# Novel Neutron Star Structures with Nucleon Mass Radius

Wei Kou<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  University of Chinese Academy of Sciences,
Beijing 100049, China  Xiaopeng Wang<EMAIL_ADDRESS>Institute of
Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China  University
of Chinese Academy of Sciences, Beijing 100049, China  Lanzhou University,
Lanzhou 730000, China  Xurong Chen  Corresponding author<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  University of Chinese Academy of Sciences, Beijing 100049, China
Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou 510006, China

###### Abstract

In order to reveal the difference between the latest neutron star observation
experiment GW170817 and the existing theory, we mainly consider the effect of
the nucleon radius on the neutron star from the existing theory. We believe
that the effect of nucleon radius in neutron star is not negligible, and the
mass radius of nucleon should be used instead of the charge radius. The
nucleon mass radius is set as $r_{m}=0.55\pm 0.09$ fm from the new
measurements. It is considered as an input to the “Excluded Volume Effects”
model in the equation of state of nuclear matter. We propose a novel neutron
star mass-radius relation by using proton mass radius is consistent with the
observation GW170817.

###### pacs:

24.85.+p, 13.60.Hb, 13.85.Qk

##  I Introduction

The properties of neutron stars (NS) are determined by their internal
components. The formation of NS is a hot spot problem in which the nuclear
physics, particle physics and astrophysics and their combinations. The
discovery of GW170817 Capano  _et al._ (2020), a pair of NS, raises the climax
of the research. In general, NS are the densest observable objects in the
universe, and they are excellent natural laboratories for studying the state
of matter under extreme conditions in nuclear physics and Quantum
Chromodynamics (QCD). As a key input to determine the structure of a NS, the
equation of state (EOS) of dense nuclear matter is particularly important.

EOS of nuclear matter is a good means to explore the structures and properties
of NS. A number of models have been developed in recent decades to explain the
astrophysical observations Özel and Freire (2016). Most models are based on
Thomas-Fermi model (See Ref. Shen  _et al._ (2011)) or Relativistic Mean Field
(RMF) model Serot and Walecka (1986) and Hartree-Fock model Machleidt (1989).
To obtain the EOS of the nuclear matter, the RMF model is based on the single
boson exchange mechanism of the nuclear force, and the relativistic Lagrangian
that satisfies the basic symmetry is constructed. The meson fields are
approximated as the mean fields in the multi-body problems, and the ground
state and excited state properties of medium-mass nuclei are described
successfully (See the Sec. II.1 for more).

As a typical application of RMF, the NL3 Lalazissis  _et al._ (1997)
parameters set gives a good description of the EOS of nuclear matter, but it
has a slight deviation when calculating the structures and properties of NS.
The prediction for a NS radius by NL3 which have 1.4 solar masses is about
14-15 km. But in the most recent experimental observation, GW170817, the
radius of a NS with the same mass was measured which is about
$R_{1.4M_{\odot}}=11.0_{-0.6}^{+0.9}$ km (90$\%$ credible interval) Capano
_et al._ (2020). In the Ref. Fattoyev  _et al._ (2010); Mart and Sulaksono
(2013); Suparti  _et al._ (2017), the authors used a new parameters set,
IUFSU, and considered the effects of the nucleon radius on the structure of
the NS. Their results are close to the latest observations. But the past two
years ago, other authors Zhu and Li (2018); Zhu  _et al._ (2019) have used the
quark mean field bag model to calculate the NS radius on the basis of previous
studies. The latter thought that the nucleon radius had little effect on the
structures of NS. There are many models and experiments for NS radius
predictions. For more information, please refer to the review article Özel and
Freire (2016).

Nucleons radii are the elementary physical quantity of themselves. They have
been measured for a long history. The current data about the “Charge” radius
of the proton is fixed at $r_{ch}=0.8409\pm 0.0004$ fm and “Magnetic” radius
of the neutron at $r_{mag}=0.864\pm 0.008$ fm Zyla  _et al._ (2020). In
previous work we discussed in last paragraph, the authors introduced the
charge or magnetic radii of the nucleons into EOS and obtained different
results. But in dense nuclear matter, the nucleons are very close together at
a narrow space. The strong interaction is dominant and the “Range” of the
medium particles transmitting the strong interaction is smaller than the
nucleon charge radius. Neutron stars, on the other hand, are dominated by
neutrons. Neutrons are electrically neutral and therefore do not participate
in electromagnetic interactions. The authors of Ref. Zhu and Li (2018) believe
that the viewpoint in Ref. Mart and Sulaksono (2013); Suparti  _et al._ (2017)
is not accurate enough, and they believe that the free radius of nucleons has
little influence on the calculation results of the whole NS. In the latter
literature Suparti  _et al._ (2017), RMF parameters were re-fitted with
different nucleons charge radii.

As stated in the previous discussions, the effects of nucleons charge and
magnetic radii are small in dense nuclear matter, which are consistent with
the conclusions in Ref. Zhu and Li (2018). But they are not all stories. In
recent years, the problems of the origin of proton mass has attracted much
attention Ji  _et al._ (2021); Ji and Liu (2021); Ji (2021); Kharzeev (2021);
Wang  _et al._ (2021); Kou  _et al._ (2021a, b). The mass radius of the proton
is introduced into the researches of QCD. In our previous work Wang  _et al._
(2021), we used the vector meson photoproduction data near the threshold Barth
_et al._ (2003); Mibe  _et al._ (2005); Ali  _et al._ (2019) to obtain the
gravitational form factor and the mass radius of proton. It is about
$r_{m}=0.55\pm 0.09$ fm and obviously smaller than charge one. On this basis,
we believe that the nucleon radius has an influence on the structure of the
NS, but it should be dominated by the mass radius of the nucleon rather than
the charge one.

This paper is divided into the following parts: In Sec. II, we mainly
introduced the idea of RMF to calculate the EOS of nuclear matter, and adopted
excluded volume effects (EVE) model Rischke  _et al._ (1991); Kouno  _et al._
(1996) to add the mass radius of nucleons from Ref. Kharzeev (2021); Wang  _et
al._ (2021) as the correction. At the end of Sec. II, we briefly explained how
to get the radius and mass information of NS from EOS. In the Sec. III we
present the results of our work and the discussions. The last section is the
summary.

##  II FORMALISM

###  II.1 Relativistic Mean Field Theory

Using the relativistic mean field (RMF) theory Serot and Walecka (1986) has
worked well in describing the EOS of NS. To carry out the study of nuclear
matter, the meson-exchange model should be introduced in standard Lagrangian
to describe the nucleon-nucleon interaction. The starting point of RMF theory
is a standard Lagrangian density Gambhir  _et al._ (1990); Lalazissis  _et
al._ (1997)

$\displaystyle\mathcal{L}=$
$\displaystyle\bar{\psi}\left(\gamma\left(i\partial-g_{\omega}\omega-
g_{\rho}\vec{\rho}\vec{\tau}-eA\right)-m-g_{\sigma}\sigma\right)\psi$ (1)
$\displaystyle+\frac{1}{2}(\partial\sigma)^{2}-U(\sigma)-\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega^{2}$
$\displaystyle-\frac{1}{4}\overrightarrow{\mathrm{R}}_{\mu\nu}\overrightarrow{\mathrm{R}}^{\mu\nu}+\frac{1}{2}m_{\rho}^{2}\vec{\rho}^{2}$
$\displaystyle-\frac{1}{4}\mathrm{~{}F}_{\mu\nu}\mathrm{F}^{\mu\nu},$

which contains nucleons mass $m$; scalar and vector mesons $\sigma$, $\omega$
and $\rho$; the electromagnetic field; and nonlinear self-interactions of the
scalar $\sigma$ field,

$U(\sigma)=\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{3}g_{2}\sigma^{3}+\frac{1}{4}g_{3}\sigma^{4}.$
(2)

The total antisymmetric tensors of the vector mesons can be written as

$\displaystyle\Omega_{\mu\nu}$
$\displaystyle=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu},$ (3)
$\displaystyle\overrightarrow{\mathrm{R}}_{\mu\nu}$
$\displaystyle=\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu}.$

Generally speaking, each parameter in Lagrangian is given by fitting the
experimental data of nuclear physics. Over the last few decades, physicists
have used some of the following parameter sets most frequently: NL1 Reinhard
_et al._ (1986), NL3 Lalazissis  _et al._ (1997), NL-SH Sharma  _et al._
(1993), PL-40 Sharma and Ring (1992), FSU Todd-Rutel and Piekarewicz (2005)
and its update version IU-FSU Fattoyev  _et al._ (2010). The last one
considered the vector mesons self-interactions and $\rho-\omega$ couplings, so
it has more parameters to describe the interactions in nuclear matter system.
Since our focus is on how much the radius of the nucleons corrects the EOS of
nuclear matter, in this work we only consider simple scalar meson self-
interactions by using the NL3 parameters set. In NL3 parameters set, we use
$m=939$ MeV, $m_{\sigma}=508.194$ MeV, $m_{\omega}=782.501$ MeV and
$m_{\rho}=763$ MeV. Table 1 represents the fundamental parameters by NL3 model
Lalazissis  _et al._ (1997) which describe the coupling constant of meson-
exchange interactions.

Table 1: Parameter sets for the NL3 models discussed in the text. $g_{2}$ (fm-1) |  $g_{3}$ (MeV) |  $g_{\sigma}$ (MeV) |  $g_{\omega}$ (MeV) |  $g_{\rho}$ (MeV)
---|---|---|---|---
$-10.431$ | $-28.885$ | $10.217$ | $12.868$ | $4.474$

Based on the previous discussions, we easily obtain the energy density of the
nuclear matter system by Legendre transformation of the Lagrangian. In the
same ways we can also get the filed equations of nucleons and mesons by using
the Euler-Lagrange equation. We will discuss them in Sec. II.2.

###  II.2 EOS by Correction from Nucleons Mass Radii

Before calculating the EOS of nuclear matter, we first consider the
thermodynamic properties of nuclear matter. In multi-body physics, of course,
we cannot solve the motion states of every particle. Therefore, in the multi-
body nuclear matter, a large number of nucleons are treated as approximate the
ideal Fermi gases. One can use thermodynamic quantities to describe the
properties of nucleons behaviors, such as chemical potential and pressure and
so on. In this work, we are inspired by previous researches Rischke  _et al._
(1991); Kouno  _et al._ (1996); Mart and Sulaksono (2013); Suparti  _et al._
(2017); Zhu and Li (2018); Zhu  _et al._ (2019) and we also think that the
radii of the nucleons have effects on the structure of the NS – mass-radius
relations.

The effect of nucleon volume on the EOS of nuclear matter is introduced by
Ref. Rischke  _et al._ (1991); Kouno  _et al._ (1996). In the mean field
theory for point-like nucleons, at zero temperature, the baryon density
$\rho_{B}$ is given as

$\rho_{B}=\frac{p_{F}^{3}}{3\pi^{2}},$ (4)

where $p_{F}$ is the Fermi momenta of nucleon $B$ and $B=n,\ p$. In general
case the nuclear matter has $N$ nucleons and a volume $V$ or $\rho=N/V$. If we
consider the volume effect of the nucleons then we must make a correction for
the density. In the EVE model Rischke  _et al._ (1991); Kouno  _et al._
(1996), the volume of nuclear matter $V$ for a system of $N$ nucleons in
configurational space is reduced to an effective one, $V^{\prime}=V-NV_{n}$,
where $V_{n}=4\pi r_{m}^{3}/3$ is the volume of a nucleon and $r_{m}$ is the
nucleon mass radius which can be set as $0.55\pm 0.09$ fm Kharzeev (2021);
Wang  _et al._ (2021). In other words, nucleons of volume $V_{n}$ in volume
$V$ are described as a “point-like particle” and are moving thermally in an
effective volume $V^{\prime}$. In this case, we can redefine the particle
density as $\rho^{\prime}=N/V^{\prime}$ which is equal to the density
$\rho_{B}$ for the given $k_{F}$. In this way, the true nucleon density in
nuclear matter is written as follows:

$\rho=\frac{\rho^{\prime}}{1+V_{n}\rho^{\prime}}.$ (5)

Because of the consistency of thermodynamic quantities, the scalar density of
the nucleons in the whole system can also be written as

$\rho_{s}=\frac{\rho_{s}^{\prime}}{1+V_{n}\rho^{\prime}},$ (6)

where

$\displaystyle\rho_{s}^{\prime}$
$\displaystyle=\frac{1}{\pi^{2}}\sum_{i=n,p}\int_{0}^{p_{F}^{i}}dpp_{i}^{2}\frac{M_{N}^{*}}{\sqrt{M_{N}^{*2}+p_{i}^{2}}}$
(7)
$\displaystyle=\frac{M_{N}^{*}}{2\pi^{2}}\left(p_{F}^{i}E_{F}^{i}-M_{N}^{*2}\ln\left|\frac{p_{F}^{i}+E_{F}^{i}}{M_{N}^{*}}\right|\right),$
$\displaystyle E_{F}^{i}$
$\displaystyle=\sqrt{M_{N}^{*2}+\left(p_{F}^{i}\right)^{2}},$

and the effective mass $M_{N}^{*}=m+g_{\sigma}\sigma$ for nucleons come from
Eq. (1), $p_{F}^{n}(p_{F}^{p})$ come from Eq. (4) but have the corrections
with EVE model Rischke  _et al._ (1991); Kouno  _et al._ (1996).

In the Lagrangian density corresponding to nuclear matter, the meson fields
exist in the form of quantum fields. Because it is very difficult to calculate
the quantum fields of multi-body, the mean field approximation method is
needed to replace the meson fields with the mean fields and change the field
equations into an algebraic equations to solve easily. Applying the Euler-
Lagrange equation

$\frac{\partial\mathcal{L}}{\partial\phi}-\partial^{\mu}\left[\frac{\partial\mathcal{L}}{\partial\left(\partial^{\mu}\phi\right)}\right]=0,$
(8)

we can then take the different fields and substitute them into Eq. (8) to get
the equations of motion for each mean field Zhu and Li (2018); Zhu  _et al._
(2019):

$\displaystyle m_{\sigma}^{2}\sigma+g_{2}\sigma^{2}+g_{3}\sigma^{3}$
$\displaystyle=-\frac{\partial M_{N}^{*}}{\partial\sigma}\rho_{s},$ (9)
$\displaystyle m_{\omega}^{2}\omega$ $\displaystyle=g_{\omega}\rho_{N},$
$\displaystyle m_{\rho}^{2}\vec{\rho}$ $\displaystyle=g_{\rho}\rho_{3},$

where $\rho_{N}=\rho_{p}+\rho_{n}$, and $\rho_{3}=\rho_{p}-\rho_{n}$ that
equals 0 in symmetric nuclear matter.

After the mean field approximation, Eq. (9) has become algebraic equations,
solving them and directly obtaining the information of the meson fields. To
calculate the nuclear matter EOS, we should get the energy density and
pressure from Lagrangian. They can be generated by Legendre transformation
from Eq. (1) Zhu and Li (2018); Zhu  _et al._ (2019):

$\displaystyle\mathcal{E}=$
$\displaystyle\frac{1}{\pi^{2}}\sum_{i=n,p}\int_{0}^{p_{F}^{i}}\sqrt{p^{2}+M_{N}^{*2}}p^{2}dp$
(10)
$\displaystyle+\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{3}g_{2}\sigma^{3}+\frac{1}{4}g_{3}\sigma^{4}-\frac{1}{2}m_{\omega}^{2}\omega^{2}$
$\displaystyle-\frac{1}{2}m_{\rho}^{2}\vec{\rho}^{2},$ $\displaystyle P=$
$\displaystyle\frac{1}{3\pi^{2}}\sum_{i=n,p}\int_{0}^{p_{F}^{i}}\frac{p^{4}}{\sqrt{p^{2}+M_{N}^{*2}}}dp-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}$
(11)
$\displaystyle-\frac{1}{3}g_{2}\sigma^{3}-\frac{1}{4}g_{3}\sigma^{4}+\frac{1}{2}m_{\omega}^{2}\omega^{2}+\frac{1}{2}m_{\rho}^{2}\vec{\rho}^{2}.$

From Eq. (10) and (11), the properties of nuclear matter can be determined.

In the earlier part of this section we discussed the standard method for
calculating the EOS of nuclear matter. We also introduced the form that the
nucleon number density should satisfy under the EVE model. If we consider the
nonnegligible nucleon volume, then the modified nucleon density in EVE model
should be used instead of the original Fermi gas model:

$p_{F}=(3\pi^{2}\rho^{\prime})^{\frac{1}{3}},$ (12)

Eq. (6) and (7) should also be considered.

###  II.3 Neutron Star Structure

The nuclear matter EOS we considered only have the nucleons and mesons
interactions. In order to get the NS matter, the leptons contributions must be
considered to balance the positive charge of the proton and chemical potential
of nuclear matter. The Lagrangian density of leptons can be written as

$\mathcal{L}_{\text{lep
}}=\sum_{l=e,\mu}\bar{\Psi}_{l}\left(i\gamma_{\mu}\partial^{\mu}-m_{l}\right)\Psi_{l}.$
(13)

Since $\tau$ lepton has too large mass and decays easily, we only consider the
first and second generation leptons – $e,\ \mu$. The contributions from $e$
and $\mu$ for energy density and pressure are in the same way with Eq. (10,
11):

$\displaystyle\mathcal{E}_{\text{lep
}}=\frac{1}{\pi^{2}}\sum_{l=e,\mu}\int_{0}^{p_{F}^{l}}\sqrt{p^{2}+m_{l}^{2}}p^{2}dp,$
(14) $\displaystyle P_{\text{lep
}}=\frac{1}{3\pi^{2}}\sum_{l=e,\mu}\int_{0}^{p_{F}^{l}}\frac{p^{4}}{\sqrt{p^{2}+m_{l}^{2}}}dp.$

If we consider the conditions of $\beta$ equilibrium and charge neutrality, so
we have

$\rho_{p}-\rho_{e}-\rho_{\mu}=0,$ (15)

and

$\displaystyle\mu_{p}+\mu_{e}$ $\displaystyle=\mu_{n},$ (16)
$\displaystyle\mu_{e}$ $\displaystyle=\mu_{\mu},$

where Kouno  _et al._ (1996)

$\mu_{i}=E_{F,i}^{*}+V_{i}P_{i}^{\prime}+g_{\omega}\omega\pm\frac{1}{2}g_{\rho}\vec{\rho}$
(17)

is the chemical potential of nucleon with $V_{p}=V_{n}=4\pi r_{m}^{3}/3$,
“$+$” for proton and “$-$” for neutron. The function $P_{i}^{\prime}$ can be
described as Kouno  _et al._ (1996); Mart and Sulaksono (2013); Suparti  _et
al._ (2017)

$\displaystyle P_{i}^{\prime}=$
$\displaystyle\frac{1}{12\pi^{2}}\left[E_{F}^{i}p_{F}^{i}\left(E_{F}^{i2}-\frac{5}{2}M_{N}^{*2}\right)\right.$
(18)
$\displaystyle\left.+\frac{3}{2}M_{N}^{*4}\ln\left(\frac{p_{F}^{i}+E_{F}^{i*}}{M_{N}^{*}}\right)\right].$

Since leptons are point particles, their chemical potential can be written as

$\mu_{e(\mu)}=\sqrt{m_{e(\mu)}^{2}+p_{F}^{e(\mu)2}}.$ (19)

According Eq. (15) to (19) the number density of nucleons and leptons could be
obtained to generate the NS matter EOS.

To calculate the NS properties such like mass and radius of NS, we should
consider the slowly rotating system must have spherical symmetry
representation. Assuming that the NS matter is a perfect fluid, one can obtain
the Tolman- Oppenhaimer-Volkoff (TOV) equation Tolman (1939); Oppenheimer and
Volkoff (1939) which can describe the Einstein equation with slowly rotating
NS metric Idrisy  _et al._ (2015),

$\displaystyle\frac{dm}{dr}=4\pi\mathcal{E}r^{2},$ (20)
$\displaystyle\frac{dP}{dr}=-G\frac{\mathcal{E}m}{r^{2}}\left(1+\frac{P}{\mathcal{E}}\right)\left(1+\frac{4\pi
r^{3}P}{m}\right)\left(1-\frac{2Gm}{r}\right)^{-1},$

By substituting the EOS of the dense matter, i.e. $P(\mathcal{E})$, into the
stellar structure equation (20), the mass-radius relationship of the NS and
other properties can be obtained. Of course, we assume that the pressure at
the NS radius $R$ is zero, which is $P(R)=0$. Using this condition we can
determine the mass for the radius $R$ of the NS Li  _et al._ (2020).

##  III Results and discussions

In the previous section, we have discussed the idea of calculating the EOS of
NS, and presented the results of revising the mass radius of nucleons in the
EVE model. It is important to note that since NS have the shell structures, we
do not use the uniform EOS, although such works would be interesting. RMF
method is used to calculate the core part of the NS, while BPS Baym  _et al._
(1971a) and BBP Baym  _et al._ (1971b) are used as inputs for the crust part.
The density of nuclear matter at the critical point is $\rho_{c}=0.09371$ fm-3
Baym  _et al._ (1971a, b).

We follow Occam’s razor, the only objective is to research the effect of the
proton mass radius on the structure of NS. Parameters of the EOS of nuclear
matter are consistent with the original NL3 sets Lalazissis  _et al._ (1997).
Our calculations and results are presented in the FIG. 1 and 2.

Figure 1: (Color Online) The EOS which is calculated by EVE correction with
the mass radius of proton (neutron) and NL3 parametrization. The black solid
line is about original NL3 and red is about this work.  Figure 2: (Color
Online) The mass-radius relations of NS. Horizontal shaded band in panel is
the pulsar mass constraint from Ref. Antoniadis  _et al._ (2013). Black solid
line is about original NL3 and red is about this work. The constraint depicted
as a cyan area was taken from Ref. Özel and Freire (2016). Gray band comes
from the uncertainty of the extraction of mass radius of proton Wang  _et al._
(2021). Particularly, the purple one comes from GW170817 measurements Capano
_et al._ (2020).

Figure 1 and 2 are our final results which calculated by EVE correction with
mass radius of proton. From the results we find that our calculation is
reasonable, and is consistent with GW170817 Capano  _et al._ (2020) after
correction under the nucleon radius of $r_{m}=0.55\pm 0.09$ fm. Removing the
EOS for the neutron star crust part, the two results are different but using
the same RMF method and NL3 parameter sets (See FIG. 1). The EOS we obtained
looks a little bit “Harder” than the original one. The ideal Fermi gas model
tells us that nucleons are treated as point particles in an infinite nuclear
matter moving thermally. But the components of nuclear matter inside a NS are
complex. If the nucleon size is not negligible, the density of a finite-volume
NS should be slightly smaller than the EOS of nuclear matter predicts. Based
on previous work, we believe that the mass radius of proton is almost constant
in free space, so they can be thought of as solid spheres packed tightly
inside the NS. Compared with the ideal gas model, a NS of the same mass would
have a smaller nucleon density. As the nucleon density decreases the pressure
inside the NS decreases, too. Now we can go back to Eq. (20), when we solve it
numerically we will get a smaller radius for the NS. So in FIG. 2 the radius
we calculated by nucleon mass radius correction is smaller than the original
NL3.

##  IV Summary

In this work, we applied the mass radius of nucleon extracted from our
previous work and used it to calculate the mass and radius information of the
NS. According the results we obtained, the size of the nucleons in NS must be
considered. On one hand, we believe that EVE model is feasible to modify the
EOS of nuclear matter of NS. But the premise is that the mass radius of the
nucleon is used instead of the charge one. On the other hand, the calculation
result of the radius of NS which has the 1.4 times solar-mass is consistent
with the experiment–GW170817 Capano  _et al._ (2020). The gray error band in
FIG. 2 of our calculation results depends on the uncertainty of the extracted
nucleon mass radius. At present, it seems that the error is relatively large.
In the future, more precision experimental data from Electron-ion Collider
(EIC) Accardi  _et al._ (2016); Abdul Khalek  _et al._ (2021) for US and
Electron-ion Collider in China (EicC) Chen (2018); Chen  _et al._ (2020);
Anderle  _et al._ (2021) will reduce the uncertainty of the results.

###### Acknowledgements.

We are very grateful to Profs. Jianmin DONG and Ang LI for their fruitful
suggestions and the discussions. This work is supported by the Strategic
Priority Research Program of Chinese Academy of Sciences under the Grant NO.
XDB34030301.

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# How Computer Science can Aid Forest Restoration

G. Gordon1, A. Holcomb2, T. Kelly1, S. Keshav1,3, J. Ludlum1, and A.
Madhavapeddy1

1University of Cambridge
2University of Waterloo
3Corresponding author: email<EMAIL_ADDRESS>
The world faces two interlinked crises: climate change and loss of
biodiversity. Forest restoration on degraded lands and surplus croplands can
play a significant role both in sequestering carbon and re-establishing bio-
diversity. There is a considerable body of research and practice that
addresses forest restoration. However, there has been little work by computer
scientists to bring powerful computational techniques to bear on this
important area of work, perhaps due to a lack of awareness. In an attempt to
bridge this gap, we present our vision of how techniques from computer
science, broadly speaking, can aid current practice in forest restoration.1[]
The world faces two interlinked crises: climate change and loss of
biodiversity. Forest restoration on degraded lands and surplus croplands can
play a significant role both in sequestering carbon and re-establishing bio-
diversity. There is a considerable body of research and practice that
addresses forest restoration. However, there has been little work by computer
scientists to bring powerful computational techniques to bear on this
important area of work, perhaps due to a lack of awareness. In an attempt to
bridge this gap, we present our vision of how techniques from computer
science, broadly speaking, can aid current practice in forest restoration.1

##  1 Introduction

We are emerging into a post-COVID world where attention is keenly focused on
the next two global crises: climate change and biodiversity loss. Recognizing
that some critical aspects of modern society can never be fully decarbonised,
a shift to a net zero-carbon economy will necessarily require some level of
carbon capture and sequestration (CCS). Many approaches to CCS have been
proposed, including pumping CO2 into oil wells [56] and trapping it as chalk
precipitate [40]. These solutions, however, do not address the biodiversity
crisis. In contrast,the conservation of existing forests and the restoration
of forests to degraded lands and surplus croplands can play a significant role
both in sequestering carbon and re-establishing biodiversity. In this paper,
we focus only on forest restoration.

To give an idea of the scale involved, the median carbon sequestration
potential of an agroforestry system, which combines agriculture and forestry,
is estimated to be about 95 metric tons (T) per hectare [1] and, if left
undisturbed, can trap this CO2 for decades or even centuries [52]. This is
roughly the equivalent of the emissions from 10 UK households in one year
[29]. Recent work suggests that the earth has the potential to sustain a
trillion more trees planted in approximately 900 million hectares of currently
denuded land [7], with a carbon sequestration potential of 205 GT, which is
comparable to anthropogenic emissions of 1500 GT to date [46, 8]. Thus,
technological interventions to promote reforestation is an attractive approach
to mitigate anthropogenic climate change.

In addition to carbon sequestration, native forests provide many other
benefits both to the environment and to humans. Environmental benefits include
increased biodiversity, air purification, flood prevention, and slope
stabilization. Human benefits include jobs in forestry and agroforestry,
improving public health, providing food, and providing a source of sustainable
building materials. For these reasons, reforestation has become a topic of
great current interest. For example, the World Economic Forum announced in
January 2020 an initiative to plant 1 trillion trees [49].

Unfortunately, reforestation is not simply a matter of planting saplings or
seeds in the ground and walking away [12, 28, 49].

  * •

Reforestation guided purely by economic criteria results in monoculture
plantations that can neither sequester carbon for the long term nor support
biodiversity [28, 34]. They also lack resilience to disease, pests, and
climate change.

  * •

Successful tree-planting, but of fast-growing species such as pine or
eucalypts, can result in not only draining of the water table [37] (and
depletion of soil nutrients in the case of eucalypts) but also increased fire
risk [25].

  * •

A top-down technocratic approach that ignores the land’s owners and other
local actors (who may even have been responsible for deforestation in the
first place) is frequently doomed to failure [14].

  * •

Even multi-species, long-term reforestation motivated by climate credits may
encourage fraud due to lack of trustworthy monitoring [51].

  * •

There is ongoing debate about the accuracy of the models used to project the
CCS benefits from forests [59].

Fortunately, over the past two decades, practitioners in the discipline of
Forest and Landscape Restoration (FLR) have developed a deeper understanding
of reforestation approaches that balance the needs of the environment and
local actors [28, 13, 9, 38, 12]. At the risk of oversimplification, this work
can be summarized as follows.

First, for ecological restoration and carbon sequestration, as opposed to mere
tree harvesting, it is important to combine appropriate policies (such as a
guarantee of long-term land tenure) with institutional frameworks (such as
participatory management of communal forests). Importantly, reforestation
outcomes must be economically sustainable for all stakeholders. To ensure
this, local actors must be engaged participants in the planning process.

Second, there is a continuing need for the improved monitoring of trees, other
vegetation, soil, and wood products at all spatial scales. This monitoring can
not only help identify forests under threat of degradation and deforestation
but also help evaluate forest health, raise the alarm for interventions and
provide a basis for reliable administration of carbon credit payments.

These two areas are precisely where computer science, broadly speaking, can
help FLR practitioners, especially when there is a need for very large scale
and resources are limited. In the remainder of this paper, we present an
outline of the some potential areas for computer scientists to be involved in
forest restoration (Section 2), along with a handful of near-term research
opportunities (Section 3) to demonstrate the great potential for work in this
area. Note that some interventions can reuse existing computational tools,
while others will require fundamentally new techniques.

##  2 FLR lifecycle

Figure 1: The Forest and Landscape Restoration lifecycle.

A forest restoration project goes through three distinct phases [12] (Figure
1). In the planning phase (§2.1), critical decisions need to be made about
where to plant trees (location) and what species to plant (species mix) so as
to restore woodlands in a way that best mimics pre-existing ecosystems. In the
deployment phase (§2.2) trees are planted, potentially along with ground-based
monitoring equipment such as sensors. Finally, in the sustenance phase (§2.3),
the health of both the trees themselves as well as the FLR organization need
to maintained.

We discuss these phases in more detail next, outlining how computer science
can help in each phase, and subsequently cover research opportunities (§3).

###  2.1 Planning

In the planning phase, forestry experts and local actors collaborate to plan
the tree locations, species mix, and time of planting.

Choice of location: Trees would ideally be planted to rehabilitate degraded
lands that are otherwise unsuitable for agriculture or human habitation,
making best use of existing rootstock and undisturbed pockets of mature forest
that can provide animal, plant, and fungal colonizers [28]. At a macroscopic
level, AI-based analysis of satellite data and LiDAR can be used to create 3D
elevation maps to identify potential areas of forest degradation [60, 3].

Once suitable land has been identified, the local climate and micro-climate
must be considered in choosing exact planting locations. Significant factors
include the land slope, levels of rainfall and groundwater flow, wind speed
and flow, different types of soil and different native species of vegetation
[24]. Given the complexity, participation by local actors can help avert poor
decisions (Section 3.2).

At a microscopic level, careful consideration of the existing vegetation and
soil state is required, since disturbing soil actually releases existing
stores of greenhouse gases, as well as increasing the risk of erosion,
particularly if protective soil crusts are destroyed [23]. Moreover, forests
that soak up rainwater for their own growth may result in shriking water
supplies downstream. Hence, mathematical optimization can be used to choose
locations that balance the cost of breaking ground and loss of water supply
with long-term benefits from planting in a particular location. Note that
successful FLR projects need to consider the human element of land function,
social participation, and natural resources management [18, 6]. Optimization
models must, therefore, include the socio-environmental factors attached to
the physical locations, such as the type of land tenure, the current functions
of the land and potential impacts associated with the land in question.

Choice of species mix: Poor choice of trees can further damage the environment
[10]. Moreover, a diverse population of trees is important for the health of a
forest; even a 10% loss of diversity can lead to a 3% loss of productivity
[35], as well as a higher risk of disease [36]. A productive species mixture
emerges if there is complementary resource usage among different trees [26],
and is influenced by the micro-climate [5] and expected future climate [43].
Species diversity complicates the planning process due to the need for more
sophisticated analysis of mapping data [17]. There is some evidence that the
correlation between biodiversity above ground and soil carbon sequestration
rates is dependent on the landscape and climate context [45]. Hence, there is
a need for accurate modelling of multi-species forests (discussed in §3.4).

###  2.2 Deployment

In this phase, both trees and monitoring equipment must be deployed safely
across the restoration area.

Planting: Trees can be planted as saplings, seeds, or pelletised seeds, i.e.,
seeds embedded in a pellet that contains soil, fertilizer, anti-grazing
chemicals, and sometimes a dye for visibility [28]. Planting can be done by
humans, or by helicopters and planes. In many cases, planting must be preceded
by burning weed cover or scarifying the earth using tractors or other
equipment.

Given the labour intensive nature of planting, it is natural to consider the
use of automation. Handling, storage and delivery practices can affect
seedling establishment, and environmental factors and the planting microsite
directly affect the physiological response of saplings immediately after
planting [22]. If the species is one that can be grown from seed, drone fleets
can be used to seed large areas rapidly [16]. It is conceivable that a form of
robotics may be used to plant either seeds or saplings, as well as for weeding
in the first few years111E.g. The Milrem tree planter from Estonia:
https://milremrobotics.com/product/robotic-forester-planter/; this is a
potential topic for future research.

Sensor co-deployment: In addition to planting the trees, sensor technology
should be simultaneously deployed. Remote sensing technologies such as RFID
and LoRA can be used to gather microscopic level data such as soil salinity,
which can aid the identification of vulnerable areas or vegetation that
requires immediate intervention but also play a vital role in ongoing data
collection, feedback and extrapolation. We discuss the types of sensors that
might be deployed in this forested environment later (§3.3).

###  2.3 Sustenance

While many socio-economic and socio-environmental factors determine the long-
term success or failure of an FLR project [31, 32], it is clear that planted
forests need to be monitored for several decades to ensure long-term carbon
sequestration. Organizations responsible for forest management also need to be
financially sustainable.

Forest sustainability: Once seeds or saplings are established and thriving, we
need to ensure their continued health for maximum carbon storage, as well as
broader markers such as biodiversity. Environmental success indicators
include: a suitably high survival rate; the structure of the planted
vegetation (canopy, ground, litter and shrub cover); species diversity; and
ecosystem functions (overall soil health, quantities of groundwater and of
course, carbon sequestration) [31]. All of these indicators need to be
appropriately monitored (§3.3).

Organization viability: Organizations responsible for forest management need
to be financially sustainable and free of fraud. Creating a system of trusted
proofs for carbon certification may be a viable source of funding for the long
term. Local sensor data and global satellite sweeps combined with digitally
verified contracts and transactions could go some way to eliminate the fraud
present in existing carbon certification programs [19]. We discuss some
options later (§3.1).

##  3 Research opportunities

Computer Science has an important role to play in scaling FLR programs to the
global scale required to plan, deploy and sustain a trillion trees over the
coming decades. We now discuss several research opportunities based on current
understanding of FLR (§2).

###  3.1 Trustable carbon sequestration credits

Forest-based carbon sequestration has recently started to attract substantial
funding. For example, the UK Forestry Commission announced the Woodland Carbon
Guarantee [58] in 2019, a £50m scheme to provide landowners with the option to
sell sequestered CO2 from newly planted trees back to the government for a
guaranteed price for the next 30 years. As schemes such as this grow
worldwide, the opportunity for fraud has also followed [19]. Hence, there is
an opportunity to use distributed ledger technology to inject trust into the
system [54]. Automation via trustworthy distributed ledgers [4, 2] would
ensure that such schemes are economically sustainable over the decades needed
to realise the carbon sequestration potential of newly planted forests.

A distributed ledger for carbon sequestration credits would need to register
ownership of a a reforestation project and then reliably track its carbon
sequestration over time. The project owner would be permitted to generate
annual sequestration certificates for the prior year of forest regrowth; if
the project were to fail, then revenues from the sale of these certificates
would dry up, incentivizing the appropriate behaviour [52]. Satellite
observations could be used to reliably observe project outcomes. Moreover,
cloud-based data archiving, curation, and analysis of satellite observations
would provide ongoing detailed information to certificate purchasers about the
progress of the scheme. The design of such a global, scalable, and trustworthy
system is an open area for research.

###  3.2 Visualization and collaboration technology

It is critical to the long-term success of an FLR project that local actors
and policy makers are happy with the eventual outcomes. FLR practitioners must
also elicit knowledge of local conditions and constraints from local
stakeholders, including indigenous communities, during the planning phase
[20]. These two objectives can be met by presenting visualizations of
potential outcomes and using collaboration technology to allow planning
participants to contribute their knowledge.

While technologies for collaboration are well known [48], there are research
opportunities in developing innovative visualization technology specifically
for FLR. For example, state-of-the-art visualization using Augmented Reality
might be beneficial in this context [57]. We believe that AR can help in
creating common ground and make the outcomes more real to participants, policy
makers, and stakeholders. Once local stakeholders are involved in the process,
it is natural to crowdsource data from them to scale our data collection.
Mobile apps that encourage citizen science (such as Seek by iNaturalist and
WWF [50]) could provide ongoing crowdsourced data feeds to augment the
planning and sustenance processes, as well as educate participants about the
progress and benefits they are realizing as the newly planted forests grow.

Visualizations can also be used to present large amounts of data that has been
gathered from satellite- and ground-based observations. One useful direction
may be for computer scientists to construct an online knowledge elicitation,
communication, and collaboration platform to involve all the stakeholders and
work across disciplines [30].

###  3.3 Monitoring forests

Monitoring forests using sensor systems is critical from many perspectives:
deciding on interventions, measuring forest health, and validating carbon
credits. Macroscopic monitoring from satellites in well known, and more
recently mesoscopic monitoring, such as LiDAR scans from planes or drones,
provide an excellent source of data for recognising individual trees [33],
which is vital for trustable sequestration credits and estimating current and
potential carbon sequestration. Further data can be gathered at the
microscopic scale from on-the-ground sensors helping us to determine the
health of individual trees (e.g. identifying disease, insect or animal damage,
or failure to thrive) and enabling rapid intervention if required. As trees
grow, the micro-climate changes, strongly affecting the plant communities [5],
therefore ongoing measurements of temperature, humidity and other
environmental parameters are also needed [61].

However, many sensors in use today in agricultural contexts are not suitable
for long-term mass-scale forest deployment. Forests environments are less
controlled and consistent than agricultural ones, and access to deployed
sensors is comparatively limited. Thus, the sensors need to be self-managing,
robust and long lived. Sensors may make different trade-offs between energy
usage and lifespan. For example, active medium-lifespan sensors (e.g.
spectroscopy for measuring soil quality) used during sapling establishment
could be deployed alongside passive long-lived sensors designed to last for
the expected lifetime of the trees (e.g. temperature and humidity via chipless
RFID [11]). Additional sensors may also be considered outside of those
normally associated with agriculture, for example to account for dead organic
matter, estimate carbon content of soil and litter, and to monitor for
poaching, theft, vandalism and possible fraud. Thus, there is ample scope for
research into appropriate sensor system design.

Another direction of sensor system design is for monitoring, reporting, and
verification (MRV) of forest-based carbon projects. The existing protocol to
measure forest carbon sequestration, for example in the UK Woodland Carbon
Code (WCC) [55], is almost entirely manual, with a suggested material list
that includes a smartphone, GPS or compass, printed paper spreadsheets, and
measuring tape. The process is time-intensive and may be quite costly, with
surveyors recommended to begin their work 6-12 months before the scheduled
verification deadline. Moreover, the UK is not alone: both a 2016 review of
research into REDD+ programs and a 2018 report on forest-based carbon projects
highlighted MRV as a key challenge [41, 21].

In the short term, camera-mounted, GPS-enabled drones with a connection to the
cloud, working alongside human surveyors, might offer a dramatic improvement
for data collection, management, and analysis. Longer term, these labelled
data sets could further the development of machine learning and vision
algorithms to measure tree growth or infer species and health. Eventually,
this data might be integrated with existing satellite or LiDAR imagery, and
could be collected completely autonomously.

###  3.4 Agent-based models for forest simulation

An agent-based model (ABM) can simulate the growth of a multi-species forest
from seed to maturity, taking into account the local soil condition,
environmental variability, and inter-species interaction [39, 44]. It can play
a key role throughout the lifecycle of tree restoration projects, from
planning to sustenance. In the planning phase (§2.1), an ABM allows competing
forest restoration approaches to be evaluated ahead of planting. It may help
determine what species mix to plant in a given area or which land area is most
likely to be successful in reforestation, incorporating concerns spanning
multiple disciplines, such as the level of stakeholder engagement and the soil
characteristics. Robustness stress testing could also be conducted for fire
risks and potential climate changes. Such a simulation tool can also be used
to help sustain a forest project (§2.3), for example by allowing foresters to
simulate interventions for disease management or understand how a forest will
respond to extreme weather events. It can also play a key role in the
financial element of project sustenance. Many carbon credit programs require
demonstrating increased sequestration against a hypothetical baseline [42],
which can be more easily and reliably established through trusted, peer-
reviewed models.

There are existing agent-based modelling packages that allow large-scale
forest simulation, for example TROLL [39], but new CS techniques will be
needed to scale up the simulations and utilize cloud-scale clusters. An ideal
ABM would also allow large scale evaluation of emergent system behaviour while
automatically integrating macroscopic (satellite), mesoscopic (LiDAR) and
microscopic (soil condition) data to calibrate and verify the simulations. For
example, satellite images could be used to identify species types in existing
forest and re-forested areas. These areas could have their growth from
saplings simulated, using our knowledge of growth processes [27, 15, 53],
while controlling error propagation by feeding in high-res satellite imagery
[47]. By adjusting the model’s parameters, the simulation should show the same
crown cover (for some metric of similarity) as reality. This combination of
growth processes with multiple data scales would allow new reforestation
techniques to be accurately developed.

##  4 Conclusions

The role of reforestation in carbon sequestration is well recognized. A naive
approach to reforestation, however, can lead to numerous undesirable
consequences, which can be averted by relying on proven methodologies from the
discipline of Forest and Landscape Restoration. We have identified several
broad areas where computer science tools and techniques can help FLR
practitioners. We also identify CS research problems in the specific areas of
forest simulation, trustable carbon credits, visualization, and monitoring. We
hope that the new tools and techniques created from this research will give
FLR practitioners new degrees of freedom in restoring degraded lands and
combating climate change. We conclude by quoting the UK’s Committee on Climate
Change, which in its recent assessment criterion for new potential land-use
policies asks “Is the [reforestation] policy feasible to implement, monitor
and verify over time?” [29]. By extending the scope of what is feasible,
computer scientists can aid in developing effective reforestation policies not
just in the UK but around the world.

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