B and
𝐄_W𝐬̇
are of order 10^-7 Nm at the most strenuous point of a flyover, whilst the gyroscopic terms are
at least two orders smaller. This validates that, in the case of no external
torque,

    𝐈_Totω_𝐁≈
      -𝐄_W𝐬.

This is expected as a result of Newton's third law, stating that
torques generated by the flywheels must be exactly matched by an opposite
torque from the body.
This is an approximate interpretation of the equation but outlines the general
principles that are being utilised to control the satellite's orientation.
More detail will be given, as well as discussion of the gyroscopic terms, in Section
<ref>.

Another important conclusion to take from (<ref>) is that the torque
delivered by the flywheels is shown to be independent of their position within the
satellite body. The torque that the flywheels directly produce is represented
by the term, 𝐄_W𝐬̇. This term is a vector of wheel accelerations, multiplied
by the inertia-like matrix, which is comprised of the mass, radius, and
orientations of the wheels within the satellite. The wheel positions
are represented by the 𝐀_Wi matrices, from
(<ref>). These are then incorporated into the
𝐈_Tot term as shown in (<ref>). Hence
the positions only factor into the distribution of the satellite's total mass.
This validates the assertion that the wheels can be placed anywhere and it will not impact
the torque that they deliver.

To describe the rate of change of the satellite's orientation, a quaternion
approach is used.
Since quaternion multiplication is noncommutative, the
ordering of the operands results in two different definitions of the quaternion
derivative. For the above case, where the body's angular velocity is expressed in the
rotating frame, the equation is written as

    q̇ = 1/2q⊙ω_B.

A full derivation and explanation of this equation is in Appendix
<ref>.



 §.§ Four Wheel Configuration


The use of either three or four flywheels is being considered for the PROVE mission
satellite. A minimum of three wheels are required, with linearly independent
axes of rotation, to rotate the body in three
dimensions. Using four flywheels, however, provides security in the case that one wheel
fails, allowing the system to maintain three axes of control.

The equations of motion have been kept in a general form to
accommodate the variable configuration of the flywheels. Their inertia-like 
matrix, 𝐄_W, comprises the rotation axes of the
flywheels, multiplied by the moment of inertia about the rotation axes of the
flywheels. The column vectors denoting the axes of rotation are concatenated
together when forming this matrix, hence the four wheel configuration will
result in 𝐄_W being of shape 3×4. The
equation that is solved to acquire the wheel accelerations is

    𝐄_W𝐬̇ = τ_W,

where τ_W is the torque vector that will be
calculated from the control law discussed in Section <ref>. The
rank of 𝐄_W is 3 in either wheel configuration. When
four wheels are used, the columns representing their axes of rotation will not
be linearly independent. This corresponds with having four variables to solve
for and only three equations with which to do it, hence the solution to the
matrix equation is not unique. This means that the solution for
𝐬̇ will be of the form 

    𝐬̇ = 𝐚 + μ𝐛.

This describes a line in 4 dimensions which contains all valid solutions to
(<ref>). Parameters 𝐚 and 𝐛 are
constant, whilst μ can be varied to change the distribution of wheel
accelerations. A simplified diagram of this line with the minimised
𝐬̇ vector is shown in Figure <ref>.

This case requires an additional condition to find an explicit
value for wheel accelerations. This condition was for the sum of
squared wheel accelerations to be minimised, written as

    _μ |𝐬̇|^2,subject to  𝐄_W𝐬̇=τ_W

where

    |𝐬̇|^2    = 𝐬̇·𝐬̇
       = (𝐚 + μ𝐛)·(𝐚 + μ𝐛)
       = μ^2 |𝐛|^2 + μ(2𝐚·𝐛) + |𝐚|^2


Since |𝐬̇|^2 is a positive quadratic, its minimum is found by
setting

    ddμ|𝐬̇|^2 = 2μ|𝐛|^2 +
      2𝐚·𝐛=0,

and then solving for μ, giving

    μ = -𝐚·𝐛/|𝐛|^2.

Substituting this result into (<ref>) gives the optimised
solution

    𝐬̇_min   = 𝐚 -
      (𝐚·𝐛/|𝐛|^2) 𝐛
       = 𝐚 - (𝐚·𝐛̂) 𝐛̂,

where 𝐛 has been rescaled to a unit vector.



The minimum value for 𝐬̇ can now be expressed purely in terms
of vectors 𝐚 and 𝐛. The line of solutions described by these parameters is dependent on both
𝐄_W and τ_W. However, the
inertia-like matrix, 𝐄_W, is fixed. This corresponds to
a fixed gradient for the line, whilst the value of
τ_W determines its offset from the origin.
Consequently, the gradient, 𝐛, need only be calculated once. Solving
the special case of 𝐄_W𝐬̇=0,
where 0 denotes a vector of all zeros,
results in 𝐚=0. This is because, if 
τ_W=0, then the minimum wheel accelerations to
achieve this torque are 𝐬̇=0, meaning the solution
line has no constant offset from the origin.
Performing Gaussian elimination on the augmented matrix,
[𝐄_W|τ_W], yields 

    [[ 1 0 0 α 0; 0 1 0 β 0; 0 0 1 γ 0 ]],

where α, β, and γ are determined by
𝐄_W.
This result is guaranteed by the system requirement that three of the wheels must have
linearly independent axes of rotation, meaning 𝐄_W will
be of rank 3. Making 𝐬̇_4 the varied parameter, μ, gives
the solution

    𝐬̇ =
      μ[ -α; -β; -γ;  1 ]= μ𝐛,

where α, β, and γ are constants determined by the elements
of 𝐄_W.

With 𝐛 calculated, the computation for the optimum four wheel
accelerations is simplified. The fourth element of 𝐚 will always be
0 since ṡ_4 acts as the varied parameter in μ𝐛. Hence,
the product, 𝐄_W𝐚, gives the same result if the
fourth column of 𝐄_W is omitted. This now gives an
invertible matrix in the equation,

    [ E_W11 E_W12 E_W13; E_W21 E_W22 E_W23; E_W31 E_W32 E_W33 ][ a_1; a_2; a_3 ] =
    [ τ_W1; τ_W2; τ_W3 ],

from which 𝐚 can be calculated.
This equation is equivalent to solving for 𝐬̇ in the three
wheel case, where the solution is uniquely defined.

To summarise this procedure; 𝐛 is calculated pre-launch by solving
𝐄_W𝐛=0 and setting b_4=1. 𝐛 is
then rescaled to a unit vector. During operation, the control law will be used
to calculate a correcting torque, τ_W, based on
sensor inputs. 𝐚 is calculated by solving
𝐄_W^3×3𝐚^3×1 =
τ_W, where the fourth dimensions of
𝐄_W and 𝐚 have been omitted. The fourth
dimension of 𝐚 is then set to a_4=0. Finally, the optimised solution
for 𝐬̇ is given as 𝐬̇_min=𝐚
- (𝐚·𝐛̂) 𝐛̂.

An example of this approach being applied is given in Appendix
<ref>. In this demonstration the four wheels
are evenly spaced about the body's z-axis, each with a 40^∘ incline to
the x-y plane. This is the currently proposed configuration by M. Tisaev in his
evaluation of the ADCS hardware <cit.>. A similar configuration was
used for a three wheel case, with the wheels again equally spaced about
the z-axis. The sum of the wheel acceleration magnitudes for both cases are
shown to be identical. This supports that the four wheel case has been made as
efficient as the three wheel case, whilst adding a redundancy in the case of a
failure.



§ GYROSCOPIC EFFECTS


The gyroscopic effects caused by the flywheels have previously been omitted
within the PROVE mission. However, with the equations of
motion that have now been derived, the dynamics of the flywheels can be
isolated and measured to understand their significance. This is done by
analysing the magnitude of the gyroscopic torque in comparison to the other torques that
will be occurring within the satellite. The internal sources of torque from
(<ref>) are classified as follows 

    𝐈_Totω̇_B =
      τ_Ext -
      ω_B×𝐈_Totω_B_Gyroscopic
      response of body-  
      𝐄_W𝐬̇_Torque from
      flywheels -
      ω_B×𝐄_W𝐬._Gyroscopic
      response of flywheels

By isolating these terms during a simulated flyover, the
relative scale of gyroscopic effects can be understood. 
The magnitude of these terms, and the net torque that they produce, are shown in Figure
<ref> alongside the corresponding flywheel speeds.
In this case, no external torque
is present and the satellite passes directly above the ground target. 
This simulation shows a large difference in magnitudes
between intentional torque and the gyroscopic reactions that occur. As a
result, the total torque produced appears to exactly match that of the
flywheels. This is expected since external torque was omitted during
this simulation.

The gyroscopic terms depend on the angular velocity of the two components, whereas
the intentional torques come from their accelerations. This explains
why the plot for wheel torque drops close to zero when the
satellite is directly above the target. At this point the satellite's rotation
must begin slowing, resulting in a change of sign in its acceleration vector.
The torque never reaches zero since the target is also moving longitudinally,
requiring a small torque about a second axis.  
This change of sign
in the acceleration vectors correlates to the change in sign of the gradient of
the gyroscopic terms, since they depend on the integral of these acceleration
terms.

In the case that external torque is present, the magnitude of these gyroscopic terms can become
significant. Figure <ref> shows the torque magnitudes from
(<ref>) with external torque present.
The external torque is applied in the
direction opposing the satellite's rotation as in Section
<ref>. This torque is modelled to vary depending on the
orientation of the satellite as it interacts with the atmosphere. The
largest torque is felt when the satellite is directly above the target, with
its longest face presented to the oncoming drag force. At this point, the
external torque experienced by the satellite is 6×10^-7 Nm, which is the
maximum estimated value for this CubeSat design.

Although, at the peak of the flyover, the gyroscopic torque produced
by the flywheels is shown to become almost
as strong as their intentional torque, it is important to note that this is
being dealt with by the feedback loop in the controller. Due to gyroscopic
terms being a function of velocity, they will vary slowly, giving the
controller time to react and correct for the error they cause. This is why the
intentional wheel torque should always have a greater magnitude 
than the gyroscopic terms, as it is counteracting their torque in addition to
generating the desired change in angular velocity. This will only cease to be
the case when saturation occurs, preventing the necessary torques from being
produced. If desaturation could not occur at the start of the flyover, the
wheels would have some initial speed from counteracting external torques. This would 
result in larger gyroscopic terms and could cause the wheels to saturate during
a flyover. Hence the need for desaturation before each flyover starts. 

Throughout the orbit, the external torque both supports and opposes 
intentional wheel torque. On approach to the
target, the satellite must undergo angular acceleration, against the direction
of the external torque, thus requiring a larger magnitude of torque from the
flywheels. Once the satellite has passed directly above the
target, it must begin slowing its rotation speed, requiring an acceleration in
the opposite direction. During this period, the external torque is assisting
the torque produced by the flywheels, thus reducing their magnitude faster than
would occur in the case of no external torque.
After approximately 300 seconds of the two torques working cooperatively,
the total torque, which corresponds to the body's acceleration, dips below the magnitude of the
constant external torque. At this point, the constant external torque is
greater than that needed for the satellite's deceleration, prompting a change
of direction in flywheel torque.
Over a longer period, these values settle due to the
satellite's rotation slowing as its distance from the target increases. Towards
the end of the time series, satellite's acceleration is still decreasing. 
At this point, the satellite is now a quarter of the way
around the Earth from the target, requiring almost no change to its low
angular velocity. The flywheels equilibrate as they counteract the
constant external torque and gyroscopic terms.

Before this project, the gyroscopic effect caused by the flywheels within the
PROVE mission satellite was unknown. The magnitude of this torque is shown
to be less than the torque that can be created safely with the flywheels during
a flyover. This is helped by the slow rate at which they vary, since they
depend on the velocity of the components, instead of their acceleration.



§ CONTROL


With the equations of motion derived in Section <ref>, the
relationship between required rotation and necessary wheel torques can be
developed. In idealised conditions, this simply requires numerical integration
of the equations of motion, solving for the wheel speed acceleration at each
step. However, this rigid model would not work due to external factors, such as disturbance
torques and sensor noise. Therefore, a control system that
includes a feedback loop is necessary to produce a practical method of
calculating wheel torques.

The control law derived here builds off of previous work within the PROVE
mission, where a quaternion based, proportional-derivative controller was used.

For this project, an integration component was introduced. This comes from
adding the current attitude error, 𝐪_err, to the sum of
previous errors. Importantly, this sum is reduced each iteration by some
scaling factor, resulting in an exponential decay in the weighting of the errors as they
become older. Adding this term to the original control law forms a
propiortional-integral-derivative (PID) controller, taking account of the attitude
error, its integral over time, and the attitude's rate of change respectively. This gives the
full control law to be

    τ_W = -k_p𝐪_err
      - k_i𝐠_err
      - k_dω_err,

where τ_W is the torque vector required from the
flywheels, equivalent to 𝐄_W𝐬̇ from
(<ref>), 𝐠_err is the time-weighted sum of
𝐪_err terms, and the k coefficients are the gains that
will be discussed in Section <ref>. Whilst solving for these
gains analytically, it became apparent that arranging the system's Jacobian
according to
axis would greatly reduce complexity when solving for its eigenvalues. For
this to be possible, the gains were split into three components, one for each
axis. This meant that the gains became diagonal matrices, giving the explicit control
law to be

    [ τ_W1; τ_W2; τ_W3 ]
      =
      -[ k_px    0    0;    0 k_py    0;    0    0 k_pz ][ q_err2; q_err3; q_err4 ]
      - [ k_ix    0    0;    0 k_iy    0;    0    0 k_iz ][ g_err1; g_err2; g_err3 ]
      -
      [ k_dx    0    0;    0 k_dy    0;    0    0 k_dz ][ ω_err1; ω_err2; ω_err3 ].

Note that the components of 𝐪_err come directly from the
error quaternion describing the corrective rotation needed, the vector part of
this quaternion consists of q_2, q_3, and q_4.


 §.§ Terms


The proportional term of the controller uses the attitude error vector,
𝐪_err. This gives a correcting torque vector that is proportional and along
the same axis. This error vector comes from the quaternion that describes the rotation needed
from the current orientation to the target. This quaternion can be expressed
in the polar form

    q_err = cosθ/2 +
      𝐧̂sinθ/2.

From this quaternion, the unit vector, 𝐧̂, can be interpreted
as the axis about which to rotate in order to reach the target. The
sinθ/2 term scales this vector to increase with the angle of
error, θ. This can be interpreted as the minimum required angle to reach
the target, where the maximum error of θ = 180^∘, results in
a scaling term of 1. Hence, taking the last three terms of this error
quaternion gives the error vector used in the control law.

The integral term is used to correct for small constant errors that are too
small to be handled by the proportional controller. By summing the errors each
time the controller updates,
even small deviations can be registered and corrected for.
In order to avoid considering the entire error history equally, which could lead to
unpredictable behaviour, the history is continuously decayed as new terms are
added to the sum. This is done with scaling factor t_0, which decides the
duration over which significant consideration is given. The integral term is
therefore given to be

    𝐠_err(t) =
      ∫_0^t𝐪_err(t')exp(t'-t/t_0)dt',

where t, represents the time up until which the integral term should
be calculated, and t' is the integrated variable over this range. Note that the
convention here is that t≥0, where t=0 refers to the same time at which
the inertial and rotating frames first align and the satellite's operation begins. 

The derivative term causes damping as the torque will be reduced if the body is
rotating towards the target too quickly. The magnitude of this of the error
vector comes from the difference
between current angular velocity, read from the satellite's gyro, and the
target angular velocity. The direction of the vector comes from 𝐪_err.
The magnitude of the target angular velocity depends on θ in
(<ref>). In calculating
ω_tar, an upper bound must be set on the speed
to avoid exceed the operational limits of the flywheels. 
An approximate value for this was calculated using M.
Tisaev's hardware performance values <cit.>.

The maximum angular speed of the flywheels is 10,000 rpm, allowing
a safety margin, this will be treated as 8000 rpm or 48000^∘/s. The angular momentum of a
flywheel rotating with speed s is given as
m_WR_W^2 s.
With external torque omitted, the net angular momenta of the flywheels 
and the angular momentum of the body are equal.
Rotating about the body's x, or y, axis will be more difficult than
rotating about its z-axis since the CubeSat is longer about this dimension.
Using Tisaev's values for the hardware currently proposed, the satellite's moment of
inertia about its x-axis, will be 0.034 kg m^2, whilst the moment of
inertia about the flywheel's axis of rotation will be 1.1×
10^-6 kg m^2. The maximum angular speed
of the satellite can therefore be approximated as

    ω_max =
      1/2m_WR_W^2/I_x= 1.1×
      10^-6/3.4× 10^-2  48000^∘/s= 1.55^∘/s.


With this limit on angular velocity, and a knowledge of the frequency of
compute cycles in the controller, a maximum angle per cycle can be calculated,
θ_max.
This scales the θ term from 𝐪_err so that the
magnitude of the target angular velocity never exceeds ω_max. This
gives the target angular velocity to be

    ω_tar =
      min{θ/θ_max,
      1} ω_max 𝐪_err.


Revisiting how reaching the target was defined in Section
<ref>, the body-fixed positive z-axis must rotate to align with the ground
target as this axis passes through the satellite's camera. 
The axis about which to
do this rotation most efficiently will always be perpendicular to both the
initial z-axis and the resultant z-axis. This indicates that the rotation
axis will always lie in the x-y plane of satellite body frame. Hence the
𝐪_err vector will never have a z component when
expressed in the body frame as shown in Figure <ref>. Because the direction of all three terms is
governed by 𝐪_err, the satellite will have 0
instantaneous angular velocity about the body frame's z-axis at all times.
The use of instantaneous here is to account for the angular
displacement that will occur about the z-axis over time, but only due to
consecutive rotations about the other two axes.



 §.§ Gain Determination


With the terms of the controller established, a method by which to acquire the
gains is required. Initially, this was done through manual tuning. 
However, an exploratory method of solving for the gains
analitically resulted in better performance and will be discussed here. This
method uses the set of differential equations that describe the system to
derive the optimal gains for controlling it. This is done by linearly
approximating the system about the equilibrium using a Jacobian matrix. 
By treating the target as stationary in the rotating body frame, the state
variables at this equilibrium can be easily determined. From here, the
characteristic polynomial of the Jacobian can be used to calculate the
relationship between the three gains in order to achieve repeated eigenvalues,
thus optimising for stability. Finally, a discretised version of the same Jacobian 
is used in order to take account of the undetermined update time of 
the controller. By setting the repeated eigenvalue of this matrix to zero,
the value of gains that produce critical damping can be attained.

Along with the two equations of motion derived in Section
<ref>,
two additional differential equations
are required to represent all time dependent variables in the system. The
first equation describes the motion of the flywheels, which is governed by the
control law, and the second describes the
integrated element of the control law, 𝐠_err.

The integrated error is given as 

    𝐠_err(t) =
      ∫_0^t𝐪_err(t')exp(t'-t/t_0)dt',

where the integration variable is t'. The derivative of
𝐠_err, with respect to t, requires the Leibniz integral
rule. This allows for the derivative of an integral to be taken with respect
to the limit variable instead of the integration variable, giving

    𝐠̇_err(t)    = ddt(∫_0^t𝐪_err(t')
      exp(t'-t/t_0)dt') 
       = 𝐪_err(t) + ∫_0^t ∂∂
      t(𝐪_err(t')
      exp(t'-t/t_0))dt'
       = 𝐪_err(t) - 1/t_0∫_0^t 
      𝐪_err(t')exp(t'-t/t_0)dt'
       = 𝐪_err(t) - 1/t_0𝐠_err(t).


Some assumptions and simplifications have been made for this derivation in
order to reach an analytical solution that is feasible. 
The most significant of these is the omission of external torque form the
equations of motion. This is because it is an external parameter than can not be expressed
purely in terms of the system variables. This simplifies
(<ref>) to become

    ω̇_B   =𝐈_Tot^-1(
      τ_Ext -
      ω_B×𝐈_Totω_B -  𝐄_W𝐬̇ -
      ω_B×𝐄_W𝐬)
       = - 𝐈_Tot^-1(
      ω_B×𝐈_Totω_B +  𝐄_W𝐬̇ +
    ω_B×𝐄_W𝐬).

A further simplification is to treat the satellite
inertia matrix, 𝐈_Tot, as diagonal. This will prove to
simplify the characteristic equation significantly whilst maintaining a close
approximation of the real matrix. This is due to the symmetrical design of the CubeSat,
causing its principle axes of rotation almost align with the body-fixed axes.

The final simplification is to model the ground target as stationary, from the
satellite's perspective. This allows for the equilibrium values of the state
variables to be easily determined. This will prove to be an acceptable
simplification as the controller is able to maintain tracking the target,
treating its movement as another source of pointing error.

The full set of differential equations is given to be 

    ω̇_B   = - 𝐈_Tot^-1( 
      ω_B×𝐈_Totω_B +  𝐄_W𝐬̇ +
      ω_B×𝐄_W𝐬)
    q̇   = 1/2q⊙ω_B
    𝐄_W𝐬̇   = -𝐊_p𝐪
      - 𝐊_i𝐠
      - 𝐊_dω
    𝐠̇_err   = 𝐪_err -
        1/t_0𝐠_err.


Note that the wheel speeds have remained multiplied by the inertia-like
tensor, 𝐄_W, which is done to simplify later
calculations. Since the two terms only occur as a product, they can be
expressed as a new vector, 𝐬_E, which is equivalent to
the net angular momentum generated by the flywheels, and its time derivative
equal to their net torque.

These multidimensional variables can now be represented by their individual
elements. Giving the scalar state variables to be

    [ω_1  ω_2  ω_3   q_1   q_2  
          q_3   q_4   s_E1   s_E2   s_E3   g_1   g_2  
      g_3]^⊤.

The same decomposition is performed on the right side of
(<ref>) and is shown in Appendix
<ref>. This set of equations can then be differentiated
with respect to each of the 13 state variables to form the Jacobian matrix of
the system. This Jacobian is shown in Appendix <ref>. 

With this Jacobian, the values for the state variables at equilibrium can be
substituted in. Since the target is modelled as stationary, this
system will reach equilibrium when the satellite is not rotating and there is
zero pointing error. This corresponds to each state variable becoming zero,
except for the pointing error quaternion, which becomes the identity
quaternion, q=[1,0,0,0]^⊤, denoting no rotation.
Substituting in these terms yields a matrix in terms of just the principal 
moments of inertia and the controller gains.

Since the Jacobian is of shape 13×13, its characteristic polynomial will
be of degree 13.
This is not feasible for analysis as the analytical expression of the eigenvalues
would be too complex. However, rearranging the state variables by which axis
they correspond to yielded a block diagonal Jacobian, which is shown in 
Appendix <ref>.
A block diagonal matrix contains submatrices along its diagonal
and zeros everywhere else. The benefit of this is that the eigenvalues of the
whole matrix are the set of eigenvalues from each of the submatrices, resulting
in multiple characteristic polynomials with a lower degree. The Jacobian has
three 4×4 submatrices along its diagonal and a 0 for the final
element. The submatrices differ only in the axis that the gain and
inertia value correspond to. This allows for just one of the 4×4
submatrices to be analysed as the eigenvalues are the same for
the other two after changing the axis of the parameters. This is the
justification for distributing the gain values over the three dimensions,
otherwise this method would yield 3 equations for the same scalar gain
value. 

The first submatrix from the reordered Jacobian is

    [ -k_dx/I_x -k_px/I_x -k_ix/I_x         0;       1/2         0         0         0;         0         1    -1/t_0         0;     -k_px     -k_ix     -k_dx         0 ].

The characteristic polynomial of this matrix is then calculated to be

    λ(λ^3 + I_x+k_dxt_0/I_xt_0λ^2 + 2k_dx+k_pxt_0/2I_x t_0λ +
    k_ix t_0 + k_px/2I_xt_0).

Dividing by λ, whilst noting that one eigenvalue will be 0, gives a
cubic equation that can be solved for the three gains that act on the x-axis.
To achieve critical damping, the system must have 3 repeated eigenvalues,
meaning the characteristic polynomial will be of the form

    (λ - a)^3 = λ^3 - 3 a λ^2+ 3 a^2λ -
      a^3.

Matching the coefficients from (<ref>) and
(<ref>) provides three equations that can be used to solve
for a, and two of the gains, in terms of the third. Choosing this gain to be
k_dx results in the simplest equations, giving

    a    = - I_x + k_dx t_0/3  I_x t_0
    
      k_px   = 2 (I_x^2 - I_x
      k_dx t_0 + k_dx^2 t_0^2)/3  I_x 
      t_0^2
    
      k_ix   = 2(k_dx t_0- 2
      I_x)^3/27  I_x^2 t_0^3.


These equations express the values that the proportional and integrated gains
must take on in terms of the derivative gain. They also show that the repeated
eigenvalue is equal to a, guaranteeing stability as 
I_x, k_dx, and t_0 are all positive.

In order to find an analytical solution for k_dx, the linearised
system must first be discretised. This is to represent the
discrete time step of the controller, which will play a significant role in the
performance of the controller. To achieve this, Euler's method
of numerical integration is used, representing the state variables in discrete
time steps that are adjusted by the jacobian matrix scaled by step size. This
is represented as

    𝐱_n+1   = 𝐱_n+Δ t  𝐉 𝐱_n
       = (𝐈_𝐝+Δ t 𝐉)𝐱_n,

where Δ t is the time step of the controller,
and 𝐉 is the system's Jacobian matrix. 
Focussing again on just the first submatrix along the diagonal of the composite matrix shown in
(<ref>) gives

    [1 - Δ t k_dx/I_x    - 2
          Δ t(I_x^2 - I_x k_dx t_0 + k_dx^2
        t_0^2) /3 I_x^2 t_0^2    - 2 Δ t(
         k_dx t_0- 2 I_x)^3/27 I_x^3 t_0^3   
        0
    Δ t/2    1     0     0
    0    Δ t      1- Δ t/t_0   
        0
    - Δ t k_dx    - 2 Δ t(I_x^2 - I_x
        k_dx t_0 + k_dx^2 t_0^2)/3 I_x
        t_0^2    - 2 Δ t (k_dx
    t_0- 2 I_x)^3/27 I_x^2 t_0^3    1],

where the gains, k_px and k_ix, have been expressed
in terms of k_dx.

The eigenvalues of this matrix are

    λ _1,2,3   = 1 - Δ t/3t_0 - Δ t k_dx/3
      I_x
    λ _4   = 1,

where there is a repeated eigenvalue as with the original system.

With discretised Jacobian matrices, stability occurs when the eigenvalues have
an absolute value less than 1, instead of the non positivity contraint present in
the original system. Critical damping occurs when λ_1,2,3=0.
Substituting this value into (<ref>) and rearranging gives the 
solution for k_dx to be

    k_dx = 3I_x/Δ t-
      I_x/t_0.

This result is reasonable
in that the strength of the gains should increase with the satellite's inertia, and
should decrease as the time step, Δ t, increases. This avoids
overshooting before the orientation information has been updated. The value for
t_0 was found to have little effect on the magnitude of the gains when
compared to the other parameters. This is demonstrated by varying the value for
t_0 and plotting the resulting gains in one axis, shown in Appendix <ref>.

Although this value for k_dx would elicit critical damping
in a truly linear system, it will not be suitable for the nonlinear case. 
In practice, using the gains at this magnitude will cause the
controller to become unstable as it reacts to small nonlinear errors.
By reducing the magnitude of the gains, the controller will become less 
reactive and in turn become more stable. 
Hence, (<ref>) acts as an upper limit on the gain magnitudes. A
scaling coefficient for 𝐊_d is therefore introduced, allowing for all
three gain matrices to be scaled appropriately as
they are all expressed in terms the elements of 𝐊_d.

This gives the expressions for the automatic tuning of the control law gains to
be

    k_di   = ρ(3 I_i/Δ t-
      I_i/t_0)
    
      k_pi   = 2 (I_i^2 - I_i
      k_di t_0 + k_di^2 t_0^2)/3  I_i 
      t_0^2
    
      k_ii   = 2(k_di t_0- 2
      I_i)^3/27  I_i^2 t_0^3,

where i represents the respective axis and ρ is the scaling coefficient.

The scaling coefficient varies the elements of 𝐊_d
linearly whilst causing a nonlinear change in the 𝐊_p and
𝐊_i elements in order to maintain optimal tuning. This
can be seen in Figure <ref>, showing the relationship between each
x-axis element of each gain matrix and the scaling coefficient. In order to select a value
for ρ, flyovers were simulated for a range of values whilst measuring the
maximum pointing error to occur. All other parameters were set to their default
values, shown in Appendix <ref>. The results of this sweep are shown in Figure
<ref>. 

For values of ρ close to zero, the gain magnitudes cannot generate a
torque large enough to rotate the satellite as necessary. This means that the satellite
cannot catch up with the target during the flyover, causing pointing error to
increase dramatically. For larger values of ρ, errors increase due to the
gains being too strong. Since these gains are based on a linear approximation
of the system, they only work for small perturbations from zero
pointing error. The stronger these gains are, the smaller this perturbation
must be, otherwise the satellite continuously overshoots the target, resulting
in sustained oscillations. This over shoot reaches 0.1^∘ for 
the 0.3 s controller when ρ=0.2, whilst the 0.1 s
controller updates fast enough to avoid the effect. Since the update time of
the controller is not yet known, all three speeds will be considered when
analysing the controler. Therefore the gain coefficient of 0.05 was chosen, so as
to maximise the stability of all three controllers whilst maintaining the ability
to produce the necessary torque.



This coefficient safely minimises pointing error during a flyover
manoeuvre. This will also allow for attiude control during the
satellite's standby mode, when no ash cloud is present. However, 
when no ashcloud is present, the satellite will only need to maintain its orientation,
either relative to the sun or Earth. The attitude control system can therefore
prioritise power and use a controller with smaller gains, generating less
torque. 
This controller could be attained by lowering the gain coefficient to 0.01, at which point the integral
term becomes essentially zero, and the maximum torque from the controller will
be reduced. This would be adequate for the less stringent attitude requirements
during its standby mode.



§ ANALYSIS


In this section, the performance of the newly designed controler will be
assessed during flyovers of the ground based target.
The flyover manoeuvre will first be simulated under idealised conditions, for
which the sensor readings are precisely accurate, then with different
magnitudes of noise added to test the robustness of the controller.
From the PROVE mission specification, the control system should
maintain a pointing accuracy of around 0.1^∘ in perfect conditions, limited only
by operational constraints such as maximum wheel speed and controller update
frequency. This is with the intention of maintaining a
pointing accuracy of 1^∘ when sensor errors and external torques are
present.

The controller incorporates sensor readings into a feedback loop to correct for
errors in attitude. Calculating these errors and the resulting correction
relies on three inputs, which are defined in Table <ref>.
Although the satellite's position is only updated once every 12 hours, the
this is given in the form of a trajectory, meaining the position can be implicitly attained with
just a value of time. The other two inputs rely on explicitly calculated values
from on-board sensors. The frequency that these can be updated is not yet
known, as it depends on the capabilities of the chosen hardware.
Estimates for the minimum update times of each input are shown in Table
<ref>, however these will be varied in the simulations to
determine if longer update periods are feasible. This could allow for cheaper
components to be purchased, where possible, or reduce the power consumption
from the ADCS.





 §.§ Idealised Conditions

For the following simulations, unless stated otherwise, the parameter values
that will be used are shown in Appendix <ref>.
The satellite has been modelled to follow a polar orbit, with an altitude of
300 km and a period of 1.5 hours. This means the satellite will coincide with
approximately the same location on Earth every 12 hours. Throughout the
entirity of its orbit, the satellite will remain pointing at the ash could,
requiring minimal torque when in the opposite hemisphere.
An example flyover manoeuvre is shown in Figure <ref>.
The error along each of the body axes and the corresponding wheel speeds from
the same flyover are shown in Figure
<ref>. No sensor noise has been introduced at this point.
This acts as a baseline from which to compare results with noise added in.
This example flyover spans 400 seconds, the approximate
period in which valid images of the ash cloud can be taken. During the
satellite's approach to the target, its orientation must change more rapidly the closer
it becomes, meaning it becomes increasingly hard to maintain pointing
accuracy. This is consistent with the pointing errors
shown in Figure <ref>, where the satellite is directly above the
target at t=200 s. 

Although the axis error plot will strongly correlate to the measured pointing
error, they do not describe the same value. Axis error corresponds to the
components of the control law's 𝐪_err term,
which is calculated at the start of each time step, Δ t. The control law gives the
optimum torque to correct for this error in the time step. 
To gain an accurate measure of pointing error within this
discretised system, the recorded pointing error is the average of the
error at the start of the time step, when it is greatest, and the error after
the torque has been applied for Δ t seconds, when error is at its minimum. 
It is important to note that the z-axis
error will always be 0^∘ because of how pointing error is defined,
as discussed at the end of Section <ref>.
The strong correlation between axis error and wheel speeds 
is due to the default wheel orientations being used whereby they rotate about
the body frame axes. Using a different set of rotation axes would result in
different speed profiles for the wheels. An example of this case can be seen in
Appendix <ref>.



Table <ref> shows that attitude is updated at most once per
second. This is because the orientation is
determined by collecting data from the light sensors on each surface of the
satellite, and the magnetometer, which is then passed through a Kalman filter. 
This process is computationally costly and should only be done as
frequently as necessary to save power. The gyro, however, is much more
efficient. It has a minimum update time of 0.1 seconds as it is only limited by
the update time of the controller that uses it. To account for this disparity
in update frequencies, the controller extrapolates the satellite's attitude
from its angular velocity. This provides more up to date values of attitude
than can be provided by the attitude sensors alone. This extrapolation is based
on the quaternion derivative shown in (<ref>) and uses
the Euler method such that

    q_n+1 = q_n+1/2Δ t q_n⊙ω_Bn

where Δ t is the update time of the controller. This was found to increase
pointing accuracy in all simulated cases. The impact of
attitude being calculated from two separate sources is explored in Section
<ref>. Even without noise, the performance of the
controller is affected by the size of Δ t, as shown in Figure
<ref>. Here, the attitude update time is varied whilst
measuring both the mean and maximum pointing errors of three controllers.

Each flyover in this plot lasts 400 seconds and had the satellite pass
directly above the ground target, which requires the sharpest turn and thus
causes the largest error. The best performance shown here achieves
0.075^∘ maximum pointing error. Comparing this to the example flyover, where the
satellite passes 4^∘ west of the target, the maximum error is just
0.041^∘. The impact of flyover angle on pointing accuracy is shown in Figure
<ref>. With this in mind, the
worst-case scenario of flying directly overhead will be used for any
simulations used to test controller performance to confirm that the results are applicable to
any manoeuvre.


The max error values in Figure <ref> show that all
controllers can achieve approximately 0.1^∘ pointing accuracy as long as the
orientation update time is shorter than 5 seconds. This suggests that
the analytical approach used to tune the gains is valid. However, despite using
a low gain coefficient, the controller
can still respond chaotically to pointing errors that are in the
range of approximately 10^∘ and above. This was solved
by gating the 𝐪_err term to limit the magnitude of
pointing error that could be given to the controller. 
Therefore, considering the case that the satellite is pointing the
opposite direction to the target. Instead of accelerating to an unsafe
speed and overshooting, a constant angular velocity is used to slowly arrive
within safe proximity of the target. This works more as a precaution since the
controller should never reach a pointing error this large.

It is worth noting the undulations that occur in the error values of Figure
<ref>. These can be explained by analysing the small
oscillations exhibited by the satellite as
a result of the sparse orientation updates. In Figure
<ref>, the case of
Δ t=0.3 s, with orientation updated every 7.8 seconds,
shows a disproportionately large maximum pointing error when compared to the 9
second case plotted to its right.
Time series plots for these two cases are shown in
Figure <ref>. Looking first at the 7.8 second case, the
transient response to the orientation being updated shows the attitude
correcting and overshooting, then overshooting again in the other direction due
to the changing location of the target. The period of this transient
oscillation coincides with the 7.8 second update time of orientation, meaning
that the overshooting behaviour is encouraged by the poorly timed updates. Then
looking at the 9 second case, the period of the transient oscillation is the
same, however the longer update time avoids the oscillation being continuous,
leading to less significant overshooting.



It should be noted that this phenomenon only seems significant in idealised conditions. The largest
deviation from expected error as a result of this occurance is around
0.03^∘, whereas variations in sensor noise will affect pointing error by
more than 10 times this amount.

Furthermore, simulations were run for orientation update times up to
30 seconds, in order to clearly show its effect on pointing
error. However, the value for this update time will realistically be in the
range of 1 to 5 seconds. Focussing on this range in the plots from Figure
<ref> shows that controller update times have a significantly
larger effect on the pointing error. This is to be expected since the gyro is
updated at this rate and can be used to approximate attitude. Consequently,
controller update time will be focussed on when assessing the controller's
performance in the presence of noise.



 §.§ Noise


The error value of the gyro, in Table <ref>, comes from its
bias instability, which can be
interpreted as the standard deviation from the correct angular rate. This comes
from the datasheet for the gyro so is assumed to be accurate. Conversely,
the attitude sensor error is the least reliable of three. The values
were attained from R. Biggs' report which only used an update time of Δ t = 0.2 seconds. This means that
the noise added to the attitude sensors when Δ t≠0.2 s may not
accurately represent the standard deviation they would experience. However,
controller performance will be assessed for a range of noise magnitudes,
meaning these standard deviations serve more as an approximate value from which
to vary noise.

Varying the noise magnitudes beyond their estimated value will
help develop safe margins of operation with respect to sensor error and provide
a measure of how sensitive the controller is to each input. It is important to
note that the errors in the attitude sensors are assumed to be independent of
each other, meaning they have zero mean. This is not the case for the position
and angular velocity readings. This is because the satellite's position will drift
over time from its estimated location due to small external forces. If the
position estimate is 1 km north of the satellite's real position, then one
minute later, the position estimate will still be approximately 1 km north and will
continue to diverge in the same approximate direction until a new TLE has
been attained. Given the long period over which this divergence happens, the position error
is modelled as constant over the 400 second flyover. Therefore, the
position of the satellite with simulated error is given as

    𝐫̅_sat = 𝐫_sat +
      𝐫_err,

where 𝐫_err is constant for each flyover and points in a
random direction.

The gyro experiences two forms of error. The first is white noise, 𝐰, which has a
high frequency and comes from thermo-mechanical events within the system.
However, this has a mean error of 0^∘/s and so does not contribute to the
gyro's bias, which is the error over a long period. 
The second cause of error is bias instability, 𝐛, which
comes from flicker noise in the electronics and is of a lower frequency. This
error depends on its previous values and so, like the position error, will not
give a mean error of 0^∘/s. Using a common formulation for gyro noise <cit.>,
angular velocity with noise is given as

    ω̅_n = ω_n+
      𝐰_n+𝐛_n,

where

    𝐰_n    = σ_wκ_n
    𝐛_n    = 𝐛_n-1+σ_bκ_n
    κ_n    ∼𝒩^3(0,1)
    σ_w   = σ_g/√(Δ t)
    σ_b   = σ_g√(Δ t),

where noise coefficient, σ_g, is varied to control the magnitude of gyro
noise, and Δ t is the time step used in the controller. This formulation
is beneficial as it allows for the use of just one parameter when controlling gyro
noise. The standard deviation of the gyro measurments from their correct value is given to be
4.5×σ_g. Hence, to model the expected standard deviation
in the gyro readings, σ_g=0.042/4.5=0.0093. The example
flyover from Figure <ref> can now be repeated but
with gyro noise added in. Three time series plots from this flyover
are shown in Figure <ref>. The
axes error values shown in Figure <ref> are far more
chaotic than before. This is a result of the extrapolation of attitude using
the noisy gyro readings. The stronger lines plotted on top show the smoothed
curves of their respective axes. This demonstrates how, although the sensor may
provide very noisy data, most of this has zero mean, so will largely cancel out
over the orientation update period of 1 second, in this case.


To simulate noise in the attitude sensors, a rotation quaternion will be
constructed such that it rotates about a random axis, 𝐧̂, in the x-y plane of
the body frame by an angle that follows a normal distribution with zero mean.
Recall that consecutive rotations are achieved by multiplying rotation quaternions
together, with the rightmost quaternion operating first.

This gives the orientation quaternion, with noise, to be

    q̅   = q_err⊙ q
    
      q_err   = cosθ_err/2 +
      𝐧̂sinθ_err/2
    θ_err   ∼𝒩(0,σ_a),

where σ_a is the standard deviation of the attitude value from its
correct value and the axis of rotation, 𝐧̂, is perpendicular to
z_B.

With expressions for the input errors now established, their magnitudes can be
individually
varied to see how the controller performs. This will provide an understanding
of the safe margins in which the satellite can still accurately orientate itself
towards the ground target.
The relationship between error in each input and the maximum pointing error to
occur during a flyover is shown in Figure <ref>

The top plot shows a strong linear relationship between position error and
maximum pointing error. This is due to the approximately constant value that
the error will have during the flyover duration, causing a constant offset to
the direction that the satellite points. 
Position error will have the greatest effect when it places the
satellite either directly ahead, or behind, its actual location in the orbit.
In this case, the angle discrepancy is largest when the estimated satellite position is
on the opposite side of the target to its actual position. This angle
discrepancy is then proportional to the magnitude of the
position error, using the small angle approximation.
The plot shows that position
error can become twice that of the estimated upper bound of 2 km, whilst maintaining a
pointing error below 1^∘. This plot also shows that the impact of
position error is independent of the update frequency of the controller.



Looking next at the impact of noise in the gyro, shown in the second plot of
Figure <ref>. A line of best fit has been used to emphasise the
linear relationship between noise magnitude and
pointing error. The expected standard deviation in the gyro readings is
0.042^∘/s. This value corresponds to a maximum pointing error of approximately
0.8^∘ for all three controllers. With the target of achieving a pointing
error of 1^∘, this leaves only a small margin of error for the other
two inputs. The significant effect that gyro noise has on the system's pointing
accuracy can be expected due to its use in extrapolating attitude between
sensor updates. This results in both the attitude, and the angular velocity, of
the satellite depending on the accuracy of the gyro. Despite this, it is
preferable to use extrapolation, since otherwise the controller 
is only using a time accurate value for attitude once per second at most.
This results in significant oscillations in
attitude throughout the flyover as the satellite periodically readjusts to the
sparsely updated attitude value.

Finally, looking at the impact of noise in the attitude sensors, shown in the
bottom plot of Figure <ref>. In this plot, a
smoothing function was used to to emphasise the relationship between errors
whilst preserving the nonlinear shape of the plots. The
steeper gradient towards the start is due to the maximum angular rate
that can be achieved by the satellite. In Section <ref>, this
was estimated at
1.55^∘/s, putting a limit on the angular displacement that can result
from a single incorrect attitude measurement. Therefore, in the case that
Δ t=0.2 s, the satellite could only rotate approximately 0.31^∘
over one cycle, thus mitigating the effect of larger errors in particular. The
pointing error will still increases with noise, however, since the errors can
compound to produce a longer perturbation from the correct attitude.
The effect of attitude sensor noise is similar to that of gyro noise.
However, within the range of their expected values, 0.25^∘ and 0.042^∘/s
respectively, attitude noise is shown to have a stronger effect on pointing
error than gyro noise. 

Additionally, towards the upper limit of estimated
noise, the maximum pointing error is shown to exceed 1^∘. In the
worst-case scenario, the controller with an update time of 0.3 seconds reaches
a maximum error of almost 2^∘. Although the proposed pointing error has
been surpassed, it is important to note that these error terms are generated
using the direct standard deviations from the individual sensors. No
additional filtering has been performed using the combined inputs to reduce the
weight given to extreme values.
In practice, a Kalman filter equiped with a model of expected motion in
will be utilised. This allows for rapidly deviating readings to be smoothed about
an estimated average, causing less errors in the inputs given to the
controller. The use of unfiltered error values provides a worst case scenario
in which to understand their effect. By aiming to attain the target pointing accuracy
under these conditions provides a safe margin for unforseen error during actual
operation.

With the effect of each individual sensor's noise quantified, their combined
effect wil now be assessed.
Figure <ref> shows the maximum pointing error to occur during
flyovers from directly above as each pair of inputs is varied from zero up to
three times their estimated upper limit. This allows for the safe margins of sensor error
magnitudes to be determined, whilst taking account of the compounding
effect that may occur when multiple sources of error are present. 


The top plot of Figure <ref> shows the maximum pointing error to occur
as gyro noise and attitude noise are varied. This
two dimensional sweep shows how a larger pointing error arises when noise is
present in both sensor inputs. As with the one dimensional sweep, attitude
sensor noise is shown to cause a steeper gradient in pointing error at lower 
magnitudes whilst pointing error increases linearly with gyro noise. The green
box shown in the bottom left corner indicates the range of noise that these
sensors are estimated to experience. Within this box, the maximum pointing errors
reach up to 1.54^∘, exceeding the proposed limit of 1^∘ shown by
the darkened contour. At the top
right corner of the plot, when both sensors experience three times that of their
expected noise, the
pointing error is shown to reach approximately 4^∘. This is similar to
the sum of pointing errors from each input's one dimensional sweep of noise
magnitude at these respective values. This suggests that there is no extra error caused
by the two inputs interacting with each other, giving a linear relationship
between net input error and resultant pointing error.

The combined effect of noise in the attitude sensors and
position error is shown in the middle plot of Figure <ref>.
Firstly, the leftmost vertical slice of this plot shows
similar pointing errors to that of the top plot, which
corresponds to the single parameter sweep of attitude noise in Figure
<ref>. However, moving horizontally, as position error is increased, 
the maximum pointing error increases only slightly. This relates to the
smaller effect that position error has on pointing accuracy,
when compared to the noise present in the other two inputs.

The third plot of Figure <ref> shows the combined effect of 
gyro noise and position error.
A similar response is shown to that of the previous plot, with position error
causing only a small increase in pointing error when compared to the effect of
gyro noise. This
can again be attributed to the relatively small effect that position error has,
within its estimated range. Once again, the maximum value of pointing error here
is approximately equal to the sum of pointing errors 
from each inputs isolated noise sweep at their respective values.



These dual parameter sweeps show that pointing error is
proportional to the sum of input errors, as opposed to their product. A
positive relationship between input errors and pointing errors was always expected,
hence identifying this to be linear, within realistic ranges, is a positive
outcome. Although the controller update time used is these sweeps was set to
the default value of 0.1 seconds, the same linear relationship was shown for update
times Δ t = 0.2 s and Δ t = 0.3 s when using their respective
error values from the one dimensional sweeps.

The two plots that include attitude sensor noise show pointing error to exceed
the target of 1^∘ within the range of expected noise values. This was
expected, given the isolated plot of attitude error also exceeded the
target pointing accuracy. However, as mentioned before, these errors
represent the inputs before any filtering has been performed on the combined
values. The degree to which input error can be reduced has not yet been
quantified for the PROVE mission. Hence, in order to avoid over-reliance on
this filtering, it was omitted completely. The expected effect of this is to
attain larger pointing errors whilst still capturing the approximate behaviour
of the controller to the inputs, i.e., a linear relationship between the sum of
input errors and resultant pointing error.



§ CONCLUSION

The primary aim of this project was to develop the current system of equations
that were being used for attitude control within the PROVE mission. This
involved the incorporation of gyroscopic torques that arise from the
individually rotating flywheels. Additionally, a method by which to optimise
wheel accelerations in the case of a redundant fourth wheel was derived.

Improvements to the state-of-the-art quaternion based controller were then
explored. Firstly, an integral term was added to the control law. This allowed
for small residual errors to be recognised and reduced by integrating them over
time. The second improvement aimed to derive equations for optimal controller
gains such that tuning could be done automatically. By defining the gains in
terms of the satellite's parameters, this controller design could still be used
once the system architecture is finalised. 

To achieve this, the system was first linearly approximated about its
equilibrium. From there, the Jacobian was rearranged to attain a block
diagonal matrix, shown in Appendix <ref>. This meant
that eigenvalue analysis could be performed on
just one submatrix, since the result is transferable to the other two. Without
this, performing eigenvalue analysis on the entire 13×13 Jacobian would not be
feasible.

By solving for the gains such that the characteristic polynomial of the system
yielded repeated eigenvalues, maximum stability could be achieved. This produced
equations for the necessary relationship between the gains in order to maintain
stability. The three gains could then be expressed in terms of just one.
Using this new representation, the system was then discretised, 
allowing for the controller's update time to be factored into the Jacobian. By
setting the repeated eigenvalue of this matrix to zero and solving for the
the gain value, a critically damped control law was attained. Treating the
system as linear, along with the assumption that the target is stationary, meant that the
controller was unstable in the presence of larger pointing errors. This was
fixed by reducing the gains proportionally using a gain coefficient.
With this adjustment, the controller performed better than could be
achieved with manual tuning methods.

After developing the controller, analysis could be performed on the
topics proposed for this project. Starting with external torque, each
potentially significant source
was assessed, arriving at an estimate for the maximum torque that will be
experienced during a flyover. Simulations were run to assess the effect this
has when at its estimated maximum value, working directly against the rotation of the satellite. The results
showed that, due to the relatively slow rate at which the external torque changes,
the controller was able to correct for it, maintaining the same pointing
accuracy as without external torque. The most notable effect that external torque had on
the system was the increase it caused in the maximum speed reached by the flywheels.
In the worst case, this was shown to result in wheel speeds reaching
8325 rpm, increasing by 12% from the case of no external torque. This
impacted the gyroscopic effect of the flywheels, as it depends on their
angular velocity.

The effect of gyroscopic torque within the system was previoulsy unexplored
within the PROVE mission. By isolating the torques during a flyover in which
maximum external torque was present, the gyroscopic effect was shown to reach a
magnitude of 10^-7 Nm at the peak of the flyover. This is approximately
two orders of magnitude smaller than the intentional torque from the
wheels during that period. Importantly, the controller maintained the
same pointing accuracy as in the idealised case when gyroscopic torque was 4
orders smaller than the intentional torque. This result showed that gyroscopic
reactions within the system are not significant enough to impact the pointing
accuracy of the satellite.

In order to validate that the automatically tuned controller could perform in
realistic conditions, a series of simulations were run to determine the
most significant factors impacting pointing error. Initial simulations, in
idealised conditions, showed that the update time of the controller had a greater effect on the pointing
error than the update time of the attitude sensors. This is because the
gyro readings can be used to extrapolate the attitude of the satellite in the
absence of an explicitly calculated value. The gyro is updated with
the controller, hence this update frequency becomes the limiting factor of performance
when sensor noise is omitted.

The controller depends on three variables to calculate flywheel torque. These
are the orientation, the angular velocity, and the position of the satellite.
These are calculated using light sensors on the exterior of the satellite, a gyro sensor,
and trajectory parameters from a TLE, respectively.
Each of the inputs will be subject to some
degree of error. A model with which to simulate the error for each input was
then developed, allowing for the magnitude of this error to be varied. From there, the effect
of each input error on pointing accuracy was compared. This showed attitude
sensor noise to be the most significant factor in causing pointing error.
Because this is updated sparsely, extrapolation from the gyro is required,
meaning any error in the attitude sensors will be propagated over the
extrapolation process until a new explicit value is attained from the attitude
sensors.

The effect of two input errors was then evaluated. The most important result
from these simulations was finding that pointing error was linearly related to
the sum of input errors, as opposed to a higher order relationship. This
allows for a linear model of total input error to be used in estimating the
maximum pointing error to occur. For example, if all inputs experience their
largest expected magnitude of error, the maximum pointing error is given to be
approximately 2.1^∘. This uses the default simulation parameters shown in
Appendix <ref>. 

Because no filtering was used on the inputs
before being fed into the controller's feedback loop, this acts as an upper limit
on expected pointing error. In reality, some degree of error reduction will be
possible by comparing the readings to a measure of expected variance for that
input, and attributing lower significance to values outside of its expectede
range.

This project has succesfully produced an improved model of the satellite's
dynamics. The model was then utilised to develop an improved control law along with a
method by which to automatically tune its gains. This controller was 
analysed through simulation, sourcing parameter values from other PROVE mission
projects and external research. A future improvement that could be made to this
controller would be to incorporate the known equations of motion into its
control law. This would consist of using the trajectory of the
satellite to calculate the expected rotation that will be needed during the
flyover. From here, sensor input could be less heavily weighted, acting more
for small corrections. This would reduce the effect of their errors
significantly. 

More research is also needed with respect to the external torques that 
will act on this CubeSat, since only approximate magnitudes have been
used so far. This is partly due to significant hardware choices in the satellite
that are unfinalised. For example, solar panels that extend from the
satellite are being considered, which  will significantly effect the
aerodynamic torque that acts on the satellite.

In addition to the physical design of the satellite, the performance of the
internal components will also need more research. By understanding the
limitations of the satellite's hardware, methods by which to reduce demand on
these components can be developed. For example, if the magnetorquer is found to be
more energy efficient than the flywheels, then it may be suitable for constant
torque generation during the satellite's standby mode in order to maintain fixed orientation.
30
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magnetorquer_control
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magnetorquer_as_magnetometer
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flywheel_motor
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  magnetorquers and static input allocation. Transactions on Control Systems
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flywheel_friction
  Tisaev, M. Design of an attitude determination and control system for
  a 3U CubeSat. Master's project, University of Bristol. 2018.
sun_sensor
  Kariniemi-Eldridge T. Coarse sun sensor design and test for attitude
  determination of a cubesat. Master's project, University of Bristol. 2018.
kalman_star_tracker
  Biggs R. Attitude control for ground-target pointing.
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kalman_sun_sensor
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  using low-cost sensors. Computational and Applied Mathematics. 2017.
normal_sensors
  French H. Agile 3 axis attitude control of a cubesat. Master's
  project, University of Bristol. 2017.
ext_torqs
  Zagrski P. Modeling disturbances influencing an Earth-orbiting
  satellite. Pomiary Automatyka Robotyka. 2012.
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  [Accessed 5 Apr. 2019].
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max_ext_torq
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grav_torq
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gyro_data
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w_err
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  navigation. IEEE Transactions on Aerospace and Electronic Systems. 2006.



§ APPENDICES



§ 



 §.§ Triple Vector Product to Matrix


Here is the full derivation through which the matrix vector product was
produced to represent the triple vector product from (<ref>). In the general case, when a vector triple product contains the same vector,
𝐚, sandwiching a different one, 𝐛, the product can be
written as

    𝐚×(𝐛×𝐚)    =
      [ a_1; a_2; a_3 ]×(
      [ b_1; b_2; b_3 ]×[ a_1; a_2; a_3 ])
       =[ a_1; a_2; a_3 ]×[ b_2 a_3 - b_3
      a_2;   b_3 a_1 - b_1 a_3;   b_1 a_2 - b_2 a_1 ]
       =[      a_2(b_1 a_2 - b_2 a_1) - a_3(b_3 a_1 - b_1 a_3); a_3(
        b_3 a_1 - b_1 a_3) - a_1(b_1 a_2 - b_2 a_1);    a_2(b_1 a_2 - b_2 a_1) -
      a_3(b_3 a_1 - b_1 a_3) ]
       =[ (a_2^2+a_3^2)b_1  -a_1 a_2 b_2  - a_1 a_3 b_3; -a_1 a_2 b_1 + (a_1^2+a_3^2)b_2  -a_2 a_3 b_3;  -a_1 a_3 b_1 -a_2 a_3 b_2 + (a_1^2+a_2^2)b_3 ]
       =[ (a_2^2+a_3^2)      -a_1 a_2     - a_1 a_3;      -a_1 a_2 (a_1^2+a_3^2)      -a_2 a_3;      -a_1 a_3      -a_2 a_3 (a_1^2+a_2^2) ][ b_1; b_2; b_3 ]
       =𝐀𝐛.



 §.§ Quaternion Time Derivative


This is a derivation of (<ref>), using the rules and conventions
discussed in Section <ref>. The orientation of the satellite at
time t+δ t is described by q(t + δ t). This is achieved by rotating from
orientation q(t) by δθ=|ω|δ t
about the unit axis
ω̂=ω/|ω|,
where ω is the angular velocity of the satellite at time
t in the inertial reference frame.
This rotation can be written as

    δ q    = cosδθ/2 + ω̂sinδθ/2
       = cos|ω|δ t/2 + ω̂sin|ω|δ t/2.

Hence the final orientation of the satellite, at time t+δ t, can be
written as the combined rotation δ q ⊙ q(t).
The derivative of a quaternion is defined, as with any other variable, as

    q̇(t) = lim_δ t → 0q(t + δ t) - q(t)/δ t.

The numerator in (<ref>) can be written as

    q(t + δ t) - q(t)    = δ q ⊙ q(t) - q(t)
       = (cos|ω|δ t/2 + ω̂sin|ω|δ t/2)⊙ q(t) - q(t)
       = (cos|ω|δ t/2-1 + ω̂sin|ω|δ t/2) ⊙ q(t)
       = (-2sin^2|ω|δ t/4 + ω̂sin|ω|δ t/2) ⊙ q(t).

The higher order term here can be neglected as δ t → 0, giving

    lim_δ t → 0q(t + δ t) - q(t)/δ t   = lim_δ t
    → 0(ω̂sin|ω|δ t/2⊙ q(t)/δ t)
       =ω̂lim_δ t
    → 0(sin|ω|δ t/2/δ t) ⊙ q(t)
       =ω̂ddt(sin|ω|t/2)|_t=0⊙
    q(t)
       =ω̂(|ω|/2cos|ω|t/2)|_t=0⊙ q(t)
       =ω̂|ω|/2⊙ q(t)
       =1/2ω⊙ q(t)

Finally, to attain the equation in the form that is used in Section
<ref>, the angular velocity vector must be transformed to the body
frame coordinate axes, giving ω_B. The
relationship between ω_B and
ω is described by the rotation quaternion q(t) such that 

    ω = q(t)⊙ω_B⊙
      q^-1(t).

Applying this to the quaternion time derivative gives

    1/2ω⊙ q(t)    =
      1/2q(t)⊙ω_B⊙ q^-1(t)⊙ q(t)
       = 1/2q(t)⊙ω_B,

where the product, q^-1(t)⊙ q(t), gives the identity quaternion. The
final equation is therefore

    q̇(t) = 1/2q(t)⊙ω_B.



§ 



 §.§ Four Wheel Time Series







§ 



 §.§ Integral Time Constant


Figure <ref> shows the relatively small effect had by the integral time constant,
t_0, on the gains. Three controller update times are shown, demonstrating how
much larger the effect of Δ t is. The integral time constant was
chosen to be 10 s, due to the time scale over which the satllite orientation
changes. However, similar results were found for different values.




 §.§ Decomposed System of ODEs


Below is the decomposed set of equations that are equivelent to
(<ref>)


    [ẇ_1
    ẇ_2
    ẇ_3
    q̇_1
    q̇_2
    q̇_3
    q̇_4
    ṡ_E1
    ṡ_E2
    ṡ_E3
    ġ_1
    ġ_2
    ġ_3]
      = 
      [1/I_x(s_E2ω_3 - s_E3ω_2 + I_y
      ω_2ω_3 - I_z ω_2ω_3 - g_1 k_ix - k_dxω_1 - k_px
      q_2)
    1/I_y(- s_E1ω_3 + s_E3ω_1 - I_x ω_1ω_3 + I_z ω_1ω_3 - g_2 k_iy - k_dyω_2 - k_py
      q_3)
    1/I_z(s_E1ω_2 - s_E2ω_1 + I_x ω_1ω_2
      - I_y ω_1ω_2 - g_3 k_iz - k_dzω_3 -
    k_pz q_4)
    -1/2(q_2ω_1 + q_3ω_2 + q_4ω_3)
    1/2(q_1ω_1 + q_3ω_3 - q_4ω_2)
    1/2(q_1ω_2 - q_2ω_3 + q_4ω_1)
    1/2(q_1ω_3 + q_2ω_2 - q_3ω_1)
     - k_px q_2- k_ixg_1  - k_dxω_1
     - k_py q_3- k_iyg_2  - k_dyω_2
     - k_pz q_4- k_izg_3  - k_dzω_3
    q_2- g_1/t_0
    
      q_3- g_2/t_0
    q_4- g_3/t_0]
 



 §.§ Jacobian


Below is the Jacobian matrix that results from differentiating the above
equations by each of the state variables. 

    [[                     - k_dx/I_x   - s_E3 + I_yω_3 - I_zω_3/I_x     s_E2 + I_yω_2 - I_zω_2/I_x                              0                     - k_px/I_x                              0                              0                              0                        ω_3/I_x                      - ω_2/I_x                   -
      k_ix/I_x                              0                              0;   s_E3 - I_xω_3 +
      I_zω_3/I_y                     - k_dy/I_y - s_E1
      - I_xω_1 + I_zω_1/I_y                              0                              0                     - k_py/I_y                              0                    -
      ω_3/I_y                              0                        ω_1/I_y                              0                     - k_iy/I_y                              0;   - s_E2 + I_xω_2 - I_yω_2/I_z     s_E1 + I_xω_1 - I_yω_1/I_z                   -
      k_dz/I_z                              0                              0                              0                     - k_pz/I_z                        ω_2/I_z                      - ω_1/I_z                              0                              0                              0                   -
      k_iz/I_z;                        - q_2/2                        - q_3/2                      -
      q_4/2                              0                        - ω_1/2                        - ω_2/2                      -
      ω_3/2                              0                              0                              0                              0                              0                              0;                          q_1/2                        - q_4/2                          q_3/2                          ω_1/2                              0                          ω_3/2                        - ω_2/2                              0                              0                              0                              0                              0                              0;                          q_4/2                          q_1/2                      -
      q_2/2                          ω_2/2                        - ω_3/2                              0                          ω_1/2                              0                              0                              0                              0                              0                              0;                        - q_3/2                          q_2/2                          q_1/2                          ω_3/2                          ω_2/2                        - ω_1/2                              0                              0                              0                              0                              0                              0                              0;                         - k_dx                              0                              0                              0                         - k_px                              0                              0                              0                              0                              0                         - k_ix                              0                              0;                              0                         - k_dy                              0                              0                              0                         - k_py                              0                              0                              0                              0                              0                         - k_iy                              0;                              0                              0                         - k_dz                              0                              0                              0                         - k_pz                              0                              0                              0                              0                              0                         - k_iz;                              0                              0                              0                              0                              1                              0                              0                              0                              0                              0                        - 1/t_0                              0                              0;                              0                              0                              0                              0                              0                              1                              0                              0                              0                              0                              0                        - 1/t_0                              0;                              0                              0                              0                              0                              0                              0                              1                              0                              0                              0                              0                              0                        - 1/t_0 ]]



 §.§ Jacobian at Equilibrium


Below is the reordered Jacobian matrix from above, now with the equilibrium
values discussed in Section <ref> substituted in. The state
variables are now ordered by which axis they correspond to, given as


    [ ω_1,    q_2,    g_1,    s_E1,   ω_2,   
      q_3,    g_2,    s_E2,   ω_3,    q_4,    g_3,
       s_E3,    q_1 ],

which gives the Jacobian to be 

    [[   - k_dx/I_x -
      k_px/I_x   - k_ix/I_x            0            0            0            0            0            0            0            0            0            0;          1/2            0            0            0            0            0            0            0            0            0            0            0            0;            0            1      - 1/t_0            0            0            0            0            0            0            0            0            0            0;       - k_dx     -
      k_px       - k_ix            0            0            0            0            0            0            0            0            0            0;            0            0            0            0 -
      k_dy/I_y   - k_py/I_y   - k_iy/I_y            0            0            0            0            0            0;            0            0            0            0          1/2            0            0            0            0            0            0            0            0;            0            0            0            0            0            1      - 1/t_0            0            0            0            0            0            0;            0            0            0            0       - k_dy       - k_py       - k_iy            0            0            0            0            0            0;            0            0            0            0            0            0            0            0   - k_dz/I_z   - k_pz/I_z -
      k_iz/I_z            0            0;            0            0            0            0            0            0            0            0          1/2            0            0            0            0;            0            0            0            0            0            0            0            0            0            1      - 1/t_0            0            0;            0            0            0            0            0            0            0            0       - k_dz       - k_pz       - k_iz            0            0;            0            0            0            0            0            0            0            0            0            0            0            0            0 ]].




§ 



 §.§ Simulation Parameters


This table describes the default values that are used for the parameters in
simulations of the controller. 





 Paper Title: Dynamical Viability Assessment for Habitable Worlds Observatory Targets.





0000-0002-7084-0529]Stephen R. Kane
Department of Earth and Planetary Sciences, University of
  California, Riverside, CA 92521, USA
skane@ucr.edu

0000-0002-4860-7667]Zhexing Li
Department of Earth and Planetary Sciences, University of
  California, Riverside, CA 92521, USA

0000-0002-0569-1643]Margaret C. Turnbull
SETI Institute, Carl Sagan Center for the Study of Life
  in the Universe, Off-Site: 2613 Waunona Way, Madison, WI 53713, USA

0000-0001-8189-0233]Courtney D. Dressing
Department of Astronomy, 501 Campbell Hall #3411,
  University of California, Berkeley, CA 94720, USA

0000-0001-5737-1687]Caleb K. Harada
NSF Graduate Research Fellow
Department of Astronomy, 501 Campbell Hall #3411,
  University of California, Berkeley, CA 94720, USA



§ ABSTRACT 

Exoplanetary science is increasingly prioritizing efforts toward
direct imaging of planetary systems, with emphasis on those that may
enable the detection and characterization of potentially habitable
exoplanets. The recent 2020 Astronomy and Astrophysics decadal survey
recommended the development of a space-based direct imaging mission
that has subsequently been referred to as the Habitable Worlds
Observatory (HWO). A fundamental challenge in the preparatory work for
the HWO search for exo-Earths is the selection of suitable stellar
targets. Much of the prior efforts regarding the HWO targets has
occurred within the context of exoplanet surveys that have
characterized the stellar properties for the nearest stars. The
preliminary input catalog for HWO consists of 164 stars, of which 30
are known exoplanet hosts to 70 planets. Here, we provide a dynamical
analysis for these 30 systems, injecting a terrestrial planet mass
into the Habitable Zone (HZ) and determining the constraints on stable
orbit locations due to the influence of the known planets. For each
system, we calculate the percentage of the HZ that is dynamically
viable for the potential presence of a terrestrial planet, providing
an additional metric for inclusion of the stars within the HWO target
list. Our analysis shows that, for 11 of the systems, less than 50%
of the HZ is dynamically viable, primarily due to the presence of
giant planets whose orbits pass near or through the HZ. These results
demonstrate the impact that known system architectures can have on
direct imaging target selection and overall system habitability.









§ INTRODUCTION



Although thousands of exoplanets have now been discovered, a full
census and exploration of nearby planetary systems has yet to be
realized. One of the primary reasons for incompleteness regarding
nearby planetary architectures are the observational biases inherent
within the two detection techniques responsible for the vast majority
of current exoplanet discoveries: the radial velocity (RV) and transit
methods
<cit.>. Both
of these methods are heavily biased toward large, close-in planets,
with strong dependencies on survey design, data precision, observing
strategy, and the time baseline of observations
<cit.>. However,
RV surveys that have been operating for multiple decades have been
able to probe beyond the snowline, providing insight into the
prevalence of giant planets, even at relatively large separations from
the host star <cit.>. The
architectures of these systems are informative for their orbital
dynamics and evolution, that can also be used to constrain the
possible presence of additional planets within those systems
<cit.>.
Of particular interest is the potential for harboring planets within
the Habitable Zone (HZ) of the system
<cit.>.
Such HZ planets are expected to be the target of extensive follow-up
observations, especially those nearby systems whose host star
brightness and angular separation of their planets may enable direct
imaging as a means to characterize the planetary atmospheres
<cit.>.

Among the recommendations of the 2020 Astronomy and Astrophysics
decadal survey (hereafter Astro2020) was the prioritization of a
space-based mission with the capability to directly image terrestrial
planets within the
HZ [https://www.nationalacademies.org/our-work/decadal-survey-on-astronomy-and-astrophysics-2020-astro2020]. The
currently adopted name for this recommended mission is the Habitable
Worlds Observatory (HWO), and considerable efforts are being
undertaken to define the mission designs that would maximize the
science yield for terrestrial planet characterization
<cit.>. Work is also being carried out to both
define and characterize the target stars that are best suited for HWO
observations. An initial list of 164 HWO target stars has been
provided to the broader community for assessment and discussion
<cit.>, the stellar properties of which have been further
refined by <cit.> and examined in the context of yield
simulations by <cit.>. Of these 164 nearby stars, all of
which lie within 24 pcs, 30 are currently known to host exoplanets,
with a total inventory of 70 known exoplanets. These exoplanet
detections have primarily been enabled through long-term RV
observations, revealing the Keplerian orbits and architectures down to
the RV precision of the corresponding surveys. Though the
architectures for these planetary systems are likely incomplete, even
partial knowledge of the planets present enables a dynamical analysis
that can inform the likelihood that each system has a HZ that contains
dynamically viable orbital locations for terrestrial planets. Given
the need to determine a high quality list of HWO targets,
deprioritizing those systems that are dynamically unsuitable is a
crucial step in the target assessment process.

In this work, we present the results of a dynamical study for the
known exoplanet hosts within the current proposed HWO target list
provided by <cit.>, that specifically provides an
assessment for the dynamical stability within the HZ of those
systems. Section <ref> provides a description of the HWO
stellar sample, with a particular emphasis on the known exoplanetary
systems within that list. The description includes a tabulation of the
stellar and planetary properties for the known exoplanet systems, and
the methodology for the calculation of the system HZ boundaries. The
structure and results from the dynamical simulations are described in
Section <ref>, with a quantification of the available HZ that
is dynamically stable. We discuss the implications and limitations of
our work in Section <ref>, and provide concluding remarks
and suggestions for future work in Section <ref>.





§ THE HABITABLE WORLDS OBSERVATORY SAMPLE



Here, we provide the details of the potential HWO target list that
harbor known exoplanets, and discuss the calculation of the HZ for
these systems.





 §.§ Known Planetary Systems



From the 164 stars identified by <cit.>, we extracted the
30 stars that are currently known to host exoplanets
<cit.>. The full list of planets for all of the included
systems, including their relevant stellar and planetary properties,
are provided in Table <ref>. The selection of relevant
parameters means those that are needed for the calculation of the HZ
and/or the dynamical simulations, described in
Section <ref>. Note that the HH Peg (HD 206860) system was
not included in our analysis since the companion is a brown dwarf at
very wide separation from the host star, detected using the direct
imaging technique <cit.>. The sources for the
stellar/planetary parameters used in all cases are included in the
final column of Table <ref>, where the source selection is
based upon the most complete data source.

The stellar multiplicity of a planetary system can have a significant
dynamical effect on HZ planets
<cit.>. Of the 30 known exoplanet
hosts in our sample, 9 of the stars are known to have stellar or brown
dwarf companions. The HD 3651 system has a brown dwarf with a
projected separation of 476 AU away the primary
<cit.>. The HD 9826 system contains a red dwarf star with
a projected separation of 750 AU from the primary <cit.>.
HD 26965 is a triple system, whereby the B and C components are a
white dwarf and red dwarf star, respectively, and orbit each other at
a distance of ∼400 AU from the primary <cit.>. The
HD 75732 system contains a red dwarf star with a projected separation
of 1100 AU from the primary <cit.>. The HD 102365 includes
a red dwarf star with a projected separation of 211 AU from the
primary <cit.>. HD 115404A is in a binary orbit with a
red dwarf star that has projected separation of 289 AU
<cit.>. HD 147513 is in a binary orbit with a
white dwarf separated by ∼5300 AU <cit.>. The
HD 190360 system includes a red dwarf star that is separated by at
least 1266 AU from the primary <cit.> and is probably at a
much larger distance <cit.>. The HD 209100 system contains a
brown dwarf binary pair that is separated from the primary by 1500 AU
<cit.>. Given the wide separations
for all of these binary/multiple systems, and the considerable
uncertainties in their orbital parameters, the stellar or brown dwarf
companions were not included in our subsequent dynamical analysis.

There are several further important notes regarding the included
systems. The vast majority of the 70 planets are not known to transit
their host star, and so the planetary masses used are minimum masses
in most cases, the implications of which are discussed in
Section <ref>. Exceptions to this include HD 39091 c
<cit.>, HD 75732 e
<cit.>, HD 136352 b, c,
and d <cit.>, and HD 219134 b and c
<cit.>, all of which are known to transit their host
stars. There are also planets in our list for which orbital
inclination measurements have been made. For HD 39091, planet b has an
inclination of ∼50^∘, resulting in a mutual inclination of
∼40^∘ with respect to planet c <cit.>, and we
assume planet d is coplanar with planet b. For HD 115404A, planet c
has an inclination of ∼25^∘, and we assume planet b is
coplanar <cit.>. For HD 140901, planet c has an inclination
of ∼170^∘, and we assume planet b is coplanar
<cit.>. For HD 160691, astrometry places an upper limit on
the inclination of ∼60^∘ for the known planets
<cit.>. For HD 209100 b, <cit.> provides an
inclination of ∼64^∘, thus establishing the actual
planetary mass. Planet masses reported in the literature as Earth mass
units (M_⊕) have been converted to Jupiter mass units (M_J)
in Table <ref> for consistency. For cases where the
eccentricity is reported as zero, the argument of periastron is set to
90^∘, corresponding to the location of inferior
conjunction. Finally, note that the existence of the planet orbiting
HD 26965 <cit.> has been disputed from RV follow-up data
<cit.>, but we elected to err on the side of caution and
included the planet in our list.



The exoplanet population included in our sample is represented in
Figure <ref>, showing the distributions of semi-major axes and
eccentricities. The ranges of these values within the sample are
0.015–11.6 AU and 0.0–0.64 for the semi-major axes and
eccentricities, respectively. The data point sizes are logarithmically
proportional to the planet mass, with a range of 0.0055 M_J (tau
Ceti g) to 12.6 M_J (pi Men b). There are several features to
consider regarding the parameter distribution shown in
Figure <ref>. There is a clear trend toward higher mass
planets at larger separations. For example, the mean planet mass for
semi-major axes less than 1 AU is 0.19 M_J, whereas the mean planet
mass beyond 1 AU is 1.19 M_J. A significant factor in this planetary
mass trend is the observational bias inherent in the RV method,
whereby the RV amplitude decreases with increasing semi-major axis
<cit.>. A similar bias impacts the observed
eccentricity distribution, since such planets produce higher RV
amplitudes, albeit during a limited period of the orbital phase
<cit.>. Moreover, for
those long-period planets with insufficient phase coverage, the
relatively slow apastron passage may result in eccentricity
misclassifications
<cit.>. Even so, the
combination of relatively high masses and eccentricities produces
numerous planetary architectures that exhibit significant dynamical
consequences within the HZ of their systems.

lrrrlrrrrl
  10
  0pc
   Stellar and planetary parameters
    for HWO known systems.
  
    Star    
    NP^†    
    M_⋆    
    T_eff    
    Planet    
    M_p^    
    a    
    e    
    ω    
    Source 

        
        
    (M_⊙)    
    (K)    
        
    (M_J)    
    (AU)    
        
    (^∘)    
    
  
  HD 3651 (54 Piscium)     1     0.799     5221     b     0.228     0.295     0.645     243     <cit.>

HD 9826 (ups And)     3     1.29     6156     b     0.675     0.05914     0.0069     0     <cit.>

                    c     1.965     0.8265     0.266     58.2     

                    d     4.1     2.517     0.294     73.8     

HD 10647 (q1 Eridani)     1     1.11     6218     b     0.94     2.015     0.15     212     <cit.>

HD 10700 (tau Ceti)     4     0.78     5333     g     0.0055     0.133     0.06     395.34     <cit.>

                    h     0.0058     0.243     0.23     7.45     

                    e     0.0124     0.538     0.18     22.35     

                    f     0.0124     1.334     0.16     119.75     

HD 17051 (iota Hor)     1     1.34     6167     b     2.27     0.92     0.14     309     <cit.>

HD 20794 (82 Eridani)     4     0.813     5401     b     0.0089     0.127     0.27     386.17     <cit.>

                    c     0.0079     0.225     0.17     31.51     

                    d     0.0111     0.364     0.25     252.67     

                    e     0.015     0.509     0.29     268.14     

HD 22049 (eps Eri)     1     0.81     5020     b     0.651     3.5     0.044     350     <cit.>

HD 26965 (40 Eri)     1     0.78     5072     b     0.0267     0.219     0.04     148.969     <cit.>

HD 33564 (kappa Cam)     1     1.25     6250     b     9.1     1.1     0.34     205     <cit.>

HD 39091 (pi Men)     3     1.07     5998     c     0.0114     0.06805     0     90     <cit.>

                    d     0.0421     0.499     0.22     323     

                    b     12.6     3.2826     0.6396     331.03     

HD 69830     3     0.86     5385     b     0.0321     0.0785     0.1     340     <cit.>

                    c     0.0371     0.186     0.13     221     

                    d     0.057     0.63     0.07     224     

HD 75732 (55 Cancri)     5     0.905     5172     e     0.0251     0.01544     0.05     86     <cit.>

                    b     0.8036     0.1134     0     90     

                    c     0.1611     0.2373     0.03     2.4     

                    f     0.1503     0.7708     0.08     262.4     

                    d     3.12     5.957     0.13     290.9     

HD 95128 (47 UMa)     3     1.06     5872     b     2.53     2.1     0.032     334     <cit.>

                    c     0.54     3.6     0.098     295     

                    d     1.64     11.6     0.16     110     

HD 95735     2     0.3899     3712     b     0.0085     0.07879     0.063     330     <cit.>

                    c     0.0428     2.94     0.132     63     

HD 102365     1     0.85     5630     b     0.0503     0.46     0.34     105     <cit.>

HD 114613     1     1.27     5641     b     0.357     5.34     0.458     196     <cit.>

HD 115404A     2     0.83     5019     b     0.097     0.088     0.232     259     <cit.>

                    c     10.319     11.364     0.211     312     

HD 115617 (61 Vir)     3     0.942     5577     b     0.016     0.050201     0.12     105     <cit.>

                    c     0.057     0.2175     0.14     341     

                    d     0.072     0.476     0.35     314     

HD 136352 (nu2 Lupi)     3     0.87     5664     b     0.0149     0.0964     0     90     <cit.>

                    c     0.0354     0.1721     0     90     

                    d     0.0278     0.425     0     90     

HD 141004 (lambda Ser)     1     1.05     5885     b     0.0428     0.1238     0.16     19.48     <cit.>

HD 140901     2     0.99     5586     b     0.0503     0.085     0.472     108     <cit.>

                    c     6.284     7.421     0.607     298     

HD 143761 (rho CrB)     4     0.95     5817     e     0.0119     0.1061     0.126     359.4     <cit.>

                    b     1.093     0.2245     0.038     269.64     

                    c     0.0887     0.4206     0.096     9.7     

                    d     0.068     0.827     0     90     

HD 147513 (62 G. Sco)     1     1.11     5883     b     1.21     1.32     0.26     282     <cit.>

HD 160691 (mu Ara)     4     1.13     5773     d     0.033     0.0923     0.16     197     <cit.>

                    e     0.439     0.9296     0.091     193     

                    b     1.665     1.5224     0.036     39     

                    c     1.873     5.0937     0.022     84     

HD 189567     2     0.83     5726     b     0.0267     0.111     0     90     <cit.>

                    c     0.022     0.197     0.16     219     

HD 190360     2     0.99     5537     c     0.0677     0.1294     0.165     322     <cit.>

                    b     1.492     3.955     0.3274     13.9     

HD 192310     2     0.8     5166     b     0.0532     0.32     0.13     173     <cit.>

                    c     0.076     1.18     0.32     110     

HD 209100 (eps Indi A)     1     0.754     4611     b     3.25     11.55     0.26     77.83     <cit.>

HD 217987     2     0.489     3688     b     0.0132     0.068     0     90     <cit.>

                    c     0.0239     0.12     0     90     

HD 219134     6     0.794     4913     b     0.012     0.038474     0     90     <cit.>

                    c     0.011     0.064816     0     90     

                    f     0.028     0.14574     0     90     

                    d     0.067     0.23508     0     90     

                    g     0.0346     0.3753     0     90     

                    h     0.34     3.11     0.06     215     

  †Number of planets.
  Minimum mass, except for 55 Cancri e and the HD 136352 planets.






 §.§ System Habitable Zones







The extent of the HZ for each system was calculated using the stellar
parameters from the sources provided in
Table <ref>. Specifically, we adopt the methodology
provided by <cit.>, in which the
boundaries of the HZ are derived from 1D Earth-based climate models
that calculate the radiative balance for which surface liquid water is
retained. This approach divides the HZ into two primary regions: the
conservative HZ (CHZ) and the optimistic HZ (OHZ). The CHZ is bounded
by the runaway greenhouse criteria at the inner edge, and the maximum
CO_2 greenhouse criteria at the outer edge <cit.>. The
OHZ is an empirical extension to the CHZ based upon assumptions
regarding retention of surface liquid water in the Venusian and
Martian evolutionary histories
<cit.>. The
CHZ and OHZ boundaries for each system are provided in
Table <ref>.

Figure <ref> and Figure <ref> show top-down views for
8 of the planetary systems, that demonstrate the diversity of
architectures in our known exoplanet host sample. The extent of the
CHZ and OHZ are shown in light green and dark green, respectively. In
most of these cases, the planetary orbit that is present within the HZ
is a giant planet with significant perturbation potential. The cases
where eccentric orbits lie within and/or cross the HZ, such as
HD 39091 (pi Men), HD 114613, HD 147513, and HD 192310, are
particularly devastating to orbital stability in that region due to
angular momentum transfer that excites other planetary orbits into
high eccentricity regimes <cit.>. The others cases shown
exhibit planetary orbits that are more circular, and can often allow
regions of orbital stability at the edges of the HZ. These scenarios
may be tested with dynamical simulations that explore particle
injection throughout the HZ region.





§ ORBITAL DYNAMICS







 §.§ Simulation Design



We conducted an exhaustive series of dynamical simulations to test the
viability of terrestrial planets orbiting within the HZ of the target
systems. The simulations were conducted using the REBOUND N-body
integrator package <cit.> that applies the symplectic
integrator WHFast <cit.>. Our simulations explored the
effect and orbital evolution of an additional Earth-mass planet in a
circular orbit that is coplanar with the other planets within each
system, where 10 equally spaced mean anomalies were chosen for the
injected planet. For the case of HD 39091 (pi Men), where there is a
substantial mutual inclination between the planets, we injected the
additional planet in an orbit coplanar with planet b, since that is
the planet that dominates the gravitational influence on the HZ for
that system (see Section <ref> and Figure <ref>). The
semi-major axes explored spanned 20% less than the inner edge of the
OHZ to 20% larger than the outer edge of the OHZ (see
Section <ref>). This region within each system was divided into
1000 equally spaced semi-major axis starting locations for the
injected planet to ensure sufficiently dynamical sampling of the
HZ. Each of these semi-major axis and mean anomaly starting locations
were integrated for 10^7 years, resulting in 10,000 simulations
performed for each system. The time step for each integration was
selected based upon the orbital period for the innermost body within
the system divided by 20, where the innermost body is either a known
planet or the injected planet.

At each of the simulation starting locations, our simulations provided
the percentage of the total simulation duration for which the planet
was able to survive, where non-survival means that the injected planet
was captured by the gravitational well of the host star or ejected
from the system. From these results, we calculated the dynamically
viable HZ (DVHZ), which is the percentage of the OHZ locations for
which the injected planet survived the full 10^7 year simulation.





 §.§ Habitable Zone Stability Results



The methodology described in Section <ref> was applied to each of
the systems shown in Table <ref>. The time required to
complete the suite of simulations for each system varied enormously,
depending upon the number of planets in the system, their locations,
and the extent of the instability induced within the HZ. For example,
the 55 Cancri system (bottom-left system depicted in
Figure <ref>) contains 5 known planets, including one with a
0.74 day orbit, resulting in the need for high time resolution within
the dynamical simulation. By contrast, systems with a single,
eccentric giant planet with high scattering potential (such as the
HD 147513 system) concluded the simulation suite relatively quickly.





The simulation results for 8 of the systems are represented in
Figure <ref> and Figure <ref>, corresponding to
the same systems shown in Figure <ref> and
Figure <ref>, respectively. The plots provide the percentage
of the full 10^7 year simulation for which the injected terrestrial
planet was able to maintain a stable orbit for each semi-major axis
location. As for Figure <ref> and Figure <ref>, the
CHZ and OHZ are shown as light green and dark green shaded regions,
respectively. In all of the 8 cases shown, there are significant areas
of instability within the HZ resulting from the known planets in the
system. Most notable is the case of pi Men (second panel;
Figure <ref>) in which the entire region explored is
unstable due to the incursion of the highly eccentric b planet into
the HZ. The HD 147513 system (second panel; Figure <ref>) is
similarly affected, although stable regions exist just outside of the
HZ. The remaining examples shown have varying amounts of stable
locations within the HZ, including regions of stability/instability
resulting from a combination of mean motion resonance (MMR) effects
and the numerical randomness intrinsic to the simulations
<cit.>. For example, the
results for HD 10647 (first panel; Figure <ref>) exhibits
numerous stability islands induced by the single giant planet in the
system. The 3:2 MMR location at 1.53 AU creates an island of stability
whereby a planet located there has the potential to avoid
gravitational perturbing interactions with the known giant planet,
similar to the 2:3 MMR of Neptune with Pluto <cit.>. Of
particular note is the 5:2 MMR location at 1.09 AU that produces a
region of instability that coincides with the inner boundaries of the
OHZ and CHZ. Multi-planet systems can exhibit significantly
complicated stability patterns due to the multitude of MMR locations
cascading throughout the system, as shown in the case of HD 75732 (55
Cancri; third panel; Figure <ref>).

lrrrrr
  6
  0pc
   HZ boundaries and DVHZ results.
  
    Star    
    OHZ_in    
    CHZ_in    
    CHZ_out    
    OHZ_out    
    DVHZ 

        
    (AU)    
    (AU)    
    (AU)    
    (AU)    
    (%)
  
  HD 3651        0.56     0.71     1.27     1.34     92.41 

HD 9826        1.35     1.71     3.00     3.16     0.00 

HD 10647       0.87     1.10     1.92     2.02     38.65 

HD 10700       0.55     0.69     1.24     1.31     20.75 

HD 17051       0.94     1.20     2.09     2.20     55.02 

HD 20794       0.72     0.91     1.63     1.72     0.00 

HD 22049       0.45     0.57     1.03     1.09     100.00 

HD 26965       0.52     0.66     1.20     1.26     100.00 

HD 33564       1.58     2.00     3.49     3.68     61.50 

HD 39091       0.95     1.20     2.11     2.23     0.00 

HD 69830       0.60     0.75     1.35     1.42     84.48 

HD 75732       0.59     0.74     1.33     1.41     56.99 

HD 95128       0.94     1.19     2.09     2.20     27.59 

HD 95735       0.13     0.16     0.30     0.32     100.00 

HD 102365      0.57     0.72     1.28     1.35     78.21 

HD 114613      1.54     1.96     3.46     3.65     33.43 

HD 115404A     0.43     0.54     0.99     1.04     100.00 

HD 115617      0.68     0.86     1.53     1.61     44.93 

HD 136352      0.77     0.97     1.72     1.82     100.00 

HD 140901      0.69     0.87     1.54     1.62     89.06 

HD 141004      1.08     1.36     2.40     2.53     100.00 

HD 143761      1.01     1.28     2.25     2.38     88.17 

HD 147513      0.74     0.93     1.64     1.73     0.00 

HD 160691      1.00     1.26     2.23     2.35     5.39 

HD 189567      1.09     1.38     2.45     2.58     100.00 

HD 190360      0.82     1.04     1.84     1.94     85.67 

HD 192310      0.48     0.61     1.10     1.16     19.14 

HD 209100      0.38     0.48     0.87     0.92     100.00 

HD 217987      0.16     0.20     0.38     0.40     100.00 

HD 219134      0.41     0.52     0.95     1.00     99.26 

  

To quantify the dynamical simulation results on a system-by-system
basis, we calculate and provide the DVHZ in each case, as described in
Section <ref>. These calculations are shown in
Table <ref>, along with the inner and outer boundaries for
the CHZ and OHZ, described in Section <ref>. For 11 of the systems,
DVHZ is less than 50%, and there are 4 cases where the DVHZ is 0%,
including the previously discussed systems of HD 39091 (pi Men) and
HD 147513. In general, the architectures for the 11 cases where DVHZ
is less than 50% are dominated by a giant planet in an eccentric
orbit that passes near or through the HZ.

On the other hand, there are 9 cases for which the DVHZ is
100%. Cross-referencing these systems with Table <ref>
reveals that their architectures are dominated by low-mass,
short-period planets with limited gravitational influence within the
HZ. Shown in Figure <ref> is a histogram of the DVHZ values
that summarizes the findings of our dynamical simulations. This shows
that half of the studied systems have DVHZ values that are greater
than 80%, ensuring that there is substantial dynamically viable space
within the HZ of those systems based on the currently known
architectures, and therefore suitable for potential direct imaging
follow-up observations.







§ DISCUSSION



Given the number of known exoplanetary systems, including those
resulting from numerous decades of RV observations around bright
stars, it is little surprise that the HWO target list contains known
exoplanet systems. The description of these 30 systems in
Section <ref> shows that the architectures of these systems is
diverse, including low-mass multi-planet systems and high-mass
single-planets in eccentric orbits. The latter category includes giant
planets beyond the snow line, the frequency of which appears to be
relatively rare
<cit.>,
and can greatly influence the volatile delivery within the system
<cit.>. The high
eccentricity cases may be the result of planet-planet scattering
events <cit.> and can dynamically eliminate the
possibility of further planets being present in the system
<cit.>. The growing understanding of such dynamically
volatile systems, including their formation and prevalence, will
greatly inform our ability to infer the possible presence of HZ
exoplanets <cit.>.

As described in Section <ref>, the integration time for each of
our simulations was 10^7 years. The integration time was chosen to
minimize the effects of orbital element error propagation
<cit.>, and to create a feasible pathway to
explore the full parameter space for all 30 targets, each of which
required thousands of simulations. However, chaotic systems are often
the result of divergent orbital eccentricities that can require
significant time to develop. It is thus possible that, even for
locations identified as stable in our simulations, instability occurs
beyond the 10^7 year integration time
<cit.>. Although our criteria for stability is a
first-order approach that necessitates all planets survive the
duration of the dynamical simulation, this criteria has successfully
been validated against chaos indicators <cit.> and applied
to numerous exoplanetary systems
<cit.>.

As noted in Section <ref>, the preponderance of planets in our
sample are not known to transit their host stars, whereby the true
mass of the planet may be realized. Thus, the planetary masses used
are generally minimum masses. A consequence of this knowledge
limitation is that the dynamical influence of the planets on each
other within a system, and their effects on the injected planet, may
be larger than that calculated and presented here. For that reason,
the results presented in Section <ref> can be considered a
lower limit on the HZ orbital instability. Furthermore, the
observational bias in the RV and transit methods mentioned in
Section <ref> means that the architectures of the known systems
are likely incomplete, with additional planets yet to be discovered in
these systems. However, it is worth repeating here that we assumed
  coplanar orbits for the injected planets in our simulations
  (coplanar with planet b in the case of pi Men). A full exploration
  of terrestrial HZ planets whose orbits are mutually inclined from
  the known planets may yet reveal stable locations that are otherwise
  unviable.





§ CONCLUSIONS



The effective transition of the exoplanet community to prioritizing
direct imaging techniques requires that the data for the expected
stellar targets be sufficiently leveraged toward a robust target
list. Of the 164 targets provided by <cit.> for direct
imaging with HWO, only ∼20% are presently known to harbor
exoplanets. Most, if not all, of the remaining stars likely also have
planetary companions, but in most cases the current database of RV
observations may be sufficient to rule out the presence of giant
planets in or near the HZ. For those that are known exoplanet hosts,
our results show that 11 of the systems have less than 50% of the HZ
that is dynamically viable (DVHZ < 50%), including 4 cases where
there is no dynamically possible orbit (ups And, eps Eri, pi Men, and
62 G. Sco). These systems should therefore be treated with extreme
caution when considered as possible HWO targets. However, over half of
the tested systems have DVHZ values greater than 80%, making them
suitable HWO targets from the perspective of orbital
dynamics. Moreover, our overall results are not necessarily
representative of what may be expected for the broader list of HWO
targets where no exoplanets have yet been detected. Giant planets are
relatively rare among the exoplanet population, and the detection
methods utilized are biased toward detecting those kinds of planets
more quickly than lower-mass planetary counterparts. This means that
current exoplanet surveys may have already detected many of the nearby
systems that contain significant perturbing agents (giant planets) for
possible terrestrial HZ planets.

With respect to exoplanet architectures, the refinement of the HWO
target sample will greatly benefit from the continued search for
exoplanets. The era of extreme precision RVs <cit.>
provides opportunities to detect previously undiscovered exoplanets
around the nearby stars. These exoplanet detections will include
low-mass terrestrial exoplanets and high-mass planets with
low-inclinations relative to the plane of the sky. Additional planets
are also likely to be found in the known systems, further constraining
the stability profiles for the HZ region, as described in this
work. Results from the Gaia mission <cit.> will aid in the
detection of long-period giant planets and allow the true mass of
known RV giant planets to be determined. All of these precursor
observational efforts will continue to form an essential component for
the success of the Astro2020 recommendations, and maximize the yield
of HWO and the characterization of HZ terrestrial planets.





§ ACKNOWLEDGEMENTS


We acknowledge support from the NASA Astrophysics Decadal Survey
Precursor Science (ADSPS) program under grant
No. 80NSSC23K1476. C.K.H. acknowledges support from the National
Science Foundation (NSF) Graduate Research Fellowship Program (GRFP)
under Grant No. DGE 2146752. This research has made use of the NASA
Exoplanet Archive, which is operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program. This research
has also made use of the Habitable Zone Gallery at hzgallery.org. The
results reported herein benefited from collaborations and/or
information exchange within NASA's Nexus for Exoplanet System Science
(NExSS) research coordination network sponsored by NASA's Science
Mission Directorate.



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 Paper Title: SQUIDs for detection of potential dark matter candidates.


    
    
    

SQUIDs for detection of potential dark matter candidates
    Siddarth Sivakumar^1, Manan Agarwal^2, Hannah Rana^1

    ^1Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA, USA

   ^2Anton Pannekoek Institute/GRAPPA, University of Amsterdam, Amsterdam, The Netherlands
    
    
====================================================================================================================================================================================================================================

§ ABSTRACT 
Superconducting QUantum Interference Devices (SQUIDs) are extremely sensitive magnetic flux sensors which render them useful in a wide array of instrumentation. SQUIDs are often paired with other detectors as a readout mechanism to obtain quantitative insight. SQUIDs have impacted many fields but much less addressed is its impact on the field of fundamental physics, particularly in the search for dark matter. Dark matter is believed to make up around 27% of all mass-energy content of the universe and will provide critical insight into understanding large-scale structures of the universe. Axions and WIMPs are the prominent two dark matter candidates whose search has been fuelled by the usage of SQUID readouts. 



§ INTRODUCTION

Superconducting QUantum Interference Devices (SQUIDs), inherently flux-to-voltage transducers, can be used to measure any physical quantity convertible into magnetic flux, used in an array of instrumentation such as signal amplifiers, magnetometers and gradiometers. SQUIDs are considered vital in instrumentation due to their quantum sensitivity (field resolution of 10 -17 T <cit.>) and enormous operational bandwidth. SQUIDs employ Josephson junctions in a superconducting loop; two in direct current (DC) SQUIDs and one in radio frequency (RF) SQUIDs. Due to their dependence on superconductors, they require cryogenic temperatures around 9 K. Although, recent advancements have brought the advent of high-T_C SQUIDs which have a relatively high operational temperature of 77 K. SQUID-based applications <cit.> extend to various fields; recently even concepts of military applications have been conceived <cit.>. SQUIDs are used as magnetometers in biology instrumentation: magnetoencephalography, magnetocardiography and gastro-magnetometry. Geophysical applications include the liquid-nitrogen-cooled SQUIDs that provide insight on low conductivity anomalies at depths from a few metres to tens of kilometres, allowing us to predict the presence and depth of oil and mineral ores. High-TC SQUIDs have piqued the interest in Non-Destructive Evaluation (NDE) of materials used to locate flaws that may compromise the integrity of structures later on; it can be used to test aircraft structures, steel-reinforced concrete, and more. Our focus in this review, however, will be their extended usage in particle detection for the detection of dark-matter candidates <cit.>, an emerging approach in the search for dark matter in fundamental physics. SQUIDs are used as readout amplifiers in the detection of axions and Weakly Interacting Massive Particles (WIMPs), potential dark-matter candidates, as they offer better sensitivity than their traditional counterparts.




§ SEARCH FOR DARK MATTER

Dark matter has been theorized to help us explain the formation and evolution of galaxies and large-scale structures in the universe; the existence of dark matter is inferred from various astrophysical observations, such as the rotation curves of galaxies, gravitational lensing, and the cosmic microwave background <cit.>. It is believed to constitute about 27% of the universe's mass-energy content. Despite no interaction with light and being detectable only through its gravitational effects, our search for dark matter continues through multiple experiments pushing the boundaries. Researchers have proposed several hypothetical candidates for dark matter particles, with WIMPs and axions being two prominent possibilities.


    
  * WIMPs interact only through the weak nuclear force and gravity, making them difficult to detect. WIMPs are predicted by extensions of the Standard Model of particle physics, such as supersymmetry <cit.>. Their mass is expected to be in the range of 10 to 1000 times that of a proton (10 to 1000 GeV/c^2), and interact with normal matter at a rate much lower than that of photons or other particles.
    
  * Axions, proposed as a solution to the strong CP problem in quantum chromodynamics, interact extremely weakly with normal matter and could contribute to the dark matter density in the universe. They are expected to have masses much lower than WIMPs, potentially in the microelectronvolt (μ eV) range <cit.>.




§ ADMX - AXION DETECTION



 §.§ Background

The Axion Dark Matter eXperiment (ADMX) at Lawrence Livermore National Laboratory has been searching for axions since 1996 <cit.>. The primary challenge in axion detection is their extremely weak interaction with ordinary matter, requiring highly sensitive detection methods. Traditional detection systems, such as those using the GaAs field-effect transistor (HFET) amplifiers, have inherent noise limitations. The novel approach of using SQUID amplifiers, with their near quantum-limited noise performance, significantly enhances the sensitivity and scan rate of the search. 



 §.§ Experimental Setup



The experimental setup includes:


    
  * A 7.6 Tesla superconducting solenoid with a 0.5-meter diameter bore.
    
  * A cylindrical, copper-plated microwave cavity inside the magnet bore.
    
  * A DC SQUID amplifier placed above the magnet to minimize interference from the magnetic field.

Dark matter axions passing through the cavity are expected to convert into microwave photons, detectable by the resonant cavity. The experiment scanned for axion signals by tuning the cavity frequency and integrating the signal over time to improve the signal-to-noise ratio. The cavity's frequency is tunable by moving copper-plated axial rods, allowing the experiment to scan different axion masses. The expected power from axion-photon conversion, given by the inverse Primakoff effect, is extremely low, requiring ultra-sensitive amplification. The power generated by axion-photon conversion depends on several factors, including the magnetic field strength, cavity volume, axion density, and the quality factors of the cavity and axion signal.



 §.§ Role of SQUIDs




In the ADMX experiment, SQUID amplifiers replace conventional HFET amplifiers, due to several key advantages:


    
  * Noise Temperature: The SQUID amplifier operates with a noise temperature close to the quantum limit, significantly lower than that of HFET amplifiers, allowing for better sensitivity and faster scan rates. The SQUID amplifier has achieved noise temperatures as low as 47 mK, close to the quantum noise limit of 33 mK at 700 MHz.
    
  * Thermal Stability: SQUIDs maintain their low noise temperature down to very low physical temperatures, unlike HFET amplifiers which plateau at a few Kelvin.
    
  * Gain Performance: The specific design of the SQUIDs used in ADMX includes a resonant microstrip input coil, which mitigates the gain roll-off issue at microwave frequencies.

 


 §.§ Results and Analysis


    
  * Noise Reduction: In the ADMX experiment, the SQUID amplifiers demonstrated a noise temperature of approximately 50 mK at 1 GHz, which is close to the quantum limit of 33 mK. This low noise temperature is crucial for enhancing the signal-to-noise ratio (SNR) and, consequently, the sensitivity of the axion search. 
    
  * Sensitivity: The experiment's sensitivity is quantified by the Dicke radiometer equation, which links SNR to system noise temperature, integration time, and bandwidth. With the introduction of SQUID amplifiers, the system noise temperature is minimized, leading to a higher SNR and allowing for a more efficient scan of axion masses.
    
  * Improvement: The use of the SQUID amplifier significantly improved the scan rate compared to previous runs using HFET amplifiers.
    
  * Findings: The ADMX experiment was able to rule out certain models of dark matter axions with masses between 3.3 and 3.53 μeV, with 90% confidence, assuming a local dark matter density of 0.45 GeV/cm^3. This new result extends previous research that had only ruled out axions up to 3.3 μeV. While the hunt for dark matter candidates and axions in particular will continue, current findings suggest that they are less likely to have masses in the 3.3 to 3.53 μeV range. 




 §.§ Future Directions and Conclusion

Future experiments will use dilution refrigeration to further lower the temperature of the SQUID and cavity, potentially increasing the scan rate and sensitivity. The goal for the future is to cover the entire plausible axion mass range with higher sensitivity, even for the most pessimistic axion-photon couplings. The application of SQUIDs in the ADMX experiment showcases its potential in particle detection instrumentation. The improvements in sensitivity and scan rate afforded by SQUID amplifiers represent a critical step forward in the ongoing quest to detect dark matter axions. 



§ CRESST - WIMP DETECTION




 §.§ Background

The Cryogenic Rare Event Search with Superconducting Thermometers (CRESST) experiment <cit.> uses cryogenic detectors that leverage SQUIDs, which play a huge role in enhancing the sensitivity and precision of WIMP detection. The experiment is set up in the Laboratori Nazionali del Gran Sasso (LNGS) underground laboratory in Italy, which provides a shielded environment to minimize background noise from cosmic rays and other sources. 



 §.§ Experimental Setup

The core idea of the 2004 data run of the CRESST-II phase of the experiment is to identify WIMP interactions by detecting the tiny amounts of energy deposited by the phonon (heat) signal and scintillation light, both of which need to be detected simultaneously and with high precision by sensitive tungsten phase-transition thermometers, when a WIMP collides with a nucleus in the scintillating CaWO4 crystals used as the detector material. The SQUID readout system subsequently measures resistance changes that allow us to infer changes in temperature. The setup includes:


    
  * Detectors: Two 300 g CaWO4 scintillating crystals were used as targets in their respective modules with reflecting housings, with associated cryogenic light detectors that use the phonon-light technique, which measures both phonon and light signals to identify nuclear recoils indicative of WIMP interactions. Using both channels help distinguish nuclear recoils from background events.
    



    
  * Phonon Channel: The phonon signal is the energy transfer to a nucleus from a WIMP-nucleus elastic scattering event, which is measured using a cryogenic calorimeter read out by a SQUID. The amount of energy deposited by a WIMP is expected to be very low and SQUID's sensitivity allows it to detect the minute temperature changes resulting from the phonon signal with high accuracy and precision.
    
  * Light Channel: Alongside the phonon channel, a cryogenic light detector measures the scintillation light. This dual measurement system—phonon and light—discriminates between nuclear recoils (potential WIMP signals) and electron recoils (background noise) and enables the identification and suppression of background events, improving the signal-to-noise ratio crucial for WIMP detection. 




    
  * Readout: The system uses SQUID readout electronics to measure resistance changes in the tungsten thermometers attached to the detectors.
    
  * Stability: Temperature regulation and periodic calibration pulses ensure stability and accurate energy measurements throughout the experiment.



 §.§ Role of SQUIDs

    
  * Sensitivity: SQUIDs provide unmatched sensitivity in detecting the minute signals produced by WIMP interactions.
    
  * Noise Reduction: The high precision of SQUIDs helps in significantly reducing background noise, which is crucial for identifying rare events.
    
  * Dual-channel Detection: By enabling simultaneous detection of phonons and scintillation light, SQUIDs enhance the ability to discriminate between different types of interactions, improving overall detection accuracy. 

    





 §.§ Results and Analysis




    
  * Data Collection: Data was collected from January 31 to March 23, 2004, with a net exposure of 20.5 kg days.
    
  * Nuclear Recoil Events: A total of 16 events in the nuclear recoil acceptance region (12-40 keV) were observed, corresponding to a rate of (0.87 ± 0.22) events/(kg day), which is consistent with expected neutron background. This demonstrates the effectiveness of the SQUID-based detection system in distinguishing between neutron-induced recoils and potential WIMP signals. 
    
  * Background Suppression: The phonon-light technique provides strong suppression of non-nuclear recoil backgrounds, enhancing the experiment's sensitivity to WIMPs. The combination of phonon and light signals from the scintillating cryogenic detector allowed for a strong suppression of non-nuclear recoil backgrounds, enabling accurate identification of potential WIMP interactions.
    
  * Quenching Factor: It is the measure of the reduction in light yield for nuclear recoils compared to electron/gamma events. For CaWO4, the quenching factor was found to be 7.4 for neutrons and 40 for tungsten recoils.
    
  * Findings: The experiment set significant limits on the WIMP-nucleon cross-section. The experiment set exclusion limits on the same, with the best module showing no events in the region corresponding to tungsten recoils. Hence, no events could be conclusively attributed to WIMPs and the search for dark matter candidates continues.




 §.§ Future Directions and Conclusion

Challenges remain, such as the need for extremely low temperatures and the complexity of the SQUID system. Future upgrades will include additional shielding and more detector modules to further reduce background and improve sensitivity, with a focus on enhancing the resolution of the light detectors and studying quenching factors to better understand WIMP interactions. The application of SQUIDs in the CRESST-II phase of the experiment showcases its potential in particle detection instrumentation. The improvements in sensitivity and background suppression due to SQUIDs represent a critical step forward in the ongoing quest to detect WIMPs.














§ CONCLUSION

The integration of SQUIDs has significantly advanced dark matter research by enhancing the detection of weakly interacting massive particles (WIMPs) and axions. In the ADMX experiment, SQUID amplifiers achieved near quantum-limited noise temperatures, improving sensitivity and scan rates, and ruling out axion models with masses between 3.3 and 3.53 μeV. In the CRESST experiment, SQUID readouts enabled precise dual-channel detection of phonon and scintillation light signals, improving background suppression and setting exclusion limits on the WIMP-nucleon cross-section. These advancements underscore SQUIDs' critical role in reducing noise and increasing detection accuracy, showing great promise for facilitating future discoveries in the search for dark matter and bringing us closer to understanding the universe's unseen mass.


§ ACKNOWLEDGEMENTS

Special thanks goes to our supervisors H. Rana and M. Agarwal for their incredible mentorship and moral support, and to the CCIR team for giving me an opportunity to connect with incredible people and provide extensive support.

 unsrt   
 



 Paper Title: Determination of electron screening potential of 6 Li(p,α)3 He reaction using MultiLayer Perceptron based neural network.














    [ dipayanchattopadhyay90@gmail.com^1Department of Physics, The ICFAI University Tripura, Agartala 799210, India§ ABSTRACT 
          Background Understanding the nuclear reactions between light charged nuclei at sub-coulomb energy region holds significant importance in several astrophysical processes. Determination of the precise reaction cross-section within the astrophysically important Gamow range is difficult because of electron screening. Various polynomial fits, R-Matrix and Indirect Trojan horse method estimate much higher electron screening energies as compared to the adiabatic limit.
        
          Purpose Obtain the bare astrophysical S-factor of ^6Li(p,α)^3He using Multi-Layer Perceptron based Artificial Neural Network based analysis and extract the electron screening energies. 
        
          Methods Experimental S-factor of ^6Li(p,α)^3He, available in literature, are reanalyzed using the Multi-Layer Perceptron based Artificial Neural Network based algorithm to obtain the energy dependent astrophysical S-factor. Bare astrophysical S-factor is also calculated using the same Feed-forward Artificial Neural Network from the data range above 60 keV where the electron screening effect is expected to be negligible. Electron screening potential is then obtained by taking the ratio of total shielded S-factor with the bare S-factor.    
        
          Results and Conclusions The electron screening potential obtained from the Present work through the Artificial Neural network based algorithm is found to be 220 eV. The extracted electron screening potential through the present analysis indicates that the Artificial Neural Network might be an alternative tools for estimation the electron screening potential involving light nuclei.      
        25.70.Gh, 25.70.Jj, 25.60.Gc
                                      
        Determination of electron screening potential of ^6Li(p,α)^3He reaction using Multi Layer Perceptron based neural network
             D. Chattopadhyay^1
            August 5, 2024
        =========================================================================================================================
        
        
        
        § INTRODUCTION
        
        Understanding the nuclear reactions mechanism between light charged nuclei at ultra low energies relevant to astrophysical energies is emerging as a burning topic in recent times. Measurement of precise cross-section is essential to understand the stellar evolution as well as abundances of various nuclides observed in the universes <cit.>.  Low energy nuclear reaction process involve transfer and capture processes, with the former primarily influenced by the nuclear force and the latter predominantly driven by the electromagnetic interaction. Determination of reaction cross-section at ultra low energies (much below the Coulomb barrier) of astrophysical interest requires considerable effort. Direct determination of cross-section in many cases is found to be very difficult or even impossible owing to small cross-section because of Coulomb repulsion of the interacting particles <cit.>.  Extrapolation of direct cross-sections down to stellar energies requires extra theoretical effort. Different Indirect method has been evolved in recent times to overcome this problems <cit.>. 
        
        In this context, the low-energy cross sections governing reactions that produce or deplete lithium isotopes are pivotal. These cross sections are indispensable for addressing unresolved astrophysical questions, including the intricacies of Big Bang nucleosynthesis and the observed lithium depletion in celestial bodies like the Sun and other stars within our galaxy <cit.>. The precise estimation of bare astrophysical cross-sections from experimental measurement is essential as they serve as key inputs in the stellar evolutionary codes. However, as the Gamow energy region is far below the Coulomb barrier, the cross-sections are very small and also very difficult to predict due to presence of electrons inside the target. Electrons inside the target nucleus effectively screen the coulomb potential at ultra low energies and thus reducing the height of the coulomb barrier. As a result the cross-section is enhanced by a factor exp(U_eπηE) <cit.> and is given by 
        
            σ(E)=σ_b(E)exp(U_eπηE)
        
        
        where, U_e is the electron screening potential U_e(=Z_1Z_2e^2R_a where, R_a is atomic radius) arising from the interaction of the charged nuclei of the projectile with the electrons inside the target. In terms of astrophysical S-factor, the above expression can be written as  
        
        
            S(E)=S_b(E)exp(U_eπηE)
        
        
        Similarly, in stellar plasma, ionized atoms are also surrounded by a sea of electrons within the Debye-Huckle radius and there also screening effect is important which depends on the temperature and density of the plasma. This plasma screening factor is different from the lab screening factor. Hence, it is essential to determine bare cross-section as accurate as possible by eliminating the electron screening effect from laboratory experiment. From the bare cross-section, the actual cross-section in stellar plasma can be determined through
        
            σ_pl(E)=σ_b(E)f_pl
        
        
        where, f_pl(E) is the plasma screening enhancement factor and is related with the plasma screening factor by 
        
            f_pl(E)=exp(U_plπηE)
        
        
        U_pl(E) is the plasma screening potential and can be calculated within the framework of Debye-Huckel theory.  
        
        Several attempts by different authors have been taken to understand the electron screening factor. Engstler  et al. <cit.> have extracted screening potential for both the reactions ^6Li(p,α)^3He  using polynomial fit S(E)=a+bE+cE^2+dE^3. The sensitivity of the ratio of E to U_e has also been addressed by them. Extracted screening potential for both the reactions are found to be higher than the adiabatic limit (U_e=175 eV) <cit.>. Later Cruz et al. <cit.> have used lithium implanted targets and attempted to extract the screening potential for both ^6Li(p,α)^3He reaction. They have extracted potential as U_e=273±111 eV  for ^6Li(p,α)^3He. The values are lower than those obtained by Engstler  et al. <cit.> but still higher than the adiabatic limit. The uncertainty in the extracted electron screening potential is too large to make any feasible comment. In a systematic work by Barker <cit.> have also pointed out the higher value of lab screening potential with respect to the atomic adiabatic limit. 
        
        Indirect Trojan Horse Method has been taken by Lamia et al. <cit.> to obtain electron screening potential. They have also determined much higher potential than the predicted value of adiabatic limit. It is also interesting that the screening potential thus obtained for ^6Li(p,α)^3He (U_e=350±100 eV) is less than that for ^7Li(p,α)α (U_e=425±60 eV) which is also in contrary to the results obtained by Engstler et al. <cit.> . Phenomenological R-Matrix method has been applied by Angulo et al. <cit.> for ^6Li(p,α)^3He to obtain electron screening for both solid and gas target. Extracted electron screening potential is also found to higher for both cases.  
        
        Artificial neural networks (ANN) serve as a powerful mathematical tool for estimating various values in science and technology, including nuclear physics studies. This method is particularly important in providing accurate results for highly nonlinear relationships between dependent and independent data variables. Recently, ANN has found application in numerous areas of nuclear physics, such as constructing consistent physical formulas for detector counts in neutron exit channel selection <cit.>, determining one and two proton separation energies <cit.>, developing nuclear mass systematic <cit.>, determining ground state energies of nuclei <cit.>, identifying impact parameters in heavy-ion collisions <cit.>, determining beta-decay energies <cit.>, and estimating nuclear rms charge radius <cit.>.
        
        In this investigation, the Multi-Layer Perceptron based Feed-forward Artificial Neural Networks (ANN) <cit.> using scikit-learn is utilized to estimate the bare astrophysical S-factor, total S-factor and subsequently the electron screening potential for ^6Li(p,α)^3He reaction. The experimental datas were extracted from existing literature <cit.>. Through training the ANN with known experimental cross-section values for various combinations of beam, target, and bombarding energy, bare S-factor and total S-factor is successfully predicted. From the ratio of total S-factor with bare S-factor, the electron screening potential has been calculated. The details of Artificial Neural Network method will be discussed in next section followed by Calculations of Screening potentials and Conclusions.
         
        
        
        
        § ARTIFICIAL NEURAL NETWORK
        
        An artificial neural network (ANN) is a computational model that draws inspiration from the structure and functionality of biological neural networks found in the human brain. It serves as a crucial element in machine learning and artificial intelligence systems, aiming to emulate the information processing mechanism of the human brain. Artificial neural networks are constructed with neurons as their fundamental units. Each neuron receives inputs, undergoes computation, and generates an output. These neurons are arranged into layers, comprising an input layer, one or more hidden layers, and an output layer. In the case of a feed-forward neural network, information progresses unidirectionally from the input layer to the output layer without loops. The network includes multiple layers of nodes (neurons), featuring at least one hidden layer positioned between the input and output layers. Neurons establish connections through edges, each associated with a weight representing the strength of the connection. Additionally, a bias term is often employed to adjust the output of a neuron. Throughout the training phase, the weights and biases undergo adjustments to optimize the overall performance of the neural network. The activation function plays a crucial role in determining a neuron's output based on its input, introducing non-linearity and enabling the network to grasp intricate patterns. With a single output node, the network is well-suited for regression tasks, where the objective is to predict a continuous variable. During the feed-forward operation, input data progresses through the network layer by layer. Each neuron's output is computed by evaluating the weighted sum of its inputs and applying the activation function. Training a feed-forward neural network entails fine-tuning the weights and biases by considering the disparity between the predicted output and the actual target. This is typically accomplished through supervised learning using an optimization algorithm like gradient descent. The gradients of the loss concerning the weights and biases are computed, indicating how much the loss would change with adjustments to each weight and bias. The weights and biases are then modified in a direction that minimizes the loss, often employing optimization algorithms such as stochastic gradient descent or Adam. In the current analysis, a grid search will explore both `Adam' and `Stochastic Gradient Descent,' determining the most effective solver during the hyperparameter tuning process. These processes iterate for a specified number of training epochs (maxiter), progressively refining the model to enhance its predictive capabilities.
        
        The subsequent discussion explores the mathematical framework of a straightforward feedforward neural network, specifically a Multi-Layer Perceptron (MLP) based for regression. I will start by breaking down the network layer by layer, and then proceed to introduce the training process, which incorporates gradient descent and adam optimization. 
        
        
         §.§ Forward Pass:
        
        
         §.§ Input Layer:
        
        The neural network takes information about a single data point through a vector called X. If there are m features, X is like a vertical list with m rows and 1 column, often represented as:
        
            X    = [ x_1; x_2;   ⋮; x_m ]
           
        Here, each x_i represents a feature of the input data point. The information (x_1,x_2,...,x_m) passes to the nodes in the hidden layer. 
        
        
        
         §.§ Hidden Layer:
        
        For a single hidden layer consisting of n neurons, the output Z_hidden is calculated as:
        
            Z_hidden=(W_hidden· X+b_hidden)
         
        where,
        
        
        
        W_hidden is the weight matrix of size n × m.
        
        b_hidden is the bias vector of size n × 1.
        
        After computing Z_hidden, it is passed through an activation function σ:
        
            A=σ(Z_hidden)
        
        Where, A represents the output of the hidden layer. If the weighted sum of inputs to a neuron exceeds the threshold or activation layer, the neuron fires. In this analysis, the activation function for the hidden layers is being investigated with two choices: `ReLU' (Rectified Linear Unit) and `tanh' (Hyperbolic Tangent). The model will undergo training and evaluation using both activation functions to determine which one yields better performance for the provided data.
        
        So, the overall computation can be written as:
        
            A=σ(W_hidden· X+b_hidden)
        
        
        This represents the forward propagation through the first hidden layer of a neural network. The activation function σ introduces non-linearity to the model, allowing it to learn complex patterns in the data.
        
        In the present analysis four hidden layers are used in the neural network, the computation of the output at the first hidden layer Z_hidden1 with n_1 neurons in the layer and the subsequent hidden layers Z_hidden2 with n_2 neurons, Z_hidden3 with n_3 neurons and Z_hidden4 with n_4 neurons can be expressed as follows:
        
        For the first hidden layer:
        
            Z_hidden1=σ(W_hidden1· X+b_hidden1)
         
        Here:
        
            W_hidden1 is the weight matrix of the first hidden layer with dimensions n_1× m,
            
            b_hidden1 is the bias vector of the first hidden layer with dimensions n_1× 1,
        
            X is input vector with dimensions m× 1, 
            
            σ represents the activation function already mentioned above.
        
        For the second hidden layer:
        
            Z_hidden2=σ(W_hidden2· Z_hidden1+b_hidden2)
         
        
        Here:
        
            W_hidden2 is the weight matrix of the second hidden layer with dimensions n_2× n_1,
            
            b_hidden2 is the bias vector of the second hidden layer with dimensions n_2× 1.
        
            Z_hidden1 is output of the first hidden layer with dimensions n_1× 1.  
        
        For the third hidden layer:
        
            Z_hidden3=σ(W_hidden3· Z_hidden2+b_hidden3)
         
        
        Here:
        
            W_hidden3 is the weight matrix of the second hidden layer with dimensions n_3× n_2,
            
            b_hidden3 is the bias vector of the second hidden layer with dimensions n_3× 1.
        
            Z_hidden2 is output of the first hidden layer with dimensions n_2× 1.    
        
        For the fourth hidden layer:
        
            Z_hidden4=σ(W_hidden4· Z_hidden3+b_hidden4)
         
        
        Here:
        
            W_hidden4 is the weight matrix of the second hidden layer with dimensions n_4× n_3,
            
            b_hidden4 is the bias vector of the second hidden layer with dimensions n_4× 1.
        
            Z_hidden3 is output of the first hidden layer with dimensions n_3× 1.        
        
        This process enhances the neural network's ability to learn hierarchical features and intricate representations from the input data.
        
        
        
         §.§ Output Layer:
        
        For regression, there is typically no activation function in the output layer, and the output Y is given by:
        
            Y=W_output· Z_hidden4+b _output
         
        
        where, W_output is a weight matrix of size 1 × n_4
        
        Z_hidden4 is the output from the fourth hidden layer,
        
        b_output is the bias term for the output layer.
        
        
        
         §.§ Training with Stochastic Gradient Descent:
        
        
         §.§ Loss Function:
        
        
        The Mean Squared Error (MSE) is commonly used for regression:
        
            MSE=∑_i=1^N (Ŷ_̂î-Y_i)^2N
         
        where,
        N is the data points.
        Ŷ_̂î is the predicted output for data point i.
        Y_i is the actual output for data point i.
        Normalized Mean Squared Error(NMSE) is defined as:
        
            NMSE=MSES_max-S_min
         
        where, S_max and S_min is the maximum and minimum value of astrophysical S-factor.
        Furthermore, a regularization term (R) is incorporated into the loss function during training to counteract overfitting by imposing penalties on large weights in the model. In the case of L2 regularization, the regularization term is directly linked to the square of the weights:
        
            R_L2=α2∑_i=1^n W_i^2
         
        Here, α represents the strength of the regularization term. Higher values of α result in stronger regularization.
        
        
         §.§ Back Propagation:
        
        
        Backpropagation (short for “backward propagation of errors”) is a widely used supervised learning algorithm for training artificial neural networks. It is a gradient-based optimization algorithm that aims to minimize the error between the predicted output and the actual target output. The key steps in the backpropagation process for an artificial neural network (ANN) are as follows:
        
        (1) Compute the gradient of the regularization term with respect to the output layer weights: ∂ R∂ W_output.
        
        (2) Compute the gradient of the mean squared error with respect to the output layer weights: ∂ MSE∂ W_output.
        
        (3) Compute the gradient of the mean squared error with respect to the output layer biases: ∂ MSE∂ b_output.
        
        (4) Propagate the gradients through the network to compute the gradient with respect to the fourth hidden layer weights and biases: ∂ R∂ W_hidden4,∂ MSE∂ W_hidden4, ∂ MSE∂ b_hidden4 .
        
        (5) Propagate the gradients through the network to compute the gradient with respect to the third hidden layer weights and biases: ∂ R∂ W_hidden3,∂ MSE∂ W_hidden3, ∂ MSE∂ b_hidden3 .
        
        (6) Propagate the gradients through the network to compute the gradient with respect to the second hidden layer weights and biases: ∂ R∂ W_hidden2,∂ MSE∂ W_hidden2, ∂ MSE∂ b_hidden2 .
        
        (7) Propagate the gradients through the network to compute the gradient with respect to the first hidden layer weights and biases: ∂ R∂ W_hidden1,∂ MSE∂ W_hidden1, ∂ MSE∂ b_hidden1 .
        
        The process involves the backward propagation of gradients through each layer of the network. The chain rule is applied to calculate the gradients at each layer based on the gradients computed in subsequent layers. This iterative process helps update the weights and biases of the network, refining its parameters for improved performance over time.
        
        
        
        
         §.§ Update Weights and Biases:
        
        
        
        Update the weights and biases using the gradients and a learning rate η:
        
            W_output← W_output-η(α∂ R∂ W_output+∂ MSE∂ W_output)
        
            b_output← b_output-η∂ MSE∂ b_output
        
            W_hidden4← W_hidden4-η(α∂ R∂ W_hidden4+∂ MSE∂ W_hidden4)
        
            b_hidden4← b_hidden4-η∂ MSE∂ b_hidden4
        
            W_hidden3← W_hidden3-η(α∂ R∂ W_hidden3+∂ MSE∂ W_hidden3)
        
            b_hidden3← b_hidden3-η∂ MSE∂ b_hidden3
        
            W_hidden2← W_hidden2-η(α∂ R∂ W_hidden2+∂ MSE∂ W_hidden2)
        
            b_hidden2← b_hidden2-η∂ MSE∂ b_hidden2
        
            W_hidden1← W_hidden1-η(α∂ R∂ W_hidden1+∂ MSE∂ W_hidden1)
        
            b_hidden1← b_hidden1-η∂ MSE∂ b_hidden1
         
        where, R represents the regularization term added to the loss function during training. 
        
        This process is repeated for maximum iterations until convergence.
        The training with Stochastic Optimization using Adam(Adaptive Moment Estimation) has been described in  <cit.>. 
        
        This is a simplified overview, and the actual scikit-learn implementation may involve additional details. 
        
        
        § CALCULATIONS OF ELECTRON SCREENING POTENTIAL
        
        Multi Layer Perceptron based ANN algorithm has been used to find the S-factor from the available experimentally measured literature data. Those data sets are chosen in present calculations which are experimentally well established and have better accuracy. Reliable experimental data are crucial as it will contribution to the robustness of the present ANN model calculations, that can help in drawing meaningful conclusions. Table <ref> summarizes the different data sets used for the present analysis. The data sets are directly taken from  EXFOR database. Some datasets in the existing literature exhibit notable discrepancies with the majority of previous findings are excluded in the present analysis. The utilized data sets cover an energy range in which both electronic screening effects are expected to be present(below ≈ 60 keV) and in which they are not expected to be present(above ≈ 60 keV). The collecting dataset has then been splitted randomly into train and test part. The train part contains 70% and test part contains 30% of total data set. Model is trained using the train dataset. During the initial phase of the study, following numerous trials, the optimal number of hidden layer neurons was identified and employed in a consolidated four-in-one layer which demonstrated the most effective outcomes for the present problem. Present analysis employs RandomizedSearchCV with a comprehensive grid search for optimal hyperparameters. Hyperparameters play a crucial role in machine learning algorithms by controlling the learning process and shaping the final model's behavior. Optimized hyperparameters are used to obtain the best correlation coefficient. Higher correlation coefficient indicates the lower uncertainty and vice versa.  In the present calculations, one input layer, four hidden layers with size (128,64,32,16) and one output layer is used after obtaining this configuration from several trials for optimization. The number of hidden layer and hidden units are so chosen to get the minimum loss as much as possible. The maximum iteration is used as 5000. In the Adam algorithm, the initial learning rate is 0.001 and decay constants are 0.9 and 0.999. Once the model are trained, the same set of Hyperparameters are used to predict the performance of the model on test data-set.  In the second part, data's are above 60 keV are used to find the bare S-factor using the same ANN code with same hyperparameters. Once the bare S-factor is obtained the electron screening potential is then obtained by taking the ratio of the total S-factor with the bare S-factor. The total S-factor and bare S-factor obtained from the present ANN based model is shown in Fig. <ref>. Bare astrophysical S-factor at zero energy calculated from the present model is 3.36 Mev-barn and it is tabulated in Table <ref> and compared with the value obtained by the different methods in Literature. The values obtained from the present calculation is in excellent agreement with that obtained from the R-MATRIX calculation <cit.> and THM calculation <cit.> by taking into account the systematic uncertainty. The variation of bare astrophysical S-factor with that obtained from R-MATRIX and THM has been compared in Fig. <ref> and Fig. <ref> respectively. It has been observed from the comparison that the present calculations reasonably matches with the well established R-MATRIX and THM calculation. Electron screening potential obtained from the present calculation (U_e=220 eV) has been tabulated in Table <ref> and compared with the electron screening potential obtained from the other methods as shown in Fig. <ref>. The electron screening potential thus obtained from the present ANN based model nicely matches with the other methods within the quoted uncertainty. This results indicates that Multi-Layer Perceptron based Artificial Neural Network model can be used an alternative tools to find the the electron screening potential in light charged particle induced reactions of astrophysical interest.   
        
        
        
        § CONCLUSIONS
        
        In this study, the electron screening potential is obtained for ^6Li(p,α)^3He reactions by using the Multi-Layer Perceptron based Artificial Neural Network based algorithm for the first time. From the present analysis, the electron screening potential is obatined as 220 eV. The electron screening potential matches resonably well with other welll estrablished techniques like R-MATRIX calculation and THM method indicating the ANN method as an alternative powerful tool for extraction of such astrophysical quantity. The variation of the bare S-factor with R-MATRIX and THM calculation also supports the claim.   
        
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 Paper Title: Study of Wide-Field-of-View X-ray Observations of the Virgo Cluster Using the Lobster Eye Imager for Astronomy.





§ INTRODUCTION




Lobster-eye micropore optics (MPO) is an novel X-ray focusing technology that is known for its ability to observe a wide field of view and its true imaging capabilities. Since Angel <cit.> first introduced a design of an X-ray all-sky monitor (ASM) based on lobster-eye MPO, several teams have conducted extensive research on lobster-eye optics  <cit.> and have developed conceptual designs for lobster-eye X-ray ASMs  <cit.>. Real missions using lobster-eye ASMs have been proposed <cit.>. The first instrument constructed for a formal mission was the MIXS <cit.> aboard BepiColombo, which consists of a 1^∘ field-of-view (FoV) Wolter telescope and a 10^∘ FoV collimator, both constructed using MPO. Its debut observations are expected within a few years, when the mission reaches Mercury. Several X-ray telescopes based on lobster-eye MPO are currently in development and scheduled for launch in the coming years, including SVOM-MXT <cit.> and SMILE-SXI <cit.>.

The Lobster Eye Imager for Astronomy (LEIA), launched in July 2022 onboard the SATech-01 satellite, is the first lobster-eye wide-field focusing X-ray telescope <cit.>. It features a mostly unvignetted FoV measuring 18.6^∘× 18.6^∘ in the energy band 0.5–4.5keV. It serves as a pathfinder for the wide-field X-ray telescope in the Einstein Probe mission and was designed to verify the in-orbit performance of the EP-WXT and optimize its instrumental parameters and operational conditions. LEIA achieves a spatial resolution ranging from 4^' to 7^' of the full width at half maximum (FWHM), and its effective area of 2–3 cm^2 shows only mild variations across almost the entire FoV <cit.>.



LEIA has been operating in orbit for more than one year, with Virgo being the first galaxy cluster observed. The Virgo Cluster is the closest (∼16.1 Mpc <cit.>) bright irregular cluster <cit.>. The brightest cluster galaxy is the giant elliptical galaxy M87 (NGC 4486), which contains a super-massive black hole. The Virgo Cluster was first captured by the ROSAT all-sky survey <cit.> over the course of three months from November 1990 to January 1991. With an average exposure time of ∼450 ks, the X-ray image of the Virgo Cluster is clear at the 0.1 to 2.4 keV band observed by ROSAT. The large-scale structure of the Virgo Cluster was first studied in detail by <cit.>, and the results revealed  an extended emission in the 12.8^∘× 12.8^∘ area of the Virgo region, with most of the cluster's mass being concentrated in galaxy M87. Using ASCA, <cit.> was the first to derive a full temperature map of the Virgo Cluster, covering an area of 19 deg^2. <cit.> analyzed a mosaic comprising thirteen XMM-Newton images observed from the center of the cluster to the north with a radius of approximately 1.2 Mpc and obtained a mean temperature of ∼2.3 keV.


In this study, we present an X-ray image of the Virgo Cluster as observed by LEIA. The image is affected by LEIA's PSF, but the main radiation results are consistent with the results obtained from ROSAT. We also analyzed the spectra taken on the Virgo Cluster and obtained temperature fitting results consistent with those obtained by XMM-Newton. This work aims to validate LEIA's capabilities in the study of extended sources.
The paper is arranged as follows: in Section <ref>, we describe the observation data used. The pictures of the Virgo Cluster and the spectrum fitting are described in Section <ref>. The Discussion and Conclusions are given in Section <ref> and Section <ref>, respectively. The redshift (z) adopted for M87 is 0.00428. In this work, H_0 = 67.8   km  s^-1  Mpc, Ω_m=0.308, and Ω_Λ=0.692 <cit.>.




§ OBSERVATIONS AND DATA REDUCTION




 §.§ LEIA Observations

A total of 36 observations of the Virgo Cluster were used. The observations were carried out between 19 December 2022  and  26 February 2023. The telemetry data from LEIA were downloaded to the ground station each day and sent to the data center at the National Space Science Center (CAS), where the telemetry data were decoded, unpacked, verified, and processed to remove duplicates. Then, the data were sent to the science center at the National Astronomical Observatories (CAS), where the data were converted into standard FITS format and processed by a software pipeline and the corresponding calibration database (CALDB) to calibrate the events and extract high-level products in the following steps <cit.>:


  
  * Since the digital number (DN) of each event was only subtracted by the corresponding row's median value on-board, the ground processing pipeline further subtracted the bias residual of each pixel stored in CALDB.
  
  * The position of each event was converted to the celestial coordinates (J2000) using the satellite attitude and alignment matrix between the detector and the satellite. The nonlinear distortion due to the non-spherical shape of MPO was also corrected for in this step according to the result of the on-ground calibration performed at IHEP.
  
  * The anomalous pixels stored in CALBD and their nearby pixels were flagged. New anomalous pixels (hot and flickering) were also searched for and flagged.
  
  * A grade and a single PHA value were assigned to the event according to the DNs of the 3×3 pixels around the event.
  
  * The Pulse-Invariant (PI) value of each event was calculated according to the gain stored in CALDB.
  
  * Single, double, triple, and quadruple events without any anomalous flag were selected for further processing. The geomagnetic cut-off rigidity (COR) was employed to remove the high background interval in the high-latitude region, i.e., COR>5. The earth elevation angle was required to be larger than 10 degrees to remove the effect of earth occultation. The angular distance to the nominal pointing should be smaller than 0.2 degrees to ensure the stabilization of the satellite.
  
  * The pipeline generated an exposure map that accounts for bad CMOS pixels and columns, attitude variations, and telescope vignetting (optional) for event files.
  
  * An image in the 0.5–4.0 keV band was accumulated for each CMOS using the cleaned events, and source detection was performed on each image to generate a source catalog.
  
  * The light curve and spectrum of each source in the catalog were extracted. The pipeline also generated the corresponding Response Matrix File (RMF) and Ancillary Response Function (ARF) file, which accounted for the effective area, vignetting correction, and PSF correction.

	
The above procedures were conducted on each CMOS separately. The resulting event files and images can be further merged to search for faint sources.
 


§ RESULTS




 §.§ The X-ray Image 

We used the Montage Image Mosaic Engine
Available from <http://montage.ipac.caltech.edu/> 
 to combine LEIA's observations of the Virgo Cluster and its surroundings to obtain the exposure-corrected mosaic image in the 0.5–4.5 keV band shown in Figure  <ref>. The exposure time of the center position is ∼17.8 ks. The green crosses indicate the direction of each observation. The brightest core is the central galaxy M87. Its emissions extend several degrees outward in the shape of a cross. We then created a white box (shown in ) that was 12^∘× 12^∘ in size and centered around M87 to frame the Virgo Cluster to further investigate its structure. The right panel of  is an enlarged image of the box region. The white contours were obtained from the smoothed ROSAT Position-Sensitive Proportional Counter (PSPC) image of the medium X-ray emission of the Virgo Cluster's intracluster in the 0.5–2.0 keV band <cit.>. The contour lines represent being 3σ, 5σ, 11σ, and 25σ above the background <cit.>.
The elliptical galaxy M86 <cit.> lies northwest of M87. Another prominent feature in Figure  <ref> is the X-ray halo around M49 <cit.>, which is located south of M87. Some structures that extend beyond the ROSAT contour in the far north and west in the LEIA images are affected by the PSF effect <cit.> and not the real cluster structure. We provide a detailed discussion on the arm structure in Section <ref>.






As the ROSAT PSPC observations only offer an energy band that extends from 0.1 to 2.4 keV, LEIA's wide-field X-ray imaging up to 4.5 keV provides us with an opportunity to see the features of the Virgo Cluster. 
In order to better reveal the radiation distribution of the Virgo Cluster, Figure  <ref> presents a LEIA image of the source in the emission ranges of 0.5–1.5 keV, 1.5–2.4 keV, and 2.4–4.5 keV from the central 6^∘ point. For comparison, we also provide a contour in each panel which shows 5σ levels above the background noise for the images in the 0.5–1.5 keV range and 3σ levels for the images in the 1.5–2.4 keV and 2.4–4.5 keV bands. In the 0.5–1.5 keV band, the diffuse arm structure around M87 is clearly visible. The scale of arm extension to each edge is up to 5^∘; above the 1.5 keV band, the thermal gas radiation is barely visible, overshadowed by the central point source and the influence of its PSF. This is the first time a lobster-eye focusing telescope has been used to see a large-scale image of the Virgo Cluster that shows that diffuse thermal emission exists around M87 in the 0.5–4.5 keV band.







 §.§ LEIA Spectrum


We selected five observations containing the majority of the thermal gas emissions and relatively high-quality data for spectral analysis. To extract the spectra of the Virgo Cluster, we adopted a box region that was 4^∘× 6^∘ in size and included M87 (shown in Figure  <ref>, the same size as the source region for every observation). To eliminate the influence of the cosmic background, we circled an area with a 1^∘ radius and where there were no resolved point sources in the background region. The spectra were extracted in XSELECT (version 2.4). We merged these spectra together using the  command to enhance the data statistics. Then, we binned the merged spectra to ensure that 1 bin contained more than 200 photons. 

We used XSPEC (version 12.11.1) to model the spectrum. Most regions of the Virgo Cluster can be well fitted by a single temperature model <cit.>, while the central region (<2 arcmin) may be affected by non-thermal radiation <cit.>. The LEIA spectrum can be characterized by an absorbed APEC model, allowing the model and the spectrum to be fit together with the WABS*APEC fitting model. The hydrogen column density was fixed as 2.13 × 10^20  cm^-2 <cit.>. The redshift was fixed to 0.00428. Due to the inability to determine the metal abundance effectively from the spectra, we set the abundance ratio to 0.3Z_⊙. 
The fitting results are listed in Table  <ref> and shown in Figure  <ref>. 
The blue and red data points are from the source and background components, respectively.
The energy spectrum above 1.6 keV is dominated by the background caused by the aperture and non-aperture cosmic X-ray background (CXB), as well as the particle-induced background <cit.>.
The reduced chi-squares of ∼1.24 show that our model is reasonable. 
The temperature of the model fitting is 2.1^+0.3_-0.1 keV, which is consistent with the overall temperature (2.4^+0.3_-0.2 keV) within the 1^∘ radius of M87 measured by ROSAT <cit.> and
the average temperature of ∼2.3 keV <cit.> derived by <cit.> obtained from XMM-Newton.













§ DISCUSSION


The LEIA image is generally consistent with the ROSAT contours, with slight differences in the north and west, as described in Section <ref>. It is necessary to determine whether the extra structures in the LEIA image are the real components of the Virgo Cluster. Due to the design principle of the lobster-eye optical device, LEIA's PSF exhibits a cross shape. LEIA's mirror is composed of numerous small square apertures that are densely arranged on a spherical surface and point towards the common curved center of the sphere. When X-rays enter these apertures, they undergo grazing incident reflection on the inner walls of the apertures, ultimately focusing on the focal plane and forming a cross-shaped pattern.

To simulate the imaging of ROSAT images under the influence of PSF, it is necessary to first generate a PSF model, which can be an ideal model or modeled based on the actual PSF data of LEIA. Then, using the ROSAT observation image as the source model, the source model is convolved with the PSF model to obtain the blurred PSF imaging result. LEIA observed Sco X-1, the brightest known persistent X-ray source <cit.>, which is in great agreement with the PSF measured during on-ground calibration <cit.> (shown in the right panel of Figure  <ref>). Therefore, the X-ray image of Sco X-1 can be treated as an ideal PSF image for LEIA. The left panel in Figure  <ref> shows the ROSAT PSPC image of the Virgo Cluster in the 0.5–2.0 keV band. After rotating the PSF image to the average characteristics of the Virgo observations, we convolved it with the ROSAT image to simulate LEIA observations of the Virgo Cluster under the influence of the PSF effect. The FoV of the simulated image is 10^∘× 10^∘. 





As shown in the left panel of Figure <ref>, there is an obvious arm structure extending outward from the brightest center of the simulated image. Comparing the simulated image with the observed one (the right panel of Figure <ref>), it can be seen that their extension structures are consistent. Therefore, we infer that there are no additional large-scale structures in the Virgo Cluster according to the LEIA observations.








§ CONCLUSIONS


In this work, we present wide-field (18.6^∘× 18.6^∘) X-ray observations of the Virgo Cluster obtained using LEIA and compare these observations with ROSAT images. We also study the LEIA spectrum. The main results are as follows:


  
  * The images match the ROSAT PSPC images. Galaxies, including M87, M86, and M49, can be clearly seen from the merged LEIA image in the 0.5–4.5 keV band.
  
  * [-15]The LEIA images show an arm structure caused by the PSF effect of the LEIA instrument.
  
  * We extracted the LEIA spectrum and determined the average temperature of the FoV (2.1^+0.3_-0.1 keV ) and found it to be consistent with the overall temperature (2.4^+0.3_-0.2 keV) within a 1^∘ radius of M87, as measured by ROSAT, and the average temperature of (∼2.3 keV) obtained by XMM-Newton.
  
  * Above 1.5 keV, the spectrum is dominated by the background, with the image primarily showing the central point source and its PSF effects.


This is the first time that the ability of the lobster-eye focusing telescope to generate images and spectra has been verified; however, the images are significantly affected by PSF. Future technological progress in eliminating the impact of PSF will greatly enhance the imaging ability of this type of telescope.





Conceptualization, Shu-Mei Jia and Heng Yu; Methodology, Wen-Cheng Feng, Cheng-Kui Li, Yu-Lin Cheng and Shan-Shan Weng; Software, Wen-Cheng Feng, Cheng-Kui Li and Yu-Lin Cheng; Validation, Shu-Mei Jia, Hai-Hui Zhao and Yong Chen; Formal analysis, Wen-Cheng Feng; Investigation, Heng Yu and Yong Chen; Resources, Wen-Cheng Feng, Hai-Wu Pan, Yuan Liu,  Zhi-Xing Ling and Chen Zhang; Data curation, Wen-Cheng Feng, Hai-Wu Pan and Yuan Liu; Writing – original draft, Wen-Cheng Feng; Writing – review & editing, Heng Yu; Funding acquisition, Shu-Mei Jia, Hai-Hui Zhao and Shan-Shan Weng; Project administration, Shu-Mei Jia; Supervision, Shu-Mei Jia and Shan-Shan Weng; Visualization, Wen-Cheng Feng. 


This work was supported by the Strategic Pioneer Program on Space Science, Chinese Academy of Sciences (Grant No. XDA15310300), and the National Natural Science Foundation of China (Grant Nos. U2038104 and 11703014).
This research was carried out using Montage. It was funded by the National Science Foundation under Grant Number ACI-1440620 and was previously funded by the National Aeronautics and Space Administration's Earth Science Technology Office, Computation Technologies Project, under Cooperative Agreement Number NCC5-626 between NASA and the California Institute of Technology.

The data in this article has not been made public yet.   

The authors declare no conflict of interest.  


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 Paper Title: Approximating Rayleigh Scattering in Exoplanetary Atmospheres using Physics-informed Neural Networks (PINNs).



firstpage–lastpage
The extremes of AGN variability: outbursts, deep fades, changing looks, exceptional spectral states, and semi-periodicities
    [
    August 5, 2024
===========================================================================================================================




§ ABSTRACT 
This research introduces an innovative application of physics-informed neural networks (PINNs) to tackle the intricate challenges of radiative transfer (RT) modeling in exoplanetary atmospheres, with a special focus on efficiently handling scattering phenomena. Traditional RT models often simplify scattering as absorption, leading to inaccuracies. Our approach utilizes PINNs, noted for their ability to incorporate the governing differential equations of RT directly into their loss function, thus offering a more precise yet potentially fast modeling technique.
The core of our method involves the development of a parameterized PINN tailored for a modified RT equation, enhancing its adaptability to various atmospheric scenarios. We focus on RT in transiting exoplanet atmospheres using a simplified 1D isothermal model with pressure-dependent coefficients for absorption and Rayleigh scattering. In scenarios of pure absorption, the PINN demonstrates its effectiveness in predicting transmission spectra for diverse absorption profiles. For Rayleigh scattering, the network successfully computes the RT equation, addressing both direct and diffuse stellar light components.
While our preliminary results with simplified models are promising, indicating the potential of PINNs in improving RT calculations, we acknowledge the errors stemming from our approximations as well as the challenges in applying this technique to more complex atmospheric conditions. Specifically, extending our approach to atmospheres with intricate temperature-pressure profiles and varying scattering properties, such as those introduced by clouds and hazes, remains a significant area for future development.





radiative transfer – scattering – planets and satellites: atmospheres – methods: numerical








§ INTRODUCTION


The use of Machine Learning (ML) in exoplanetary science has grown increasingly popular during past years <cit.>. Hereby, it is mostly used to accelerate the time-consuming process of finding the posterior distributions of atmospheric parameters given some observed spectrum. Usually, this inference is done using a Bayesian scheme, like nested sampling <cit.> where many so-called forward models simulate data which can then be compared to observations.

Pioneered by <cit.>, who used a simple multi-layer perceptron (MLP) to predict the existence of molecules in exoplanetary atmospheres from their transit spectra, the approaches have diversified since then, from Random Forests <cit.> to generative adversarial networks like ExoGAN <cit.> and convolutional neural networks, often in combination with Monte Carlo dropout <cit.>.

More recent approaches achieve generating more accurate and versatile posterior distributions by employing normalizing flows in combination with variational inference <cit.>, neural posterior estimation <cit.>, or flow matching and importance sampling <cit.>.

The approaches most similar to our own are by <cit.> and <cit.>, who both try to speed up the forward model while leaving the inference scheme untouched. While <cit.> use a neural network to emulate radiative transfer by predicting the spectrum given some atmospheric parameters, <cit.> focus on accelerating computationally expensive disequilibrium chemistry calculations using an autoencoder and an LSTM-like neural network.

Most of these models rely on traditional numerical methods to generate the data that the ML model uses for training. Since all ML models cannot contain more information than their training data, they cannot generalize well beyond the used forward model.
Including the time to train a model, the respective method only offers a speed-up if enough exoplanets can be analyzed using the same model trained on the same data. This can be particularly challenging for exoplanetary atmospheres, given the diversity of exoplanets that have been observed. Additionally, as soon as a forward model is updated to include some new insight, the ML model might be completely retrained. It can also prove difficult to use one model for observations with varying instruments, that, e.g., have a different number of data points or an incompatible spectral range.

In the future, as data become more abundant and precise and the models more detailed, the computational burden of running a retrieval will further increase. Therefore, in this work, we will assess the feasibility of using pre-trained physics-informed neural networks (PINNs) to cheaply model complex phenomena in exoplanetary atmospheres. We focus on the simple example of Rayleigh scattering in the isothermal atmosphere of a transiting exoplanet.

Building upon the work of <cit.>, who used PINNs to solve the radiative transfer (RT) equation, we first model RT considering only absorption. Since for transiting exoplanets, the traditional numerical method treats scattering the same as absorption in that the light scattered out of the beam is analogous to absorbed light <cit.>, we can compare our PINN to existing models to establish its speed and accuracy.

Secondly, we include Rayleigh scattering in the RT equation. As has been shown by <cit.>, scattered light can have a significant effect on the transit spectrum at short wavelengths. Using PINNs could help us study this effect in more detail and potentially improve our estimates of the hydrogen-to-helium ratio or the existence and properties of clouds and hazes.

In Section 2, after briefly introducing PINNs, we discuss how the respective problem is presented to both the absorption and scattering PINN, such that they are able to solve it efficiently. This involves generating and normalizing the inputs, the formulation of the loss function, and the exact training procedure.
The results of both models are assessed in Section 3, while drawbacks are summarized in Section 4. Section 5 discusses key insights and possible enhancements.




§ METHODS




 §.§ PINNs


While there are many different ways of incorporating physical laws into the training of a neural network, PINNs, as introduced by <cit.>, achieve this through their loss function, which is directly based on a differential equation.

To see how a PINN is able to solve a nonlinear partial differential equation (PDE), as explained in <cit.>, let us first describe a PDE in its most general form defined on the domain Ω∈ℝ^d with its boundary ∂Ω:


    ℱ(u(z);γ) = f(z),    z ∈Ω
    ℬ(u(z)) = g(z),    z ∈∂Ω ,



where u describes the unknown solution to the differential equation, dependent on the space-time coordinate vector z := [x_1, ⋯, x_d-1, t]. ℱ refers to the nonlinear differential operator dependent on physics-related parameters γ, while f is the function identifying the data of the problem (if available). Eq. (<ref>) defines the boundary and initial conditions, where ℬ is again an operator and g is the boundary function.


While the PINN can also be used for an inverse problem of finding the parameters γ, we use it for approximating the solution u for every z, dependent on the (given) parameters γ. Hereby, the PINN itself functions as the approximation to this solution. The PINN therefore has inputs z and output û_θ(z) ≈ u(z) dependent on the network parameters θ. The corresponding loss function that should be minimized is made up of different components ℒ_i with respective weights ω_i:


    ℒ = ω_ℱℒ_ℱ + ω_ℬℒ_ℬ .



The first term ℒ is the differential equation (<ref>) itself, with all terms on one side of the equation such that it is equal to zero. Often, the following form analogous to the mean squared error is used:


    ℒ_ℱ = ∑_i=1^n ( ℱ (û_θ(z_i);γ) - f(z_i) )^2,    z_i ∈Ω ,



where the square guarantees both positive values, such that the minimum lies at zero, and differentiability (compared to the mean absolute error).

Now we calculate the value of the differential equation ℱ(u(z);γ) with automatic differentiation. We can not only use it to calculate the gradient of the loss function with respect to the weights of the network, but also calculate the gradient of the output û_θ(z) of the network with respect to the inputs z, the space and time coordinates. Hence, we can insert all the values of the variables and derivatives into the differential equation to calculate the loss.

The second term can be calculated accordingly to make sure that the boundary conditions are fulfilled:


    ℒ_ℬ = ∑_i=1^n ( ℬ (û_θ(z_i)) - g(z_i) )^2,    z_i ∈∂Ω .



In this work, our approach to employing PINNs deviates from their conventional application. Typically, PINNs are tasked with solving specific instances of differential equations, where their advantage over traditional numerical methods is most pronounced in scenarios involving complex geometries or boundary conditions. However, we leverage PINNs in a novel way by training them on a parameterized form of the radiative transfer equation. This means that the solution approximation provided by the PINN, denoted as û_θ(z), is influenced not only by the space-time coordinates z but also by a set of physical parameters γ. This unique application allows the parameterized PINN to generate solutions, û_θ(z, γ), for varying physical conditions without the need for retraining, showcasing a flexible and powerful tool for modeling radiative transfer processes in exoplanetary atmospheres.





 §.§ Setup
 

In this study, our objective extends beyond solving a PDE; we aim to tackle an integro-differential equation, specifically the stationary radiative transfer equation, which operates in two spatial dimensions:


    cos(ϕ)  u/ x   + sin(ϕ)  u/ y + (α_a + α_s) · u
    
        - α_s/4π   ·∫Φ(θ, ϕ, θ', ϕ') u(θ', ϕ') Ω' = 0 ,



where the angle ϕ = 0 is defined as the positive x-direction and lies
in the range ϕ∈ [-π, π]. α_a and α_s are the absorption and scattering coefficient, respectively and Φ the scattering phase function, depending on the incoming (θ', ϕ') and outgoing angles (θ, ϕ) of the light. In the case of Rayleigh scattering, the scattering phase function depends only on the angular distance Δ:


    Φ(Δ) = 3/4( 1 + cos^2Δ) ,



where cosΔ = cosθ cosθ' + sinθ sinθ'  cos(ϕ - ϕ') according to the law of cosines. For both the absorbed and scattered light component sensible symmetries for small angular extents of the star are exploited such that the integral can be evaluated numerically over only one angle (see Appendix <ref> for the explicit integral).

The function of interest that we want to solve for is u(x,y,ϕ), the transmission through the atmosphere. For this, we also need to define our geometry, as shown in Fig. <ref>. More specifically, we want to find the transmission u(x=x_max,y,ϕ=0), which we can then integrate over the y or radial direction to find the effective area of the planet.


This depends on a number of other parameters. So, to reduce the dimension of the problem, we assume an isothermal atmosphere and constant conditions throughout the day-night-terminator region of the planet. The models are trained on an atmosphere spanning pressures from 10^-6 to 10^2 bar and consisting of 100 log-spaced layers, where the absorption and scattering coefficient is interpolated logarithmically between them.

Furthermore, for the PINN with scattering, we assume that the planet transits its star with an orbital inclination i = 90^∘, simplifying the boundary condition. The scattering PINN also only calculates the solution for pure scattering without any absorption (α_a = 0). The total transmission function, when simulating a spectrum, is then calculated by multiplying the solution for purely absorbing species with the solution for purely scattering species, which should hold as long as one of the two is dominating.



 §.§ Data Generation
 

Before we are able to train a neural network on this problem, we have to generate and normalize the necessary input data.

For both the absorption and the scattering model, this includes the spacial coordinates x and y, which are normalized to be within the interval [-1,1] by scaling them with factors l_x and l_y, respectively. These are defined such that the innermost atmospheric layer touches the lower boundary at (x,y) = (0, -1) and the outermost layer touches the upper boundary at (x,y) = (0, 1) as well as going through the lower corners (x,y) = (-1, -1) and (x,y) = (1, -1):


    l_x = √(r_0^2 - r_99^2),    l_y = r_0 - r_99/2 ,


where r_0 refers to the outermost and r_99 to the innermost layer. Note that we have to account for these scaling factors inside of the differential equation by dividing the respective gradients by this factor.

The other two input parameters which are given to both models include the radius of the planet R_0 at the reference pressure P_0 = 10^-2 bar, spanning from 0.2 R_J to 2.0 R_J, normalized to also be within [-1,1]. Secondly, the scale height H := k_ B T / (μ  g), where T refers to the temperature, μ to the mean molecular weight, and g is the gravity at the reference pressure P_0. log_10(H[ cm]) is sampled from a normal distribution with a mean of 7.1 and a standard deviation of 0.35 and is normalized accordingly. Extreme values of the scale height can change the geometry of the problem significantly and hence, make it harder for the PINN to learn a solution. The normal distribution ensures that this does not hinder the training of the PINN, while maintaining a broad range of values in the training data, allowing the neural network to approximate regions of higher data density more quickly and accurately <cit.>. Together, H and R_0 allow us to calculate the radius profile r(P) assuming a constant temperature but variable gravity:


    r(P) = R_0 ·[ 1 + H/R_0ln(P/P_0) ]^-1 .


Although isothermality is often an inadequate approximation for exoplanetary atmospheres <cit.>, we use it to assess if the PINN can manage the simplest scenario. That is why we further assume constant mass fractions (or mixing ratios) of all species throughout the atmosphere.

For the pure-absorption PINN, we also need to generate the α_a(P)-profiles, where the absorption coefficient α_a is extended to encompass both absorption and scattering sources. In this approach, scattering is effectively treated as a form of absorption, consistent with the traditional model. Ideally, these profiles should not just consist of a database of absorption coefficients from certain molecules at certain wavelengths, but should be more general by generating them randomly on the fly during training. For this, we first investigated example α_a(P)-profiles using petitRADTRANS <cit.>, which calculates the absorption coefficient α_a = κ X ρ for every species at every layer as the product of opacity κ, mass fraction X and density ρ. Our analysis included the correlated-k opacities of H_2O and CO <cit.>, CH_4 and CO_2 <cit.> and Na and K <cit.>, as well as Rayleigh scattering from H_2 <cit.> and He <cit.> and collisionally-induced absorption from H_2-H_2 <cit.> and H_2-He <cit.>. We found that most profiles follow either α_a ∼ρ∼ P or α_a ∼ P^2 (if κ∼ P) and hence ln(α_a)/ln(P) either equates to 1 or 2. Therefore, a truncated exponential profile around 1 and 2 (≤ 2) was used to create the profiles from their logarithmic gradient. During training, we observed that the PINN improves when using more accurate but diverse profiles. Therefore, we include 6 different types of randomly generated profiles in total:


    
  * ln(α_a)/ln(P)≈ 1
    
  * ln(α_a)/ln(P)≈ 2
    
  * random transition from ln(α_a)/ln(P)≈ 1 to 2
    
  * random transition from ln(α_a)/ln(P)≈ 2 to 1
    
  * randomly switching between ln(α_a)/ln(P)≈ 1 and ln(α_a)/ln(P)≈ 2
    
  * random negative values (≥ -0.5) for ln(α_a)/ln(P) in some profiles.




In Fig. <ref>, we compare the distribution of the generated logarithmic gradients with actual ones from all opacity sources and layers of the example atmosphere shown in Fig. <ref>. The more diverse set of gradient-profiles improved the error of a spectrum from initially around 3-5% to 1-2%.

The final α_a(P)-profiles are then created by cumulatively summing up the gradients in both directions from a middle layer (with index 49), which gets assigned a log_10(α_a)-value by a uniform distribution and hence acts as a random shift of the profiles.

To inform the PINN about the profile, we give it all the absorption coefficients α_i, i ∈ [0,99] from the 100 atmospheric layers, such that it has a total of 104 inputs.

In the case of Rayleigh scattering from hydrogen and helium, the scattering cross-section does not depend on the pressure. Hence, the scattering coefficient is simply proportional to the density α∼ρ∼ P, an example of which can be seen in Fig. <ref>, and can be parameterized with just one input variable.

The scattering PINN has additional inputs, namely the angle ϕ∈ [-π, π] as well as (half) the angular extent of the star as seen from the planet Δ_* ∈ [0, 0.3π]. Both are normalized to be within [-1,1].
A summary of the parameter bounds and how they are sampled can be found in Table <ref>.






 §.§ Training


The construction of the loss function for the absorption PINN initially appears straightforward, but involves intricate considerations. The boundary condition is established by the incident stellar radiation, characterized by the function u(x=-1, y) = 1.0, representing the intensity of starlight entering the system from the left boundary. This boundary condition is coupled with the RT equation governing the system:


    u/ x + α' · u = 0 ,


where the scaling factor is taken into account inside the α'  l_x α_a.
However, utilizing this equation directly as the loss function presents significant challenges in solving the problem. To understand why this is the case, let us consider two extremes for different regions of the atmosphere: α' ≪ 1 and α' ≫ 1.

For an extremely high value of α', the RT equation effectively becomes α' · u = 0 with the simple solution u = 0. Now consider small deviations of u from 0; Because of the high α' they can result in high values of the loss function and proportionally steep gradients. This means that the PINN effectively overfits these regions to be as close to zero as possible, even though other regions might be of more interest. We can solve this by dividing the whole equation by α'. This gives us approximately u = 0 as our loss function in such regions, resulting in the PINN learning the right solution without overfitting.

In the second case where α' ≪ 1, the form of Eq. (<ref>) would be preferable, since it would effectively be u/ x = 0. Again, the neural network would learn the right solution of u = const., but would not overfit the region as would be the case if we would divide the gradient by α'.

We can combine these two forms of the same equation by calculating both and taking the (element-wise) minimum.
The RT equation, which only considers Rayleigh scattering, is normalized in a similar fashion, details of which can be found in Appendix <ref>.

Meanwhile, the boundary condition of the scattering PINN has more complexity to it. On the left side, the light from the central star enters the atmosphere, this time with an angular extent. Because of this angular extent, also light from the upper boundary could enter the atmosphere, while from the right boundary, no light should enter but only leave the considered slab:


    u(x=-1, y, |ϕ|≤π/2)    = 1  if  |ϕ|≤Δ_*, else =0
    
        u(x, y=1, ϕ≤ 0 )    = 1  if  |ϕ|≤Δ_*, else =0
    
        u(x=1, y, |ϕ|≥π/2)    = 0 .





For the lower boundary, defining an exact boundary condition poses a challenge due to the uncertainty in the amount of light that could be scattered into the observed atmospheric slab from below. However, the absence of a boundary condition can potentially lead to erroneous solutions. To mitigate this issue, we approximate the lower boundary condition by setting the scattered light component to zero. This implies that no additional scattered light enters the system from this boundary. This approximation is expected to yield an accurate solution, provided that the optical depth of the atmosphere is sufficiently high.

For the absorption component, an accurate boundary condition can be found by recognizing the spherical symmetry of the system and rotating the cord of the light ray along a path of constant radius until the point of interest reaches the x-coordinate of x=0:


    u_a(x_0 ≤ 0,y=-1,ϕ≥ 0) = u_a(x=0,y',ϕ+Δϕ),
       where  y' = y(x_0, r(x_0,y=-1)),  tan(Δϕ) = l_x x_0/r_99.


Note that this only works for x ≤ 0. Otherwise, this boundary condition is set to zero, which again should hold, if the optical depth of the atmosphere is high enough at the highest pressures. The other edge case that we have to consider is if abs(ϕ+Δϕ) > Δ_*,max (= 0.3 π), for which u_a is zero because of the inherent architecture of the PINN. As this only happens for particularly low values of r_99 and large values of |l_x x_0| (i.e., for the outer layers of the atmosphere), the boundary condition is set to u_a = 1 for these rare instances.

The final boundary condition is to ensure the periodicity of ϕ: u(x,y,ϕ=π) = u(x,y,ϕ=-π).


Because of its boundary conditions, the scattering PINN has a special architecture, as shown in Fig. <ref>. This architecture has two outputs: One (u_a) should fulfill the RTE without scattering and utilizes a hard-coded filter, such that the neural network does not have to learn the non-differentiable boundary at |ϕ| = Δ_*, where the value of u can jump from 1 to 0. The other output (u_s) should then model the diffuse component of the light such that when adding the two outputs u = (u_a + u_s), they together fulfill the complete RTE with scattering.

If we would now sample ϕ uniformly in [-π,π], most of the "absorption output" u_a would equate to zero only because of the mentioned filter. This not only wastes a lot of training points, but also effectively lowers the weight (or importance) of the corresponding residual in the loss term. We mitigate this problem by sampling half of ϕ within [-Δ_*, Δ_*].


Both the boundary conditions and the residuals enter the loss function with their mean square. To make sure that the boundary conditions are fulfilled first, a weight of 0.5 was given to the residual term (ω_ℱ = 0.5 and ω_ℬ = 1 in Eq. (<ref>)). The lower boundary condition is given a weight of 0.1, as it is reasonable to fulfill only if u_a at x=0 is already sufficiently accurate. The final loss function is then the logarithm of their weighted sum.

The used optimizer is the higher-order quasi-Newton L-BFGS. We found that initially using Adam and then switching to L-BFGS yielded no benefit, so L-BFGS was used from the start. The parameters of the optimizer differ from the default PyTorch parameters in a learning rate of 0.8, the maximal number of both function evaluations and iterations per optimization steps of 50 (or 20), the tolerance change is set to the machine epsilon and the Strong Wolfe condition is chosen for the line search. As described in many PINN papers before <cit.>, we confirmed that having a larger batch size generally improves the final model. Accordingly, we always used the maximum batch size possible on an NVIDIA A100-SXM4-40GB GPU, with double the number of points for the residual loss compared to the boundary loss. To mitigate overfitting, the batch was resampled every 50 training steps (every 20 steps for the first 8000 steps in the case of the scattering PINN). The parameters that are computationally more expensive to sample (mainly the α_a-profiles, but also the two radius profile parameters) were sampled pre-training with a size of 10^7. During training, these parameters were then sampled from this saved array and loaded onto the GPU, thereby accelerating the process.

Throughout the training of the absorption PINNs, the inputs were resampled 100 times. Considering the number of points for which the residual loss is computed (2^21 or 2^22), the total number of input combinations amounts to roughly 2.1 × 10^8 or 4.2 × 10^8, depending on the network size. For the scattering PINN, which was resampled 400 times and repeated thereafter, with 2^19 residual points, the model was similarly trained on approximately 2.1 × 10^8 unique input combinations.

For the absorption PINN, a hyperparameter grid search is used to optimize the number of layers (4 or 6) as well as the width of each layer (32, 64, or 128). A PINN with more free parameters may improve our result even more, but the 6x128 network already needed more time to calculate a spectrum than the classical numerical method, so we decided against trying out a deeper or wider network. For each configuration, 7 PINNs were trained with different random Xavier initialization. They were evaluated on 95 test spectra with the help of petitRADTRANS <cit.>.

The number of subsequent layers of the scattering PINN were not optimized, although one layer was insufficient to accurately model the correct solution.




§ RESULTS




 §.§ Absorption PINN








After trying out what maximum batch size is possible for each hyperparameter configuration, we found that the three models with the most adjustable parameters (6x64, 4x128, 6x128) are able to utilize a batch size of 3 × 2^20 while the three smaller models (4x32, 6x32, 4x64) could use double the batch size (3 × 2^21).

The distribution of the root mean squared errors with respect to each example spectrum can be seen in Fig. <ref>. It seems that the 4x64 model consistently gives us the best result, although no architecture performs significantly worse.

Given that the errors of the smallest networks with a width of 32 were not significantly higher than those of larger networks, we experimented with an even smaller architecture of 4x16 and 4x8. For the 4x16 PINN, this adjustment resulted in a slight increase in error, similar to the worst-performing 6x32 network, and a considerably larger error in the case of the 4x8 PINN, but it produced diminishing returns in terms of speed.

For one of the 4x64 models, we also show the evolution of the different loss components during training in Fig <ref>. Here we see how the weight ω_ℱ of 0.5 allows the network to quickly fulfill the boundary condition, while still giving the residual loss component sufficient weight to decrease over the epochs. The final value of this residual loss log_10(ℒ_ℱ) = -4.0 already gives us a first order of estimate for the errors of the PINN of 10^-2.

We also checked for possible correlations of the error with any atmospheric parameters. We only found that for really low radii, our formula for the radius profile r(P) breaks down for the defined pressure range, yielding inaccurate results.

Fig. <ref> presents an example spectrum, notably incorporating sodium (Na) and potassium (K). These elements serve as robust indicators for assessing the PINN's capacity to generalize beyond its initial training data, due to their potential for exhibiting abrupt increases in absorption coefficients at specific atmospheric layers. During the training phase, the α_a(P) profiles we generated typically did not showcase logarithmic gradients ln(α_a)/ln(P) > 2. However, for these atomic species, the gradient between two layers can surge to values well above 2, occasionally surpassing 10. Such dramatic shifts can result in the optical depths of subsequent layers abruptly increasing from negligible to substantial values. Although α_a(P) profiles with such non-differentiable characteristics were not included in the training due to the complexity they introduced in minimizing the residual loss, the PINN demonstrates a commendable ability to approximate accurate solutions, even though this results in slightly larger errors than for known α_a(P) profiles. This success underscores the network's adaptability and effectiveness, showcasing its capability to handle complex scenarios that extend beyond the initial training data.

While the observed accuracy gives us confidence in our method, it does not offer any improvement in performance. Using the larger models of width 128 can even significantly slow down the simulation of a transit spectrum compared to the numerical method.

This contrasts with the results of <cit.>, who could significantly speed up their forward model using a neural network. But their method directly calculates the spectrum (in a fixed spectral range and spectral resolution) from the atmospheric parameters (also fixed in their number and range). Our PINN, on the other hand, calculates the transmission function u(y) for every individual α_a(P) profile, so for every wavelength and every source of absorption. Depending on the number of wavelength points and absorbing species, this can mean that the PINN has to calculate the solution in multiple batches, significantly reducing its speed. The advantage of our model lies within its versatility, as it is not trained on, e.g., specific molecules.




 §.§ Scattering PINN


Since we cannot easily compare the spectra obtained from the scattering PINN to the ones from a traditional numerical method, which do not take scattering into account, evaluating its performance is harder to do.

In Fig. <ref>, we can see an example solution with the two components, the absorbed u_a and scattered light u_s separated, as well as the corresponding residuals. These residuals show how well the RT equation is fulfilled and give us an estimate of the error. In the example, we can see residuals on the order of 10^-2 spanning a significant fraction of the considered slab.

Additionally, we compared our method to a fixed PINN for the same example, the result of which can be seen in Fig. <ref>. By fixed PINN, we mean that all the parameters of this PINN have been fixed to certain values during training, apart from the spacial coordinates x, y, ϕ (and Δ_*, which cannot be fixed because of the boundary condition at y=-1). Leaving the batch size the same should yield a solution of higher accuracy, as the fixed PINN has to explore a significantly smaller parameter space. Similarly to the absorption PINN, we can see errors of around 1%.

Finally, we want to explore the usefulness of this method by plotting an example spectrum (see Fig. <ref>). Even for the extreme case of Δ_* = 0.15π, which would correspond to an exoplanet orbiting a sun-like star (R_* = R_⊙) at a distance of a = 0.01 AU, the Rayleigh scattering barely influences the spectrum. This is because our assumptions for calculating the scattering integral only hold for small angles. Thus, while the scattering PINN is able to find an accurate solution, the approximation of parameterizing our problem with only one angle is not sufficient to accurately model Rayleigh scattering for large angular extents of its host star, which is when it becomes relevant.









§ LIMITATIONS
 

The assumptions under which both the absorption and scattering PINN operate include isothermality of the atmosphere and a radius range of R_0 ∈ [0.2, 2.0] R_ J.

In the case of the scattering PINN we further assume an orbital inclination of i = 90^∘. It also focuses narrowly on pressure-independent scattering cross-sections, characteristic of Rayleigh scattering from hydrogen and helium, and does not consider any absorbing species simultaneously being present in the atmosphere. Additionally, it lacks an exact lower boundary condition. The most significant shortcoming is the small angle approximation for the calculation of the scattering integral, which is ultimately insufficient to accurately model Rayleigh scattering.

Lastly, the fully-connected layers of both PINNs offer no interpretability, although the special architecture of the scattering PINN at least simplifies differentiating the diffuse light component from the total solution.




§ CONCLUSIONS


In this study, we have assessed the effectiveness of PINNs in simulating complex atmospheric phenomena like scattering, which are traditionally challenging and resource-intensive to model.

The absorption PINN demonstrated in this research exhibits a high level of precision in forecasting atmospheric transmission for transiting exoplanets, achieving an error margin within a few percent, even when confronted with unfamiliar non-differentiable α_a(P) profiles. This accuracy underscores the model's robustness and reliability in complex atmospheric simulations. However, it is worth noting that, in terms of computational speed, this method does not yet surpass the efficiency of existing numerical techniques. This opens up opportunities for further optimization and development in the realm of PINN methodologies to enhance their speed performance while maintaining their high accuracy. The use of other architectures than the simple MLP chosen in this work, could aid in achieving this. A 1D-CNN for the α_a inputs could, for example, significantly reduce the number of weights in the network.

The Rayleigh scattering PINN in this study, while showcasing potential, operates within certain constraints, as summarized in Section <ref>, that currently limit its broader applicability. Having these caveats in mind, nevertheless, a parameterized PINN could prove effective in modeling scattering processes in the atmosphere with a minimal additional computational cost beyond the training phase.

In this work, we have laid out one possible path of how to present this complex problem to a neural network. Here we want to emphasize the key ingredients for successfully training a PINN:

    
  * especially when dealing with a complicated differential equation, one has to carefully consider all possible configurations of parameters, how they affect the absolute values of the respective residual and normalize all cases accordingly
    
  * similar to the insight from <cit.>, it is important to respect causality when choosing weights for different parts of the loss function
    
  * having a larger batch size generally improves the accuracy of a PINN
    
  * incorporating physical knowledge into, e.g., the architecture of the PINN can significantly simplify finding the correct solution, especially if any non-differentiable boundaries are involved


However, further enhancements to this approach are necessary to calculate the correct solution to the RTE with scattering, where it becomes relevant for transit spectra. E.g., auxiliary PINNs <cit.> could aid in expanding the problem to a second angular dimension (θ), while making it possible to evaluate the scattering integral without any numerical integration. Experimenting with alternative architectures like FNOs <cit.> could also make addressing these challenges more feasible.

In the future, the inclusion of specific pressure-temperature profiles can be achieved by simply extending the input parameters, though this might slightly impact the model's accuracy due to the increased dimensionality of the problem. Integrating other scattering sources, such as clouds with their variable α(P) profiles and complex scattering phase functions Φ(θ, θ'), presents a more significant challenge. One major concern is the possible non-differentiability of cloud decks. But, as the absorption model's predictions for Na and K show, a PINN might be able to overcome such difficulties.




§ ACKNOWLEDGEMENTS


We thank the referee for the helpful comments.
We also thank Lukas Heinrich, Nicole Hartman and the ORIGINS Data Science Lab (ODSL) for valuable discussions and hardware support during the development of the project, which was made possible through the funding of the ORIGINS Excellence Cluster by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2094 - 390783311; in particular through ORIGINS Excellence Cluster's Seed Money Program.

This research has made use of the NASA Astrophysics Data System,
and the Python packages NumPy, SciPy, Matplotlib, and PyTorch.




§ DATA AVAILABILITY


The code, based on <cit.>, is available at <https://github.com/DavidDahlbudding/AtmosphericScatteringPinn> (commit https://github.com/DavidDahlbudding/AtmosphericScatteringPinn/tree/3c805f5f4a67be9caf76c644c91144fda956d6ea3c805f5). The respective models and other data can be downloaded from <https://zenodo.org/doi/10.5281/zenodo.10727497>.





mnras
 



















§ THE MULTI-LAYER PERCEPTRON


A feedforward neural network or multilayer perceptron (MLP) is a versatile approximator that maps x to y = f(x,θ), where θ defines the free network parameters that are adjusted during training.

What makes these standard MLPs so versatile is just simple linear algebra: a linear mapping W (the weights), a displacement b (the biases) and, most importantly, a non-linear activation function σ. The output y of one such a layer f_1 can then be computed with:


    y = f_1 (x) = σ(Wx + b),

where the activation function σ is applied element-wise: σ((x_1, x_2)^T) = (σ(x_1), σ(x_2))^T. Multiple of these layers can be stacked on top of each other to be able to represent highly complex functions between the input x and the output y = f_n∘⋯∘ f_2 ∘ f_1 (x) (where ∘ denotes a function composition: f_2 ∘ f_1 (x) = f_2(f_1 (x))) <cit.>. The layers between the input and the output layer are called the hidden layers and the number of hidden layers is called the depth of the network, while the number of nodes per layer is called the width. This simple architecture is also mostly used in this paper.

To train a neural network, we need another important function, the so-called objective or loss function. This function should be minimized to achieve a desired goal (e.g., the mean squared error when fitting a function to given data points). Then, the gradient of the loss function with respect to the weights and biases can be cumulatively calculated for a batch of data using automatic differentiation, and the neural network can "move" towards a minimum by adjusting its parameters in small steps in the negative direction of this gradient.




§ THE SCATTERING INTEGRAL
 

As mentioned in Section <ref> we need to make a few approximations valid for small angular extents of the star for the problem to be parameterizable by only one angle (ϕ). Here, we choose different symmetries for the two components of the radiant flux: u_a and u_s. Because they are added together to get the total solution to the RTE, we can calculate the total integral by adding the two integrals for each component:


    ∫Φ  u  Ω = ∫Φ  u_a  Ω + ∫Φ  u_s  Ω


For the absorption component u_a we choose the conventional spherical (local) coordinate system, such that the angle ϕ∈ [-π, π] lies in the x-y-plane and θ∈ [0,π] is the angle relative to the z-axis. If we now take a fixed ϕ, let θ vary and expand this into a plain, the intersection of this plain with the sphere of an atmospheric layer is a circle. For small angular extents of the star each light ray for varying θ travels roughly the same path through this circle, which is why we assume the symmetry of u_a(ϕ, θ) ≈ u_a(ϕ, θ=π/2).

We can parameterize θ' in terms of ϕ' such that we can integrate over the complete solid angle of the star with angular extent 2 Δ_*, for which u_a ≠ 0, by using the law of cosines with θ = π/2:


    cosΔ = sinθ' cos(ϕ-ϕ').


Inserting this into the scattering phase function Φ(Δ) (<ref>) and the integration limits, yields:


    ∫_-Δ_*^Δ_*ϕ'   u_a(ϕ') ∫_-θ'(ϕ')^θ'(ϕ')θ'   |sinθ'|  Φ(Δ) =
       ∫_-Δ_*^Δ_* u_a(ϕ')  3/2√(1-a^2)( cos^2(ϕ-ϕ') ( 1-a^2/3 - 1) - 1 ) ϕ',
       where  a := cosΔ_*/cosϕ'


For exploiting a symmetry of the scattering component we have to rotate our local coordinate system. The ϕ_s, coordinate now lies in the y-z-plane and θ_s is the angle relative to the x-axis. The absorption component u_a therefore is a concentrated beam of light around θ_s = 0 for small Δ_*. Since the scattering phase function is only dependent on the angular distance Δ, relative to θ=0, we get Δ = θ. Hence we can approximate, while allowing for some slight asymmetry: u_s(θ_s, ϕ_s ≶ 0) ≈ u_s(θ_s, ϕ=±π/2). The integral can therefore be calculated using:


    ∬Φ(Δ)   u_s(θ_s'   , ϕ_s')   |sinθ_s'|  θ_s'  ϕ_s' =
    ∫_0^π |sinθ_s'|  θ_s'    [ u_s(θ_s', ϕ_s'=-π/2) ∫_-π^0 Φ(Δ) dϕ_s' + .
       .   u_s(θ_s', ϕ_s'=π/2) ∫_0^πΦ(Δ) dϕ_s' ] .


The integral over ϕ_s' can again be integrated analytically by inserting the law of cosines (assuming ϕ_s = ±π/2) into the scattering phase function Φ(Δ) (<ref>):


    ∫_0^±π   Φ(Δ) dϕ_s' =
    ±3/4   [ π( 1 + cosθ_s cosθ_s' ±1/2sinθ_s sinθ_s') .
       . ∓sgn(ϕ_s) sin(2θ_s) sin(2θ_s') ] .


The numerical integration employs the Trapezoidal rule with a total of 19 sample points in the interval [-π,π], where 5 of these points are equally spaced in the interval [-Δ_*, Δ_*] and the remaining intervals [-π, -Δ_*] and [Δ_*, π] each have another 7 equidistant sample points.




§ RESIDUAL LOSS FOR SCATTERING PINN
 

As mentioned in Section <ref>, we have to divide the gradients in the RTE (Eq. <ref>) with the respective scale lengths l_x and l_y. Given in cm, these parameters are quite large and can therefore diminish the value of the residual loss, especially for low values of α. Hence, we multiply the whole equation with l_x:


    cos(ϕ)  u/ x   + sin(ϕ) l_x/l_y u/ y + l_x α_s · u
    
        - l_x α_s/4π   ·∫Φ(θ, ϕ, θ', ϕ') u(θ', ϕ') dΩ' = res_1.


The remaining problem is that the value of l_x/l_y can be on the order of 10, possibly inflating the value of the residual loss for a certain set of parameters. Because of this, we further normalize this formulation of the residual with norm = |cos(ϕ)| + l_x/l_y |sin(ϕ)|.

The second formulation is similar to the one for the absorption PINN, where we divide the RTE by α_s:


    cos(ϕ)/l_x α_s u/ x   + sin(ϕ)/l_y α_s u/ y + u
    
        - 1/4π   ·∫Φ(θ, ϕ, θ', ϕ') u(θ', ϕ') dΩ' = res_2.


The complete residual is then again the minimum of the two formulations:


    res = min( res_1/norm, res_2 ).











