# Gamma-Ray Burst Pulses and Lateral Jet Motion

Jon Hakkila  University of Alabama in Huntsville, Department of Physics and
Astronomy, Huntsville, AL 35899, USA  Geoffrey N. Pendleton  DeciBel Research,
Huntsville, AL 35806, USA  Robert D. Preece  University of Alabama in
Huntsville, Department of Space Sciences, Huntsville, AL 35899, USA  Timothy
W. Giblin  511 Space Innovation LLC, Colorado Springs, CO 80919, USA

(Accepted February 28, 2024)

###### Abstract

We propose that gamma-ray burst pulses are produced when highly-relativistic
jets sweep across an observer’s line-of-sight. We hypothesize that
axisymmetric jet profiles, coupled with special relativistic effects, produce
the time-reversed properties of gamma-ray burst pulses. Curvature resulting
from rapid jet expansion is responsible for much of the observed pulse
asymmetry and hard-to-soft evolution. The relative obliqueness with which the
jet crosses the line-of-sight explains the known GRB pulse morphological
types. We explore two scenarios: one in which a rigid/semi-rigid jet moves
laterally, and the other in which a ballistic jet sprays material from a
laterally-moving nozzle. The ballistic jet model is favored based upon its
consistency with standard emission mechanisms.

Gamma-Ray Bursts (629) — Light Curves (918) — Relativistic Jets (1390) —
Special Relativity (1551)

††journal: ApJ

##  1 Introduction

Despite decades of study, the mechanisms responsible for producing gamma-ray
burst (GRB) prompt emission remain elusive. One reason for this is the
inability of theoretical models to provide consistent explanations of GRB
light curves characteristics. In recent years, improved data analysis
techniques have allowed greater insights into GRB light curve behaviors. Some
of the newly-identified behaviors, not found in other types of high energy
transients, place peculiar constraints on GRB jet structure and thus also on
GRB emission mechanisms.

###  1.1 Pulses

Pulses are the basic units of GRB prompt emission (e.g., Norris et al. (1996);
Hakkila et al. (2018b)). FRED (Fast Rise Exponential Decay) pulses were first
recognized in GRBs because they describe emission episodes that most closely
conform to the traditional definition of a pulse. FREDs are generally smooth,
simple, monotonically-increasing and decreasing monotonic pulses. In fact,
modeling approaches have almost exclusively assumed that all smooth pulses are
monotonic (e.g., Norris et al. (1996, 2005); Stern & Svensson (1996); Lee et
al. (2000a, b); Kocevski et al. (2003); Bhat et al. (2012); Nemiroff (2012)),
with shapes ranging from symmetric (equal rise and decay times) to highly
asymmetric (short rise and long decay times). GRB pulses having longer rise
than decay times are so rare that the Norris et al. (2005) model does not
account for them.

Monotonic models have been successfully applied to FRED-like pulses belonging
to GRBs of different duration classes (Norris et al., 1986, 1996; Pendleton et
al., 1997; Norris et al., 2005, 2011; Hakkila & Preece, 2014; Hakkila et al.,
2018a, b), x-ray flares seen in the GRB afterglow phase (Margutti et al.,
2010; Hakkila et al., 2016), and bursts detected by instruments ranging from
BATSE to Suzaku to Swift to Fermi GBM. The approaches have been more
successful describing faint smooth pulses than bright complex ones.

Most GRBs exhibit a bright rapidly-varying component (e.g., Reichart et al.
(2001)) that is separate from background noise. It can be difficult to
separate this variable component from noise in the low photon count
environments in which gamma-ray burst detectors operate. However, in bright
GRBs variability can significantly exceed noise levels.

Over the years, many authors have attempted to describe how variability (or
structure following Hakkila et al. (2018b)) might develop in GRB lightcurves.
These have included models in which prompt emission is produced within
internal shocks (e.g., Paczynski & Xu (1994); Rees & Meszaros (1994);
Kobayashi et al. (1997); Daigne & Mochkovitch (1998); Bošnjak et al. (2009)),
via relativistic turbulence (e.g.,Kumar & Narayan (2009); Lazar et al. (2009);
Narayan & Kumar (2009)), in mini-jets (e.g.,Lyutikov & Blandford (2003);
Yamazaki et al. (2004); Zhang & Zhang (2014)), in fluctuations in black hole
hyperaccretion disks (e.g.,Lin et al. (2016)), and from fluctuations in the
magnetic fields of accretion disks (Lloyd-Ronning et al., 2016). However, the
lack of a consistently-applied variability definition in these studies, along
with the lack of a consistent pulse definition, means that many properties
attributed to variability in one study were attributed to pulses in another
study, and vice versa.

Pulse modeling is made difficult not only by instrumental noise and by the
different pulse fitting algorithms used, but also by attempts to apply models
to data that the algorithms do not adequately fit. The data used in various
studies have been collected from instruments having different threshold
sensitivities, spectral ranges and resolutions, and temporal resolutions. Flux
that is not recognized as belonging to a fitted pulse has been treated
variously as structure, as belonging to other (unfitted) pulses, as
instrumental noise, or as some combination of these. The treatment of pulses
has even been inconsistent within individual studies, especially when data
from multiple instruments have been examined. In order to overcome these
issues, a meaningful delineation between a monotonic pulse and its structure
must be found, and this requires identification of some universal relationship
that can be used to separate the two components.

###  1.2 Time-reversed Structure

In FRED pulses, where the underlying monotonic pulse is easy to model and
subtract out, structure is found to take the form of a three-peaked wave
structure in the residuals (Hakkila & Preece, 2014). The three peaks of the
wave occur during the pulse rise, the pulse maximum, and the pulse decay, such
that the distribution of peaks aligns with the asymmetric shape of the
underlying pulse. The features are clearly seen in the residuals once the
underlying smooth pulse has been fitted and removed using a model such as that
of Norris et al. (2005). These peaks are also associated with spectral
evolution of the pulse. The monotonic components of a FRED pulse undergoes
hard-to-soft evolution while the overall pulse (monotonic pulse plus
structure) experiences a re-hardening at the time of each peak (Hakkila et
al., 2015).

Structure is not only a component of simple FRED pulses; it is also found in
GRB pulses associated with more complex emission episodes. Because pulses
identified in high signal-to-noise ratio data typically contain more structure
than those identified in low signal-to-noise data, brighter, more complex GRB
pulses provide better insights into the nature of structure than do fainter,
simpler FRED pulses (Hakkila et al., 2018b). Structure is washed out at low
signal-to-noise ratios, even when the underlying smooth pulse is not.

A study of bright GRB pulse light curves shows that most exhibit time-
reversible characteristics (Hakkila et al., 2018b). These strange
characteristics occur for both the smooth and structured pulse components.

Monotonic pulse components are all to first order time-reversible. Pulse decay
is the reverse process of pulse rise, with pulse peak intensity providing the
delineation between the two processes. The main thing preventing a pulse’s
decay from appearing to be a time-reversed version of the pulse rise is its
asymmetry. However, an adjustment to the rise time made by stretching it can
cause the rise and decay to appear similar. The similarity can be seen by
folding the stretched pulse light curve at the time of the pulse peak and
aligning the folded rise profile with the decay profile.

GRB pulse structure shares similar time-reversed properties to the monotonic
pulse. Structure occurring prior to a time of reflection ($t_{\rm mirror}$)
appears in reverse order following it Hakkila et al. (2018b); Hakkila &
Nemiroff (2019); Hakkila (2021) and Liu and Zou (2023, preprint). This reverse
ordering of features may also be accompanied by a temporal expansion
(stretching) such that identifiable features found in the structure prior to
$t_{\rm mirror}$ (denoted by the function $f(t_{\rm mirror}-t$)) match closely
with those following it after accounting for a temporal stretching $s_{\rm
mirror}$. In other words, $f(t_{\rm mirror}-t)\approx s_{\rm mirror}f(t-t_{\rm
mirror}))$. This effect is demonstrated in the right panel of Figure 1. The
time of reflection generally occurs near or just after the time that the
monotonic light curve peaks; this time difference is referred to as the
offset. Hakkila (2021) finds that $86\%$ of BATSE GRB pulse light curves can
be fitted by a model of a monotonic smooth pulse combined with a time-reversed
structured component.

Time-reversed structure is demonstrated in Figure 1. Here, the data shown in
the left panel (solid black line) have been fitted by an asymmetric monotonic
pulse model (Norris et al. (2005); shown as a dashed blue line), and the
residuals of this fit shown in the right panel have been fitted with a time-
reversed and stretched version of the same data to produce a combined model
(solid red line) in the left panel. The process is described in detail in
Hakkila (2021).

Figure 1: Example of fitting a GRB pulse and its time-reversed structure. The
left-hand panel demonstrates a smooth, asymmetric fit (dashed blue line) to
the BATSE 64-ms 4-channel counts data of GRB pulse 659 (solid black line)
using the Norris et al. (2005) model. The right-hand panel demonstrates the
time-reversed fit to the residual pulse structure (solid black line) that is
obtained by reversing in time the structure prior to the time of reflection
(dashed red line) and matching it to the structure following the time of
reflection after stretching it by an amount $s_{\rm mirror}$, such that time
intervals align. This pulse, BATSE 659, is an example of the rollercoaster
pulse morphology.

The amount of stretching in the time-reversed structure correlates with the
amount of monotonic pulse stretching associated with the pulse asymmetry.
Using the Norris et al. (2005) pulse model in which symmetric pulses have
$\kappa=0$ and asymmetric pulses have $\kappa=1$, the stretching of the
temporal structure is found to closely align with the pulse asymmetry.
(Hakkila et al., 2018b) find that when the pulse profile is symmetric, the
time-reversed structure is distributed symmetrically around the time of
reflection. In other words, $s_{\rm mirror}\approx 1-\kappa$. The $p-$value of
this anticorrelation is $4.7\times 10^{-20}$ for a sample of 160 BATSE GRB
pulses for which $s_{\rm mirror}$ can be measured. This relationship
demonstrates that the structure and monotonic pulse characteristics are
related, and by linking them together it is clear that both components are
associated with the same event. This relationship can be used to define GRB
pulses as events having monotonic and structured components, both of which are
time-reversed and stretched (Hakkila et al., 2018b; Hakkila & Nemiroff, 2019;
Hakkila, 2021).

Despite the pronounced anti-correlation between $s_{\rm mirror}$ and $\kappa$,
individual pulses are widely dispersed from the $s_{\rm mirror}=1-\kappa$
relationship. This is demonstrated in Figure 2 (data from Hakkila (2021)).
This dispersion is not the result of uncertainty or noise in the measurements,
but is instead produced by morphological variations in GRB light curves
Hakkila (2021).

Figure 2: Morphological GRB pulse classes extracted from an unbiased set of
bright BATSE GRBs (Hakkila, 2021) Pulses exhibit similar light curves based on
relative values of asymmetry $\kappa$ and $s_{\rm mirror}$. FRED pulses (blue
squares; example is shown in the left panel of Figure 3) have large $\kappa$
values and small $s_{\rm mirror}$ values. Rollercoaster pulses (green circles;
example is shown in the left panel of Figure 1. Asymmetric u-pulses (red
crosses; example is shown in the right panel of Figure 3) have large $\kappa$
values and intermediate $s_{\rm mirror}$ values. U-pulses (red stars; example
is shown in the right panel of Figure 4) have near zero $\kappa$ values and
intermediate- to large-$s_{\rm mirror}$ values. Crown pulses (blue stars;
example is shown in the left panel of Figure 4) have near zero $\kappa$ values
and intermediate- to large-$s_{\rm mirror}$ values.

Figure 3: Examples of asymmetric GRB pulse morphologies. BATSE 332 (left
panel) is an example of a FRED pulse. BATSE 678 (right panel) is an example of
an asymmetric u-pulse. BATSE 659 (Figure 1) is an example of a rollercoaster
pulse.

Figure 4: Examples of symmetric GRB pulse morphologies. BATSE 351 (left panel)
is an example of a crown pulse. BATSE 594 (right panel) is an example of an
u-pulse.

The aforementioned morphological types, along with their associated time-
reversed characteristics, appear to be as common among pulses from the short
GRB class as they are among long GRB pulses. This has been somewhat difficult
to ascertain due to the smaller number of photons received from short GRBs.
However, FREDs can be clearly identified in short GRBs, and u-pulses were
first recognized as a morphological pulse type in a study of short GRBs
(Hakkila et al., 2018a). Despite the smaller number of short GRB pulses that
can be fitted by the time-reversed model, similar mechanisms are likely
responsible for producing pulses in both GRB classes.

GRB emission is generally assumed to occur within relativistic jets launched
from newly-formed black holes. Observations supporting jet models have
included breaks in x-ray and optical afterglow lightcurves, as these indicate
that GRBs contain highly beamed emission from relativistic jets (Rhoads, 1999;
Sari et al., 1999). Distances obtained from host galaxy redshift indicate that
GRBs release $10^{48}$ to $10^{52}$ ergs (Frail et al., 2001; Panaitescu &
Kumar, 2001; Berger et al., 2003; Liang et al., 2008; Racusin et al., 2009;
Cenko et al., 2010) of energies on timescales of milliseconds to hundreds of
seconds. Taken together, these observations have led to a standard model in
which GRB prompt emission occurs within relativistic jets emanating from their
progenitors (merging neutron stars for the short GRB class and collapsars for
the long GRB class). GRB jets are generally assumed to point towards the
observer.

Few attempts have been made to incorporate the observed time-reversed
structure of GRB pulse light curves into standard jet models. Simply put, this
is because standard radial jet models do not naturally produce these features.
Although models involving forward and reverse shocks seem tailor-made for
producing time-reversed structures, they have only been shown to do so in the
simplest case of one forward and one reverse event (Hakkila et al., 2018b). It
is not obvious how they produce complex time-symmetric light curves.
Additional types of models have been developed in the framework of bulk
kinetic motion occurring within the confines of radially-evolving jets. The
models involve impactors of optically-thick material that travel through the
jet and interact with other blobs of material. Time-reversed pulse structure
might be caused by a reversal of the impactor’s motion, causing it to pass
through blobs in reverse order (Hakkila et al., 2018b), impactors moving
through symmetrically-distributed blobs structures within the jet (Hakkila et
al., 2018b), symmetrically-structured impactors moving through simple blobs
within a jet (Hakkila et al., 2018b), impactors moving through density
variations in a collapsing accretion disk (Hakkila & Preece, 2019), or density
fluctuations transitioning between subluminal and superluminal motion in an
expanding relativistic jet (Hakkila & Nemiroff, 2019). These models all
involve some ad hoc assumptions in order to be consistent with the
observations.

The inconsistency in kinematic models to match the data appears to arise from
the assumption that GRB jets point unwaveringly in the direction of the
observer. This constraint places a difficult burden on models by requiring
that all time-reversed mechanisms occur within a rapidly-expanding radial jet
framework.

##  2 Lateral Motion of Relativistic Jets and Gamma-ray Burst Pulse Light
Curves

In order to examine time-reversed GRB pulse structure from a different
perspective, we propose to relax the constraining assumption that GRB jets
always point at the observer, and hypothesize that GRB jets move laterally
while they expand rather than remaining fixed in place. Allowing the observer
to see the jet cross-section as it sweeps across the line-of-sight forms the
basis of the lateral jet motion hypothesis. We hypothesize that this lateral
motion is due to motion of the jet nozzle relative to the observer. This
motion could be due to jet precession, chaotic motions as the nozzle seeks a
stable configuration, or relative motion of the nozzle’s frame relative to the
observer.

###  2.1 Lateral Jet Motion and Time-reversed Pulse Properties

A radially-symmetric jet sweeping laterally across an observer’s line-of-sight
can naturally account for time-reversed GRB pulse light curve structures. When
a jet crosses the line-of-sight, structures surrounding the jet are both the
first and the last things to be seen by the observer. If a pulse begins when
the edge of the jet first crosses the observer’s line-of-sight, then it ends
as it exits. Any radial structures within the jet will also appear as time-
reversed features in the light curve, as shown in Figure 5.

The jet structure thus likely represents regions producing more emission or
less emission than that emitted by the central core of the jet. Without
specifying the emission mechanism, we can naively assume that photon
production is proportional to the amount of ejecta emitted in specific regions
of the jet, or perhaps to the efficiency with which photons are produce in
these regions. Thus, structure could represent jet regions in which ejecta
production inordinately productive, such as in mini-jets (e.g., Lyutikov
(2006); Lazar et al. (2009); Narayan & Kumar (2009); Beniamini & Granot
(2016); Granot (2016); Gill & Granot (2024)). Conversely, they might represent
regions where mechanisms producing jet ejecta are less productive, such as
regions along the edge of the jet where it is less efficient at burning
through its progenitor star, or where less material is flowing outward, such
as in a jet’s turbulent cocoon. Regardless of the mechanism that produces
pulse structure, this model assumes that the jet contains general asymmetries
that are axisymmetric.

Figure 5: A simple model of a relativistic jet sweeping across an observer’s
line-of-sight. The jet expands outward with Lorentz factor $\Gamma$ (red
dashed line), emitting light from the jet core and the surrounding structure
as the jet sweeps (dotted arc) across the observer’s line-of-sight (dotted
line) with observed angular width $1/\Gamma$. Emission from the structured
parts of the jet appear faint relative to the core due to relativistic beaming
effects. An example of what the time-reversed light curve might look like is
demonstrated using the light curve from a BATSE crown pulse light curve shown
(from Hakkila (2021)).

The bulk motion in a relativistic jet is characterized by Lorentz factor
$\Gamma$, where

$\Gamma=\frac{1}{\sqrt{1-\beta^{2}}},$ (1)

and where $\beta=v/c$ and $c$ is the speed of light. GRBs have highly
relativistic jets, with typical $\Gamma\approx 300$. As a jet crosses the
line-of-sight, the jet’s cross-section is observed, with several relativistic
effects coming into play. First, the observer sees the jet emission confined
to a narrow cone of angular diameter $1/\Gamma$. Second, the intensity in the
beam is Doppler boosted by an amount $D^{2}$, where the Doppler factor $D$ is

$D=\frac{1}{\Gamma(1-\beta\cos\theta)}.$ (2)

The headlight effect causes emission from the line-of-sight to outshine that
from the rest of the jet. Thus, as the jet laterally traverses the line-of-
sight, each contour of the jet is captured at a specific time as part of a
tracing on the light curve. Third, the pulse duration $\tau$ indicates the
time it takes for the jet to be seen crossing the line-of-sight. Both the
opening angle of the jet and the rate at which the jet moves across the line-
of-sight contribute to the duration. Fourth, transverse motion across the
line-of-sight provides an alternate explanation for pulse asymmetry and hard-
to-soft evolution that is not limited to radial events within the jet. Lateral
jet motion might help explain differences between symmetric pulses (crown
pulses and u-pulses) and asymmetric pulses (FREDs, rollercoaster pulses, and
asymmetric u-pulses).

Slightly different features will be seen when the edge of a jet crosses the
line-of-sight than when the jet core crosses it. An off-axis pass causes an
observer to see a slightly different slice of the jet’s emission profile. This
process is demonstrated in Figure 6, where the observer sees a slightly
different profile depending on where the jet direction intercepts the line-of-
sight. Since most pulses can be characterized by only a few morphological
types Hakkila (2021), the jet structure must be fairly similar from GRB to
GRB.

Figure 6: Pulse light curve morphology as a function of off-axis viewing. An
observer can see a slightly different cross-section of the jet structure
depending on where the jet crosses the line-of-sight (left column), as
relativistic beaming causes different features in symmetric pulses (center
column) or asymmetric pulses (right column) to be magnified along each path
(a, b, c). Depending on pulse asymmetry, (a) when the edge but not the core of
the jet cross the line-of-sight, then an observer may see a u-pulse (a-center)
or an asymmetric u-pulse (a-right), (b) when the jet is viewed slightly off-
axis, then an observer may see a crown/u-pulse (b-center) or a rollercoaster
pulse (b-right), or (c) when the jet core crosses the line-of-sight such that
the core brightness overpowers the edge, then the observer might see a crown
pulse (c-center) or a FRED (c-right). The sample light curves shown are from
symmetric pulses observed by BATSE and described by Hakkila (2021). In these
light curves, data (black lines) are overlaid by the time-reversed model (red
lines), which have been folded and stretched around the time of reflection
(central dashed vertical line) between the boundaries of the pulse duration
(other vertical dashed lines).

Figure 2 demonstrates that structured GRB pulses fall into a remarkably small
number of distinctly-different light curve morphologies. If these
morphological variations are the result of jet profiles viewed from different
off-axis locations at different evolutionary times, then GRB jets must have
very similar makeups that potentially do not change much as jets expand. Armed
with this assumption, we can study GRB jet structure by considering each pulse
to be a cross-sectional structure of a jet.

The idea that a pulse light curve might be used to trace jet structure is not
new. This approach has been used to map out pulsar jets from light curves
using orbital precession in the binary pulsar system PSR $1913+16$ (Weisberg
et al., 1989). Although GRBs do not repeat, there is great potential of using
their light curves to trace out structure that seems to bear remarkable
morphological similarities from jet to jet (Hakkila, 2021).

###  2.2 Lateral Jet Motion and Pulse Asymmetry

Jet models must address more characteristics than just light curve
similarities. They need to also address the variation in pulse asymmetries, as
well as the hard-to-soft spectral evolution generally observed in GRB pulses.

Half of all observed GRB pulses have time-asymmetric profiles with faster rise
than decay times (e.g., Nemiroff et al. (1994); Hakkila (2021)). FRED pulses,
rollercoaster pulses, and asymmetric u-pulses are examples of asymmetric pulse
types. Besides the asymmetry of the underlying smooth pulse, the distribution
of structure around the time of reflection is also asymmetric. Asymmetry in
the light curve’s structure is characterized not only by an increased temporal
separation between light curve features but also by a broadening of these
features (Hakkila et al., 2018b; Hakkila & Nemiroff, 2019; Hakkila, 2021).
Thus, in time-asymmetric pulses, a relatively abrupt transition delineates the
temporal compression of events preceding the time of reflection from those
broader time-reversed events following it.

One way by which an observer may see an asymmetric jet profile is if the jet
is bent rather than straight as it crosses the line-of-sight. In order to
allow for this, viable models must consider how jet motion across the line-of-
sight might cause it to bend. This has to do with jet structure and
collimation.

There are three main types of helical jets (e.g., Steffen (1997)): ballistic
helical jets (where the flow of ejecta is along a straight line but where the
ejecta direction changes), helically bent jets (where the flow of ejecta is
along a curved jet axis), and internal helical jets (where helical structure
exists within an otherwise straight jet). For our modeling we consider the
first two of these jet types.

##  3 Model 1. Helically-bent rigid jet model

Consider a rigid or semi-rigid jet that is able to retain its structure while
moving laterally across the line-of-sight. Such a jet is a form of the
helically-bent jet model described above. Geometric jet constraints are
present that do not depend on either the mechanism that contains the jet or
the mechanism responsible for causing the jet’s lateral motion. We demonstrate
here that rigidity constrains the jet to cross the line-of-sight close to the
nozzle where the jet ejecta has been emitted by the progenitor. Figure 7 shows
how a rigid/semi-rigid jet with opening angle $\theta$ expands as it crosses
the line-of sight starting at distance $r_{0}$ from the source.

Figure 7: A rigid/semi-rigid jet model capable of explaining time-reversed GRB
pulse structure. The model also shows how asymmetry can be introduced into the
pulses created from the rapid expansion of a laterally-moving rigid/semi-rigid
jet. As the jet moves across the line-of-sight from the leading edge to the
jet center, it expands from distance $r_{0}$ to $r_{1}$ during the pulse rise
duration $\tau_{r}$, where $r_{1}=r_{0}+v\tau_{r}$ (corresponding to having
the jet traverse a lateral distance $d_{1}$ at transverse velocity
$v_{\perp}$). As the jet moves from the jet center to the trailing edge during
the pulse decay duration $\tau_{d}$, it expands from distance $r_{1}$ to
$r_{2}$, where $r_{2}=r_{0}+v\tau_{r}+v\tau_{d}r_{0}+v\tau$ (simultaneously
traversing a lateral distance $d_{2}$ at velocity $v_{\perp}$).

We define the pulse duration $\tau$ as the time it takes for the observable
jet to cross the line-of-sight. For this model, we assume that the jet must
cross at a distance equal to its diameter $d$ as it moves at lateral velocity
$v_{\perp}$ in order that an observer see the full jet cross-section. Defining
the crossing distance $r_{\rm max}$ as the maximum distance from the source at
which the trailing edge of the jet crosses the line-of-sight, we have
$r=v_{\perp}\tau/\theta$, where $\theta$ is the jet opening angle. Because the
physical upper limit is $v_{\perp}<c$,

$r_{\rm max}<c\tau/\theta,$ (3)

or

$r_{\rm max}{\rm(cm)}<1.7\ \times 10^{12}\tau/\theta(^{\circ}).$ (4)

For a GRB pulse with duration $\tau=10$ s and for jets having opening angles
in the expected range $1^{\circ}\leq\theta\leq 10^{\circ}$, crossing distances
are in the range $10^{12}\leq r_{\rm max}\leq 10^{14}$ cm (dashed curved line
in Figure 8. These crossing distances are too small by at least an order of
magnitude to be reconciled with the standard GRB emission mechanisms that
predict internal shocks to be produced at distances of $r=10^{14}$ to
$10^{15}$ cm from the source (citation). These limits are further constrained
because the total velocity $v_{\rm tot}$ of the jet’s emitting region from
which photons are produced, must be less than c. In other words,

$v_{\rm tot}=\sqrt{v^{2}+v_{\perp}^{2}-\frac{v^{2}v_{\perp}^{2}}{c^{2}}}\leq
c.$ (5)

Figure 8: Simple theoretical distance limits within which a rigid/semi-rigid
relativistic jet can cross the line-of-sight to produce a 10 s GRB pulse
having time-reversed properties. The curved dashed line represents the maximum
distance from the nozzle at which the jet can cross the line-of-sight and
correspond to a lateral velocity of $v_{\perp}=c$. The curved solid line
represents a maximum crossing distance corresponding to $v_{\perp}=0.1c.$ The
straight solid/dashed line corresponds to the minimum radial length the jet
can have and still cross the line-of-sight during its near-lightspeed
expansion.

For more reasonable limits on the lateral jet velocities ($v_{\perp}<<c$),
crossing distances must be be even closer to the progenitors (solid curved
line in 8). For a $\tau=10$ s pulse, $v_{\perp}=0.1c$ reduces the crossing
distance to $10^{11}\leq r_{\rm max}\leq 10^{13}$ cm.

The minimum crossing distance is defined by the radial expansion of the
rigid/semi-rigid jet. The minimum jet length at which the jet can cross the
line-of-sight to produce a time-reversed light curve is characterized by the
expansion velocity of the jet. If the jet starts crossing the line-of-sight
the moment it is launched, then it will have expanded to a crossing distance
$r_{\rm min}=v_{\perp}\tau$ by the time it completes its crossing. This
minimum crossing distance is defined by the lower solid/dashed straight line
in Figure 8.

Figure 9 shows the same crossing distance parameter space as is shown in
Figure 8, except for GRB pulses of $\tau=1$ s. Simiarly, Figure 10 shows the
same crossing distance parameter space as is shown in Figure 8, except for GRB
pulses of $\tau=100$ s.

Figure 9: Simple theoretical distance limits within which a relativistic jet
can cross the line-of-sight to produce a 1 s GRB pulse having time-reversed
properties.  Figure 10: Simple theoretical distance limits within which a
relativistic jet can cross the line-of-sight to produce a 100 s GRB pulse
having time-reversed properties.

These figures show that it is geometrically possible for a laterally-moving
rigid jet to produce pulses of light having time-reversed structures
consistent with those of GRB pulse light curves. However, the lateral
velocities of these jets must be relatively large in order to cross the
observer’s line-of-sight on the relatively short timescales of GRB pulses.
Since relativistic lateral jet lateral velocities seem unlikely, the jet
opening angles must be similarly constrained to $\theta\leq 5^{\circ}$. For a
typical GRB pulse, these constraints also require the jet to cross the line-
of-sight within a relatively narrow range of distances from the nozzle.

The ability of the rigid/semi-rigid jet model to explain pulse asymmetry
depends on the rigidity of collimated jet and thus on how the jet moves
laterally. For example, a rigid jet with a lateral velocity of the form
$v_{\perp}=v_{\perp 0}r$ (where $v_{\perp 0}$ is a constant) will move across
the line-of-sight at a constant angular velocity. The leading edge of the
laterally-moving jet will take the same amount of time to cross the observer’s
line-of-sight as the trailing edge, producing a symmetric pulse rather than an
asymmetric one.

If the jet is semi-rigid such that the lateral jet velocity is constant
regardless of the distance from the central engine, then the jet head will lag
behind the jet base as material flows within it. This semi-rigidity allows
pulse asymmetry to form simply due to curvature accompanying jet expansion.

Figure 7 demonstrates how pulse asymmetry may arise from a rigid or semi-rigid
GRB jet crossing the line-of-sight. The expansion can cause the lateral range
of the jet to be larger far from the central engine than near to it. If the
jet is moving laterally with a constant velocity $v_{\perp}$, then the time
$\tau_{d}$ needed for the trailing edge to cross relative to the jet center
will be longer than the time $\tau_{r}$ for the jet center to cross relative
to the leading edge.

The lateral distance $d_{1}$ crossed during the pulse rise is related to the
pulse rise duration by

$d_{1}=v_{\perp}\tau_{r}.$ (6)

Similarly, the lateral distance $d_{2}$ crossed during the pulse decay is
related to the pulse decay duration by

$d_{2}=v_{\perp}\tau_{d}.$ (7)

During the duration of the pulse rise, the jet expands at velocity $v$, and
$d_{1}$ is also given by

$d_{1}=(r_{0}+v\tau_{r})\theta/2.$ (8)

Also,

$d_{2}=(r_{0}+v\tau_{r}+v\tau_{d})\theta/2.$ (9)

Note that the pulse duration $\tau$ is given by $\tau=\tau_{r}+\tau_{d}.$
Combining Equation 6 and Equation 8 yields

$\tau_{r}=\frac{r_{0}\theta/2}{v_{\perp}-v\theta/2}.$ (10)

Similarly, combining Equation 7 and Equation 9 yields

$\tau_{d}=\frac{(r_{0}+v\tau_{r})\theta/2}{v_{\perp}-v\theta/2}.$ (11)

Substituting Equation 10 into Equation 11 results in

$\tau_{d}=\frac{v_{\perp}r_{0}\theta/2}{(v_{\perp}-v\theta/2)^{2}}.$ (12)

The pulse duration can be found in terms of $r_{0}$, $v$, and $v_{\perp}$ by
combining Equation 10 and Equation 12:

$\tau=\frac{r_{0}\theta/2(2v_{\perp}-v\theta/2)}{(v_{\perp}-v\theta/2)^{2}}.$
(13)

The stretching factor $s$ is then approximately given as

$s\approx\tau_{r}/\tau_{d}=1-\frac{v\theta/2}{v_{\perp}}.$ (14)

Equation 14 demonstrates that pulses are symmetric ($s=1$) when a jet expands
only slightly relative to its motion across the line-of-sight, e.g., when
$v\ll 2v_{\perp}/\theta$. Pulses are asymmetric ($s=0$) when a jet expands
significantly relative to its motion across the line-of-sight, e.g., when
$v\approx 2v_{\perp}/\theta$. For example, a jet expanding at $v\approx c$ and
crossing with velocity $v_{\perp}\approx c/10$ will appear to be much more
symmetric when the opening angle is small ($s\approx 0.1$ for
$\theta=1^{\circ}$) than when the opening angle is large ($s\approx 0.9$ for
$\theta=10^{\circ}$).

##  4 Model 2. Ballistic jet model

A ballistic helical jet is one in which the flow of ejecta is along a straight
line but where the nozzle ejecting it changes direction. The jet itself is not
rigid and thus does not move laterally, but particle emission from across the
jet’s cross section is funneled sequentially towards the observer as the
nozzle moves across the line-of-sight. Unlike the rigid jet model, where the
emitting region is forced to be close to the nozzle, the emitting region in
the ballistic jet model can be at much larger distances, consistent with
standard GRB shock models. As the particle stream interacts to produce
photons, these photons are produced in a sequential order corresponding to the
structure in the jet cross section. The pulse duration is a direct measurement
of the time it takes for the nozzle to cross the line-of-sight, and time-
reversed structure in the pulse light curve indicates radially-symmetric
structure of the moving jet nozzle. A simple drawing of a jet launched from a
moving nozzle is shown in Figure 11.

Figure 11: A ballistic model in which a is jet launched from a moving
axisymmetric nozzle (blue cylinder). As the jet crosses the line-of-sight,
structure from different locations in the nozzle, represented by differing
relativistic particle densities, compositions, and/or energies is sent in the
observer’s direction. Once these particles reach a certain distance, they
interact to produce photons at an emission surface (denoted by the the blue
ellipsoid. Because these events produce photons in the order they were emitted
from the nozzle, the observer sees sequential emission from the leading edge
of the laterally-moving jet (red solid line), the center of the jet (blue
dotted line, black dashed line, blue dotted line), and the trailing edge of
the jet (red solid line).

The ballistic jet model can explain some (but not all) of the pulse asymmetry
naturally. This results from the natural process of having the jet emitting
region expand with time as the jet pressure forces this region progressively
farther from the nozzle. This effect is demonstrated in Figure 12. We apply a
simple estimate to determine the amount of pulse asymmetry that this
introduces in the following fashion.

After the jet ejecta has been launched at velocity $v$, it travels a distance
$r_{0}$ before attaining the conditions necessary for a emitting region to
form. Prior to this, the jetted material does not emit photons. As the
emitting region emits, it is also pushed away from the nozzle at a velocity
$v_{p}$. Assuming the jet has radial symmetry, the pulse peak corresponds to
the time the jet center emits along the line-of-sight. During the pulse rise
phase, the nozzle turns at angular velocity $\omega$ by an amount $\theta/2$,
or half the opening angle, which also results in the emitting region expanding
to distance $r_{1}$. During the pulse decay phase, the nozzle turns by an
additional amount $\theta/2$ , which results in the emitting region expanding
to distance $r_{2}$.

Figure 12: Asymmetry from ballistic jet expanding outward with velocity $v$.
As the jet nozzle (blue cylinder) moves across the line-of-sight (angle
$\phi_{0}$) with angular velocity $\omega$, the emitting region (oval) expands
outward at velocity $v_{p}$. During the pulse rise phase, the nozzle rotates
by angle $\theta/2$ and the emitting region expands from distance $r_{0}$ to
distance $r_{1}$. During the subsequent pulse decay phase, the nozzle rotates
again by $\theta/2$ and the emitting region expands from distance $r_{1}$ to
distance $r_{2}$. Pulses appear asymmetric because it takes longer for ejecta
to travel to the emitting region during the pulse decay phase than during the
pulse rise phase.

The time $t_{0}$ that it takes for the ejecta to reach $r_{0}$ is

$t_{0}=v/r_{0}.$ (15)

The time $t_{01}$ that it takes for the emitting region to expand from $r_{0}$
to $r_{1}$ is

$t_{01}=\frac{\theta/2}{\omega}=r_{1}/v_{p},$ (16)

such that

$r_{1}=\frac{\theta v_{p}}{2\omega}.$ (17)

Similarly,

$r_{2}=\frac{\theta v_{p}}{2\omega}.$ (18)

Since the time it takes for the emission at the jet center to reach the
emitting region $t_{c}s$ is

$t_{c}=\frac{r_{0}+r_{1}}{v}=t_{0}+\frac{r_{1}}{v}=t_{0}+\frac{\theta}{2\omega}\frac{v_{p}}{v},$
(19)

the pulse rise time $\tau_{\rm rise}$ is given as

$\tau_{\rm rise}=t_{10}+t_{c}=\frac{\theta}{2\omega}(1+\frac{v_{p}}{v}).$ (20)

The pulse decay time $\tau_{\rm decay}$ is similarly

$\tau_{\rm decay}=2t_{10}+t_{c}=\frac{\theta}{2\omega}(1+\frac{2v_{p}}{v}),$
(21)

such that the pulse duration $\tau$ is given by

$\tau=\tau_{\rm rise}+\tau_{\rm
decay}=\frac{\theta}{\omega}(1+3\frac{v_{p}}{v}).$ (22)

Similarly, the stretching factor $s$ is roughly

$s=\frac{\tau_{\rm rise}}{\tau_{\rm decay}}=\frac{1+v_{p}/v}{1+2v_{p}/v}.$
(23)

Pulses are symmetric when the emitting region expands only slightly during the
jet expansion, e.g., when $v_{p}/v\rightarrow 0$, which leads to $s\rightarrow
1$. Similarly, pulses have the greatest asymmetry when the emitting region
expands a large amount during the jet expansion, e.g., when $v_{p}/v$ is
large. If the emitting region does not accelerate, then the upper limit is
$v_{p}/v\rightarrow 1$ which leads to $s\rightarrow 2/3$. The asymmetry
produced by expansion of the jet emitting region is therefore unable by itself
to produce the small stretching factors of $0.1\leq s\leq 0.6$ found in FREDs,
rollercoaster pulses, and asymmetric u-pulses. However, it is interesting to
note that a natural separation between symmetric and asymmetric pulse
morphologies occurs at $\kappa\approx 1/3$ (Hakkila, 2021), which with our
proxy of $\kappa\approx 1-s$ is consistent with symmetric pulse types having
asymmetries that are due entirely to jet expansion.

Additional pulse asymmetry can be introduced if the jet changes according to
some of the mechanisms proposed here:

  * •

The lateral motion of the nozzle can slow as it crosses the line-of-sight.
This can cause the trailing part of the jet to take longer to cross than the
leading part. Given that the jet also continues expanding during this time,
this could produce a substantial amount of additional pulse asymmetry.

  * •

The jet opening angle can increase with time. After the jet nozzle has ejected
a large amount of material, it seems reasonable that the mechanisms causing
the jet to be confined might eventually break down. These would result from
the tapering off of material that feeds the accretion disk as well as the
weakening of the toroidal magnetic field that collimates the jet. The loss of
these constraints could weaken the jet collimation at the nozzle and cause
$\theta$ to increase with time.

  * •

Curvature of the jet head may cause photons emitted at off-axis locations to
arrive at the observer later than those emitted on-axis (e.g., Ioka & Nakamura
(2001)).

  * •

The jet emitting region can accelerate outward as the jet crosses the line-of-
sight instead of expanding at a constant velocity. This can increase the time
it takes for jet ejecta to reach the emitting region during the pulse decay
relative to the time taken during the pulse rise.

We note here that not all asymmetric GRB pulses have easily-discernible
structure at the end, which we would normally take as an indication that the
trailing edge of the jet had crossed the line-of-sight. FRED and rollercoaster
$s-$values are determined primarily from the triple-peaked structure that
occurs around the middle of the pulse rise, at the pulse peak, and around the
middle of the pulse decay (Hakkila et al., 2018b). These pulses exhibit time-
reversed structure occurring towards the center of the pulse rather than
towards the end of the pulse. Thus, it may be that a slowly-decaying GRB pulse
does not have well-defined structure at its end simply because the jet that
does not finish crossing the line-of-sight until long after the main part of
the jet has crossed. This is consistent with a slowing nozzle or an
increasingly large opening angle (for a ballistic jet) or with a curved jet
(for a rigid or semi-rigid jet).

As a jet lengthens, the surface area of the jet head increases and the volume
into which jetted material flows becomes progressively larger. Thus, the
energy density decreases as the jet expands. Furthermore, the average particle
energy decreases as the jet pushes through and interacts with the surrounding
external medium. Both of these effects will contribute to a graduate weakening
of the spectrum produced at the emitting region, and this weakening should be
observed in the overall hard-to-soft spectral evolution found in both GRBs and
GRB pulses. We will explore spectral models produced by these models in a
subsequent manuscript.

##  5 Pulse Durations and Separations

There is less confusion with identifying GRB pulses once time-reversed
structure has been incorporated into the GRB pulse definition, since most
variations found within GRBs do not qualify as individual pulses. In the case
of bright bursts, most of emission can be accounted for by a small number of
time-reversed pulses, reducing the possibility that faint, unfitted pulses may
be present. Most of these bursts’ emission is contained within only one or two
pulses (Hakkila et al., 2018a, b; Hakkila, 2021).

X-ray flares, which are also scarce, have the characteristics of late GRB
pulses that occur during the afterglow phase and after the prompt emission has
ended. X-ray flare intensities decrease, durations increase, and spectra
soften with time (e.g., Chincarini et al. (2007); Kocevski et al. (2007)),
similar to what is found for GRB pulses (e.g., Ramirez-Ruiz & Fenimore (2000);
Hakkila & Preece (2014).

Separations between pulses and pulse durations can be used to place additional
constraints on mechanisms capable of producing lateral jet motion. First, as
mentioned above, pulses/flares are rare. Second, there is no indication that
GRBs exhibit any form of periodic motion. Third, there is the aforementioned
evolution of pulse properties with time.

If GRB pulses arise as a ballistic jet crosses the line-of-sight, then the
angular velocity of the nozzle can be estimated from the pulse duration
(excluding the asymmetry-producing jet expansion effects described in Section
4) as $\omega\approx\theta/\tau_{1}$ where $\theta$ is the jet opening angle
and $\tau_{1}$ is the duration of the initial pulse. Pulses produced via a
periodic process would be expected to repeat after a time ${\rm
sep}=360^{\circ}/\omega$. Therefore, relative to the duration of the first
pulse, the ratio R should be

$R={\rm sep}/\tau_{1}$ (24)

for all GRB that produce repeating pulses in this manner. Similarly, the
duration of the second pulse $\tau_{2}$ should be the same as $\tau_{1}$.

Hakkila (2021) systematically studied pulses in a sample of BATSE GRBs using
the time-reversed pulse definition which prevents pulses from being confused
with pulse structure. Although most GRBs appear to be single-pulsed, this
sample can also be used to study the evolution of pulse properties within
multi-pulsed GRBs. Unfortunately, this robust sample cannot easily be
redshift-corrected, as few redshift measurements were available for BATSE
GRBs. However, we can use a normalization process to remove the interfering
effects of redshift. By normalizing both the pulse separation $sep$ and the
duration of the second pulse $\tau_{2}$ to the duration of the first pulse
$\tau_{1}$, the redshift-independent properties $\tau_{2}/\tau_{1}$ and
$sep/\tau 1$ can be obtained. A comparison of these properties for the burst
sample allows us to determine whether or not the duration of the second pulse
changes as a function of pulse separation.

Figure 13: Properties of multi-pulsed GRBs taken from a GRB pulse sample
(Hakkila, 2021). Duration of the second pulse $\tau_{2}$ is shown relative to
the separation between pulses ($sep$), where both timescales have been
corrected for cosmological redshift by normalizing to the duration of the
first pulse ($\tau_{1}$). Pulse durations increase as the separation between
pulses increases. This increase is consistent with lateral jet motion which is
either slowing or in which the jet profile is dispersing.

Figure 13 demonstrates that pulse durations are not constant. In fact, they
increase as pulse separations increase. This behavior is not consistent with
periodic motion of a ballistic jet nozzle. Because pulse durations increase as
separations increase, and because this increase is accompanied by an apparent
decrease in pulse intensity and energy, there is a systematic evolution of the
system. This evolution appears to be consistent with expansion of the emission
surface described in Section 4, as ejecta released each time the jet recrosses
the line-of-sight will have to travel farther to reach the expanding emission
surface.

##  6 Discussion

GRB models have been around for over fifty years. In that time, the database
of GRB behaviors has grown even as the mechanism responsible for producing
their emission has remained elusive. Given the many models that have been
considered over time, the idea that GRBs might be caused by moving jets is not
a new one. For example, Blackman et al. (1996) considered the possibility that
GRBs might result from “cosmological relativistic blob emitting neutron star
jets that precess past the line of sight.” Fargion & Salis (1995) hypothesized
that GRBs resulted from precessing neutron star jets (NSJ) in binary systems
with solar-like companions. Portegies Zwart et al. (1999) developed a model in
which a precessing luminosity cone in a slaved accretion disk could be
responsible for “complex temporal behavior of bright gamma-ray bursts.” Gao et
al. (2023) have considered that jet precession may be a better explanation for
GRBs with repeating emission episodes than millilensing, but this model also
hypothesized that the mechanism responsible for the emission originates in
radial interactions of material in the jet. Fargion and various co-authors
(e.g., Fargion & Salis (1996); Fargion (2003); Fargion & Grossi (2006)) have
explored GRB light curves as being due to complex, short-term precession of
narrow jets overlaid on longer timescale precessional motions. These motions
have been designed to account for the shortest pulse structures, but not for
the more recent discover of time-reversed GRB profiles. Recently, Huang & Liu
(2021) have considered that jet precession may contribute energy injections to
GRB afterglow light curves. However, these models were not designed to account
for what appears to be the most direct evidence of a laterally-moving jet: the
time-reversed structure of GRB pulses.

The time-reversed structure observed in $86\%$ of GRB pulses, as well as the
general morphologies of GRB pulse types, can be directly explained by lateral
jet motion across the observer’s line-of-sight. For time-reversed structure to
be observed, the jet itself must produce a nonuniform brightness distribution
that is characterized by some sort of jet axial symmetry (see Section 2.1). On
top of this, both the rigid/semi-rigid jet model and ballistic jet model
described here require the jet nozzle to move laterally. The mechanism for
this motion is not specified here, but may possibly be due to the instability
of the accretion disk around the forming black hole.

Processes capable of causing relativistic jets to form and launch are
associated with energy extraction from rotating compact objects. The most
commonly accepted of these is the Blandford-Znajek process in which jets are
powered electromagnetically by the winding of magnetic field lines interlacing
a rotating central compact object (Blandford & Znajek, 1977; Komissarov, 2001;
Tchekhovskoy et al., 2010). By this process, a jet forms via conversion of the
compact object’s rotational energy into a Poynting flux-dominated outflow. For
ultra-relativistic jets like those found in GRBs, this process is accompanied
by instabilities so strong that they can cause the jet to kink (Bromberg &
Tchekhovskoy, 2016), even preventing it from penetrating the star (Mösta et
al., 2014). From a naïve perspective, the total rest mass available to a GRB
system having 100 solar masses or less is around $10^{56}$ ergs. If we
consider GRB energies in the range $10^{48}$ ergs to $10^{52}$ ergs, then
$10^{56}/10^{48}=10^{8}$. Thus, the ratio of the (energy available for a
magnetic field)/(jet power), where the magnetic field is providing
containment, would be less than what is expected for conventional gamma-ray
pulsars. As a result, the ability of the field to tightly constrain the jet
direction would be considerably diminished, and rigid/semi-rigid moving jets
seem less likely than jets with moving nozzles.

How lateral jet motion produces pulse asymmetry and spectral evolution is
model-specific. The emission surfaces of rigid/semi-rigid jets must cross the
observer’s line-of-sight within a limited range of distances from the jet
nozzle, and these distances are too close to the progenitor for generally-
accepted astrophysical mechanisms to work. Such jets naturally develop large
deviations needed to produce the pulse asymmetries and stretching factors
observed in many GRB pulses. Contrarily, ballistic jets can easily produce
photons at distances consistent with standard jet emission mechanisms because
constraints on the distance of the jet emitting region from the nozzle are not
present. Some amount of pulse asymmetry can occur naturally in the ballistic
jet model while the jet emitting region is pushed outward, and additional
asymmetry can be easily created as the nozzle’s lateral motion slows and/or as
the jet opening angle widens.

As jet ejecta loses energy in the surrounding medium, the jet volume increases
and the energy density decreases, and the jet containment weakens. These
effects should cause the emission spectrum to gradually soften, as expected
for the hard-to-soft spectral evolution observed in both GRB pulses and GRBs
overall.

Multi-pulsed GRBs are uncommon, but when they are observed they are consistent
with a process that weakens over time. In the case of a rigid or semi-rigid
jet, subsequent pulses might indicate that an existing jet is recrossing the
line-of-sight, or perhaps they might indicate that additional jets have been
launched from the nozzle. In the case of a ballistic jet, subsequent pulses
might indicate that the nozzle has reactivated, or perhaps moved back in the
direction of the observer. The increase of pulse durations with time is
consistent with emission surface expansion in ballistic jets.

##  7 Conclusions and Future Work

Laterally-moving jets, such as those described here, can explain a wide range
of previously-unexplained temporal characteristics of GRB light curves. These
include the time-reversed structures found in GRB pulses, asymmetric pulses
and their structures, the rarity of GRB pulses, and correlations between
increasing pulse durations and pulse separations. These characteristics are
not addressed in most GRB models. Furthermore, the relativistic properties of
a jet as it crosses the line-of-sight are directly tied into pulse brightness,
spectral hardness, and duration. GRB pulses may turn out to be an exceptional
laboratory for studying the effects of special relativity.

Lateral motion of the jet nozzle is more likely to occur as a jet is forming,
and when the structure of the progenitor is relatively unstable. Misalignment
of the rotation axis and magnetic field axis can occur during compact object
formation, especially in the case of a tilted accretion disk. This process can
produce a nozzle that moves in a chaotic fashion. However, a more stable
configuration is also possible where nozzle precesses. In this case, the inner
jet regions can spiral under the outer layers and turn the jet into an
expanding spiral or helix.

Scenarios by which relativistic jet curvature can develop have been examined
by various authors using 2D and 3D magnetohydrodynamic (MHD) computer codes
(Raga et al., 1993; Cliffe et al., 1996; Fendt & Zinnecker, 1998; Masciadri &
Raga, 2002; Monceau-Baroux et al., 2015; Fendt & Yardimci, 2022). The two most
common scenarios by which the moving jet might be produced are a) via a
precessing jet nozzle arising from accretion into a compact source (similar to
the massive star core-collapse model used to describe long-duration GRBs), and
b) by orbital motion of material in a close binary system (similar to the
merging neutron star model used to describe short-duration GRBs). Despite
being launched from progenitors similar to those expected for GRBs, the jets
produced in these models curve at large distances from the nozzle and move at
slow enough velocities that the crossing time should be many orders of
magnitude larger than the durations of GRB pulses. Thus, both the mechanism
needed to produce a rigid or semi-rigid jet and the mechanism needed to
produce photons from this jet are inconsistent with the close distances at
which a rigid/semi-rigid jet would need to cross the line-of-sight in order to
produce a GRB pulse. It is much more straightforward if the jet’s nozzle moves
across the line-of-sight and the jet simply sprays some of the ejecta in the
direction of the observer. Gamma-rays can be produced by any of a number of
standard processes when the ejecta is far removed from the source. Jets
produced in the manner will be curved, but observers only see emission when
the jet points at them.

It has been previously demonstrated, upon accounting for time-reversed
structure, that there are a relatively small number of GRB pulse light curve
morphologies. This argues that there is some mechanism common to all GRB
pulses. We propose that this commonality may be the jet structure itself, in
which the smooth pulse component describes the core while the structured
component describes the variations across the face of the jet. Each GRB pulse
light curve allows the observed to view a slice of the jet surface. We have
demonstrated here that many variations in GRB pulse light curves may simply
result from jets passing off-axis across the observer’s line-of-sight.

Two models have been proposed here to describe a laterally-moving jet. The
rigid/semi-rigid jet model is able to satisfy many of the geometric concerns
imposed by the observations, but the mechanism needed to produce gamma-ray
emission must occur so close to the progenitor that it contradicts what is
understood in standard GRB emission models. The ballistic jet model satisfies
most geometric concerns by simply spraying relativistic material towards the
observer as it crosses the line-of-sight, and using standard arguments that
the jet ejecta cannot produce radiation until far from the progenitor,
standard jet emission mechanisms remain viable. In both of these models,
observers briefly see jet structure as the photon-producing regions of the jet
cross the line-of-sight.

The models developed here suggest that the number of pulses within a burst is
an indicator of the level of activity in the jet’s lateral motion. The time-
reversed structure found in GRB pulses indicates that there must be only a few
line-of-sight jet crossings because the number of pulses per GRB is observed
to be small (typically one or two per burst). This implies either that GRB
jets are relatively stationary or they are active but only rarely point in the
observer’s direction. for either of these options, the temporal spacing of GRB
pulses supports the idea that jets lose energy over time. Two other pulse
spacing issues provide additional clues: 1) late flares observed during GRB
afterglows suggest that activity continues long after the main bursting event
has been completed: either the jet or the nozzle are still undergoing lateral
motion at late times, and energy is still available to power the nozzle, and
2) some GRB pulses overlap, suggesting that the nozzle or jet can undergo
rapid fluctuations in structure and/or motion.

Laterally-moving relativistic jets are able to account for a wide range of
previously-unexplained GRB pulse behaviors such as those described in this
paper. Despite the model’s success, there are still unanswered questions. For
example, we have used the pulse structure’s stretching factor $s$ as an alias
for the pulse asymmetry $\kappa$, when these values do not in fact represent a
one-to-one correlation. Rather, the $s$ varies relative to $\kappa$ in such a
way to somehow produce different pulse morphologies. Furthermore, some pulses
(primarily symmetric crown pulses and u-pulses) have $s$ values that exceed
unity, such that the structure is compressed on the decay side of the pulse
relative to the rise side. This is contrary to the time asymmetry seen in the
underlying pulse, which, when present, generally undergoes a rapid rise and a
long decay. We are currently undertaking more detailed analyses to explore how
models can explain these important second-order effects.

We are grateful to the feedback of several anonymous referees who greatly
helped improved this paper. We also thank Charles A. Meegan, Amy Y. Lien,
Istvan Horvath, and Peter Veres for helpful discussions.

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