# Analysis of CN Coma Morphology Features of Comet 21P/Giacobini-Zinner

Charles Goldberg Lunar and Planetary Laboratory, University of Arizona
1629 E University Blvd
Tucson AZ 85721-0092, USA  Cassandra Lejoly Lunar and Planetary Laboratory,
University of Arizona
1629 E University Blvd
Tucson AZ 85721-0092, USA  Nalin Samarasinha Planetary Science Institute
1700 East Fort Lowell Rd
Tucson AZ 85719-2395, USA

# Analysis of CN Coma Morphology Features of Comet 21P/Giacobini-Zinner

Charles Goldberg Lunar and Planetary Laboratory, University of Arizona
1629 E University Blvd
Tucson AZ 85721-0092, USA  Cassandra Lejoly Lunar and Planetary Laboratory,
University of Arizona
1629 E University Blvd
Tucson AZ 85721-0092, USA  Nalin Samarasinha Planetary Science Institute
1700 East Fort Lowell Rd
Tucson AZ 85719-2395, USA

###### Abstract

We analyze jet features found in the coma of Comet 21P/Giacobini-Zinner
(21P/GZ) during its 2018 perihelion passage using narrowband CN photometric
imaging in order to determine the comet’s rotational period, and constrain the
CN gas outflow velocity and rotational state through the analysis of
azimuthally enhanced morphological features. We find that 21P/GZ has a period
of either 7.39 $\pm$ 0.01 or 10.66 $\pm$ 0.01 hours. We measure a lower limit
to the outflow velocity for the Northern Jet of 730 $\pm$ 30 m s-1 and for the
Southern Jet of 740 $\pm$ 30 m s-1. We analyze the morphologies of the jet
features and determine that the Northern jet possesses a corkscrew pattern,
and utilize that knowledge to determine a rotational pole position at a Right
Ascension of ${169^{+28}_{-23}}^{\circ}$ and a Declination of
${73^{+5}_{-11}}^{\circ}$, with undetermined sense of rotation.

Comets — CN — 21P/GZ

##  1 Introduction

Since its discovery in 1900 by Michel Giacobini, and rediscovery by Ernst
Zinner in 1913, Jupiter Family Comet 21P/Giacobini-Zinner (21P/GZ) has been
monitored over its many apparitions, including throughout its 2018 perihelion
passage. During the 2018 apparition, we obtained four nights of photometric
imaging data spanning from 18 August 2018 to 12 September 2018. Utilizing
narrowband cyanogen (CN) photometric imaging to highlight 21P/GZ’s jet
features, we are able to constrain the comet’s gas outflow velocity and spin
properties, then compare to previously obtained rotational periods to confirm
standard torque effects.

The rotation period of a cometary nucleus can provide clues to the
understanding of the comet’s structure, evolution, and formation. There are
multiple methods for determining the rotation period of cometary nuclei,
including obtaining a lightcurve of the bare nucleus when it is far away from
the Sun and minimal outgassing is present, or by obtaining a lightcurve of the
coma itself when it is active and assume that variations in the coma are
representative of the nucleus’ rotational state. These methods have been used
for many comets in the past including comet 1P/Halley (e.g.: Millis &
Schleicher (1986), Leibowitz & Brosch (1986a)) or even more recently for a
variety of comets (e.g.: Kokotanekova et al. (2017), Bodewits et al. (2018)).
Other methods of determining the period of cometary nuclei include the cross-
correlation of jet features (e.g.: Ivanova et al. (2012)), the image cross-
correlation of CN shells (e.g.: Waniak et al. (2009)), repeatability of jet
features (e.g.: Knight & Schleicher (2011), Bodewits et al. (2018)), and even
radar observations (e.g.: (Howell et al., 2017)).

For the 1985 apparition of comet 21P/GZ, the lightcurve method was used on the
1.0 m reflector of the Wise Observatory to obtain a rotation period of 9.5
$\pm$ 0.2 hours from 21 July 1985 to 11 August, 1985 (Leibowitz & Brosch,
1986b). In contrast, we utilize a novel method, which measures the radial
location of the peak flux of a jet feature as a function of time. This method
is particularly effective for corkscrew-like features or morphological
features that fluctuate with time. The measured peak flux locations versus
time are transformed by a Lomb-Scargle Periodogram to find possible
periodicities within the data. This method relies upon the assumption that
there is a repeatable brightness fluctuation present. Evidence that this
assumption is justified, as well as a more in depth explanation of the method,
can be found in §2.4.

The velocity at which gas is ejected from the nucleus is essential to predict
molecular production rates and torques, among other cometary parameters.
Cochran & Schleicher (1993) showed that the outflow velocity of gases can be
estimated by the relationship $v\approx 0.85r_{h}^{-0.5}$ km sec-1, which for
typical distances at which outflow velocities are measured, produces outflow
velocities from a few hundred meters per second to greater than a kilometer
per second. Such estimates have been proven accurate over the years through
different outflow velocity measuring techniques, such as modeling sub-
millimeter cometary emission lines (Cordiner et al. (2022)), or from the OH
18-cm emission line (e.g. Tseng et al. (2007)). To measure the CN outflow
velocity we assume that the CN features are moving radially outward.

The pole vector of cometary nuclei can help interpret the rotational features
seen, and help better understand activity patterns on the surface. There are
several methods to constrain the pole vector of a comet. One such method is
analyzing a polar jet over time (e.g.: Samarasinha & Mueller (2002), Farnham &
Cochran (2002)). We use a similar method, constraining the pole position with
regards to the jet features after a deep understanding of the jet features has
been acquired (see §4.2).

Sublimating gases, whether in hemispherical outgassing or jet-like features,
create torques on nuclei that slowly change comets’ rotational periods over
time. Samarasinha & Mueller (2013) introduced a parameter, $X$, described in
§4.3, that relates the change in a comet’s rotational period per orbit to the
rotational period itself, its nucleus size, and its water production per orbit
per unit area of the nucleus. Samarasinha & Mueller (2013) claim that such a
parameter should be fairly constant for all comets within a factor of a few
unity. Jewitt (2021) empirically shows that the $X$ parameter’s approximate
constancy is due to the approximately opposing trends in the dependence on the
active fraction of features and the moment arms of torques on the size of the
nucleus. We use the rotational period we have found, in correlation with the
previously found period from Leibowitz & Brosch (1986b) to determine the $X$
parameter of 21P/GZ.

##  2 Data

We observed 21P/GZ over four nights spanning from 28 August to 12 September
2018. Comet 21P/GZ’s closest approach to Earth occurred on 10 September 2018
at 14:16 UT at a geocentric distance of 0.39 au, and its perihelion occurred
on the same day at 6:43 UT at a heliocentric distance of 1.01 au (HORIZONS-
JPL, 2021). Table 1 includes the date and time of each image frame utilized,
the peak brightness distances measured both for the Northern and Southern
Jets, the heliocentric ($r_{h}$) and geocentric ($\Delta$) distances, the
solar position angles —the skyplane-projected position angle of the sun with
respect to the comet (position angle is measured from North through East), the
solar phase angles, and the rotational phases based upon our two derived
periods. Our images were obtained using the 1.54 m Kuiper Telescope, located
on Mount Bigelow, just north of Tucson, Arizona. The Telescope has a focal
length of 9.6 meters, and the imaging system has a pixel scale of 0.42 arcsec
pixel-1 (in our 3 $\times$ 3 binning setup) with a field of view of 9.7
$\times$ 9.7 arcmin2 (Smith, 2013).

Table 1: Measurements of peak projected radial brightness for the Northern and Southern Jets for each image. Certain measurements are unavailable due to the insufficient signal-to-noise ratio (SNR). The rotational cycles are based off of periods of 7.39 $\pm$ 0.01 and 10.66 $\pm$ 0.01 hours and the integer portion refers to the number of rotations since the reference time and the decimal portion refers to the rotational phase. The reference time for both periods is 28 August 2018 at 05:12 UTC and chosen such that the minimum measured jet distance corresponds to a phase of zero for the 10.66 hour period. $\Delta$ is the geocentric distance and $r_{h}$ is the heliocentric distance for each image.  | Northern Jet | Southern Jet |  |  | Solar | Solar | 7.39 Hour | 10.66 Hour
---|---|---|---|---|---|---|---|---
| Distance | Distance |  |  | Position | Phase | Rotational | Rotational
UTC Date and Time | ($\times 10^{3}$km) | ($\times 10^{3}$km) |  $\Delta$ (au) |  $r_{h}$ (au) | Angle | Angle | Cycle | Cycle
28 Aug. 2018, 08:58 | 14.6 $\pm$ 0.4 | 13.0 $\pm$ 0.3 | 0.42 | 1.03 | 86.22 | 75.52 | 0.51 | 0.35
28 Aug. 2018, 09:03 | 16.1 $\pm$ 0.3 | 8.6 $\pm$ 0.3 | 0.42 | 1.03 | 86.22 | 75.53 | 0.52 | 0.36
28 Aug. 2018, 09:08 | 15.5 $\pm$ 0.4 | 11.3 $\pm$ 0.3 | 0.42 | 1.03 | 86.22 | 75.53 | 0.53 | 0.37
28 Aug. 2018, 09:14 | 15.7 $\pm$ 0.3 | 12.1 $\pm$ 0.3 | 0.42 | 1.03 | 86.23 | 75.53 | 0.55 | 0.38
28 Aug. 2018, 09:19 | 16.5 $\pm$ 0.3 | 12.7 $\pm$ 0.3 | 0.42 | 1.03 | 86.23 | 75.53 | 0.56 | 0.39
28 Aug. 2018, 10:08 | 18.5 $\pm$ 0.4 | 14.6 $\pm$ 0.3 | 0.42 | 1.03 | 86.25 | 75.54 | 0.67 | 0.46
28 Aug. 2018, 10:14 | 18.3 $\pm$ 0.2 | 14.0 $\pm$ 0.4 | 0.42 | 1.03 | 86.26 | 75.54 | 0.68 | 0.47
28 Aug. 2018, 10:19 | 19.2 $\pm$ 0.3 | 12.1 $\pm$ 0.3 | 0.42 | 1.03 | 86.26 | 75.55 | 0.69 | 0.48
28 Aug. 2018, 10:24 | 19.8 $\pm$ 0.3 | 14.7 $\pm$ 0.3 | 0.42 | 1.03 | 86.27 | 75.55 | 0.71 | 0.49
28 Aug. 2018, 10:30 | 19.4 $\pm$ 0.3 | 13.3 $\pm$ 0.3 | 0.42 | 1.03 | 86.27 | 75.55 | 0.72 | 0.50
28 Aug. 2018, 10:54 | 20.2 $\pm$ 0.3 | 13.8 $\pm$ 0.3 | 0.42 | 1.03 | 86.28 | 75.56 | 0.77 | 0.54
28 Aug. 2018, 10:59 | 20.4 $\pm$ 0.3 | 16.7 $\pm$ 0.3 | 0.42 | 1.03 | 86.28 | 75.56 | 0.78 | 0.54
28 Aug. 2018, 11:05 | 21.2 $\pm$ 0.3 | 14.8 $\pm$ 0.3 | 0.42 | 1.03 | 86.29 | 75.56 | 0.80 | 0.55
28 Aug. 2018, 11:10 | 20.5 $\pm$ 0.2 | 17.6 $\pm$ 0.4 | 0.42 | 1.03 | 86.29 | 75.56 | 0.81 | 0.56
28 Aug. 2018, 11:15 | 21.2 $\pm$ 0.3 | 15.3 $\pm$ 0.3 | 0.42 | 1.03 | 86.29 | 75.56 | 0.82 | 0.57
4 Sep. 2018, 09:48 | 13.1 $\pm$ 0.4 | 10.5 $\pm$ 0.2 | 0.40 | 1.02 | 89.79 | 77.56 | 23.36 | 16.19
4 Sep. 2018, 09:53 | 13.0 $\pm$ 0.2 | 10.5 $\pm$ 0.1 | 0.40 | 1.02 | 89.79 | 77.56 | 23.38 | 16.20
4 Sep. 2018, 09:58 | 12.7 $\pm$ 0.2 | 10.9 $\pm$ 0.2 | 0.40 | 1.02 | 89.79 | 77.56 | 23.39 | 16.21
4 Sep. 2018, 10:03 | 13.3 $\pm$ 0.2 | 11.8 $\pm$ 0.2 | 0.40 | 1.02 | 89.79 | 77.56 | 23.40 | 16.22
4 Sep. 2018, 10:09 | 16.2 $\pm$ 0.3 | 11.0 $\pm$ 0.2 | 0.40 | 1.02 | 89.79 | 77.56 | 23.41 | 16.23
4 Sep. 2018, 10:58 | 15.4 $\pm$ 0.4 | 11.7 $\pm$ 0.2 | 0.40 | 1.02 | 89.80 | 77.57 | 23.52 | 16.30
4 Sep. 2018, 11:03 | 16.4 $\pm$ 0.3 | 12.0 $\pm$ 0.2 | 0.40 | 1.02 | 89.80 | 77.57 | 23.53 | 16.31
4 Sep. 2018, 11:08 | 16.0 $\pm$ 0.2 | 13.1 $\pm$ 0.2 | 0.40 | 1.02 | 89.80 | 77.57 | 23.55 | 16.32
4 Sep. 2018, 11:13 | 16.4 $\pm$ 0.2 | 12.9 $\pm$ 0.2 | 0.40 | 1.02 | 89.81 | 77.57 | 23.56 | 16.33
4 Sep. 2018, 11:19 | 16.5 $\pm$ 0.2 | 13.8 $\pm$ 0.3 | 0.40 | 1.02 | 89.81 | 77.57 | 23.57 | 16.33
4 Sep. 2018, 11:41 | 17.1 $\pm$ 0.2 | 15.5 $\pm$ 0.2 | 0.40 | 1.02 | 89.81 | 77.57 | 23.62 | 16.37
4 Sep. 2018, 11:46 | 17.1 $\pm$ 0.2 | 14.7 $\pm$ 0.2 | 0.40 | 1.02 | 89.81 | 77.57 | 23.63 | 16.38
4 Sep. 2018, 11:51 | 17.9 $\pm$ 0.3 | 14.6 $\pm$ 0.2 | 0.40 | 1.02 | 89.81 | 77.57 | 23.64 | 16.39
7 Sep. 2018, 10:12 | - | 5.4 $\pm$ 0.2 | 0.39 | 1.01 | 90.50 | 77.95 | 33.16 | 22.98
7 Sep. 2018, 10:18 | - | 5.9 $\pm$ 0.1 | 0.39 | 1.01 | 90.50 | 77.95 | 33.17 | 22.99
7 Sep. 2018, 10:22 | 9.0 $\pm$ 0.2 | 7.2 $\pm$ 0.1 | 0.39 | 1.01 | 90.50 | 77.95 | 33.19 | 23.00
7 Sep. 2018, 10:28 | 10.1 $\pm$ 0.3 | 6.4 $\pm$ 0.2 | 0.39 | 1.01 | 90.50 | 77.95 | 33.20 | 23.01
7 Sep. 2018, 10:33 | 9.3 $\pm$ 0.3 | 6.7 $\pm$ 0.2 | 0.39 | 1.01 | 90.50 | 77.95 | 33.21 | 23.02
7 Sep. 2018, 11:21 | 10.7 $\pm$ 0.2 | 7.1 $\pm$ 0.1 | 0.39 | 1.01 | 90.50 | 77.97 | 33.32 | 23.09
7 Sep. 2018, 11:30 | 10.7 $\pm$ 0.2 | 8.0 $\pm$ 0.1 | 0.39 | 1.01 | 90.50 | 77.97 | 33.34 | 23.11
7 Sep. 2018, 11:36 | 11.6 $\pm$ 0.3 | 7.8 $\pm$ 0.1 | 0.39 | 1.01 | 90.51 | 77.95 | 33.35 | 23.12
7 Sep. 2018, 11:41 | 10.6 $\pm$ 0.2 | 8.5 $\pm$ 0.1 | 0.39 | 1.01 | 90.51 | 77.95 | 33.36 | 23.12
7 Sep. 2018, 11:46 | 11.2 $\pm$ 0.2 | 8.3 $\pm$ 0.2 | 0.39 | 1.01 | 90.51 | 77.95 | 33.38 | 23.13
12 Sep. 2018, 09:55 | 11.4 $\pm$ 0.2 | - | 0.39 | 1.01 | 91.05 | 77.84 | 49.37 | 34.22
12 Sep. 2018, 10:00 | 11.7 $\pm$ 0.4 | - | 0.39 | 1.01 | 91.06 | 77.84 | 49.38 | 34.22
12 Sep. 2018, 10:06 | 11.7 $\pm$ 0.2 | - | 0.39 | 1.01 | 91.06 | 77.84 | 49.39 | 34.23
12 Sep. 2018, 10:11 | 11.9 $\pm$ 0.2 | 8.2 $\pm$ 0.1 | 0.39 | 1.01 | 91.06 | 77.84 | 49.40 | 34.24
12 Sep. 2018, 10:16 | 12.1 $\pm$ 0.5 | 8.6 $\pm$ 0.1 | 0.39 | 1.01 | 91.06 | 77.83 | 49.42 | 34.25
12 Sep. 2018, 11:26 | 14.1 $\pm$ 0.2 | 10.5 $\pm$ 0.2 | 0.39 | 1.01 | 91.06 | 77.83 | 49.57 | 34.36
12 Sep. 2018, 11:31 | 15.9 $\pm$ 0.3 | 12.9 $\pm$ 0.2 | 0.39 | 1.01 | 91.06 | 77.83 | 49.59 | 34.37
12 Sep. 2018, 11:37 | 14.7 $\pm$ 0.3 | 12.2 $\pm$ 0.2 | 0.39 | 1.01 | 91.06 | 77.83 | 49.60 | 34.37

###  2.1 Image Enhancement

Figure 1: Left: Unenhanced image of 21P/GZ from 28 August 2018 that has
undergone the basic data reduction. Right: Same image enhanced by an azimuthal
median division, where white denotes regions of higher flux. The white streaks
are star trails.

All images underwent a basic data reduction process which included a
subtraction of the bias level and a division of the flat field. Dark noise is
negligible as the camera is liquid nitrogen cooled. In our analysis, we
utilize enhancement techniques of the coma to highlight coma morphology
features and determine their repeatabilities. Samarasinha & Larson (2014)
explain the benefits of enhancement through several different methods,
including: dividing the images by an azimuthal median, and dividing or
subtracting two temporally close images by one another to reveal temporal
variations. We use a combination of these techniques to observe the behavior
of CN jet features in the gas coma. Specifically, to first highlight the
cometary jet features, we use a division by the azimuthal median profile as
explained in Samarasinha et al. (2014) as seen in Figure 2. We create these
enhanced images using the online tool provided in Martin et al. (2015)
(available at https://www.psi.edu/research/cometimen). Enhancement by
azimuthal median division allows us to distinguish two distinct CN jet
features from the overarching coma: a Northern and a Southern jet, as visible
in Figure 1.

Figure 2: Median azimuthal brightness versus the radius in log-log form for
each image from each night showing a consistent and smooth relationship
between projected radial distance and median azimuthal brightness. The
brightness variations are caused by variable sky conditions.

To confirm that an azimuthal median does not affect the shape of the features
but simply removes the overarching coma, we analyze the azimuthal median
removed from each image. As is visible in Figure 2, there does not appear to
be any significant radial structure that could affect the shape of the
features, specifically the distance at which the feature peaks in brightness.
The azimuthal median division is a proxy for the overlaying coma removal
without an actual measurement of it. The nature of Figure 2 shows a good
correlation between a radially expanding coma and the azimuthal median
measurements. For this reason, and because the coma obscures faint jet
features, the measurements described in §2.2 are taken from azimuthally
enhanced images.

###  2.2 Jet feature Morphology

The morphology of jet features, made clearer through enhancement, can be
studied to constrain the activity of the coma and the nucleus. We characterize
each jet’s behavior both radially and azimuthally.

To characterize the projected radial behavior of each jet, we measure the
projected radial distance at which the flux of each jet is at a maximum. This
is achieved using a wedge centered on each jet where the total flux at each
radius is measured, then fitting a parabola to the data as seen in Figure 3
near the brightness peak to determine the location of the maximum flux. A
parabola is chosen to fit the data because it closely matches the shape of the
measured brightness profile, however, it is unclear why this is. The one-sigma
error bars quoted in Table 1 result from a combination of the goodness of the
fit and the SNR of the brightness of the feature. Measurements where stars can
not be properly removed are omitted.

Figure 3: Demonstrative fitting algorithm of Northern Jet feature showing a
parabolic trend.

To characterize the azimuthal behavior of the jet features, we choose a set of
projected radial distances from the nucleus and measure the azimuthal
locations of the jets at each distance for each image. This allows us to track
the azimuthal motion of the jet features with time. From this, we detect
patterns and behaviors within each jet feature. The analysis of these
measurements are explained in §4.1.

###  2.3 Obtaining the CN Gas Outflow Velocity

Figure 4: Projected radial distance of the peak brightness versus time (in
Julian days) for the Northern Jet taken on 28 August 2018 showing a feature
moving radially away from the nucleus. A linear fit is shown over the data
with the slope representing the measured CN outflow velocity.

During the course of our investigation, we measured the projected radial
distance of the peak brightness of each jet moving away from the nucleus,
suggesting that some amount of material was released in that projected radial
direction and was moving away from the comet. This behavior allows us to track
the material’s movement over time, and thus its projected radial outflow
velocity. If we assume the outflow velocity of the material to be constant for
each jet over the course of a given night, we can fit a straight line to the
projected radial distance of the peak brightness versus time, where the slope
is the projected outflow velocity of the CN gas for that feature. Figure 4
shows the linear fit for 28 August 2018 for the Northern Jet. The errors of
the outflow velocities are determined using a Gaussian distribution of the
errors on the measurements and a Markov Chain Monte Carlo approach. In this
method, the data points are randomly placed along their error bars according
to a Gaussian distribution centered on the measurement point. For each
iteration, the outflow velocity is measured and recorded. After 100,000
iterations, the standard deviation of the recorded velocities is taken as the
error of the velocity measurement. Because of the projection effects, our
measured outflow velocities are all lower limits on what the actual CN outflow
velocities may be provided the jet widths are narrow (cf. Samarasinha, 2000).

###  2.4 Determination of the Periodicity

To determine the periodicity, we rely on the assumption that there exist
repeatable brightness fluctuations consistent with radial expansion. This
could consist, for example, of continuous jet features with rotationally
driven variations in outflow, or even line-of-sight enhancements due to the
geometry that specific directions of outflow make with the observer. In any
case, we observe radially expanding features that are periodic in nature with
brightness fluctuations, where the brightest location of the features are
shown in Figure 4.

To determine the periodicity, we perform a Lomb-Scargle Periodogram (which
specializes in discrete and uneven intervals of time) with the peak brightness
location versus time data for each jet (see §2.2) to generate a power density
spectrum (PDS) of the Fourier transform of our data for each jet individually.
From each PDS, we extract several most probable periodicities (VanderPlas,
2018). Once we determine the periodicities, we phase our data by them to find
the most probable period. The phasing equation as described in Stellingwerf
(1978) is given by:

$Phase=\frac{\text{mod }(t,P)}{P},$ (1)

where $t=JD-JD_{\circ}$, the Julian date ($JD$) minus some referenced or
starting point date ($JD_{\circ}$), and $P$ is the period in days. In this
equation, we divide the time - which is an adjusted day - by the period, and
take the remainder. This remainder is then divided by the period again to give
us a number between zero and one. This is a common method for obtaining the
rotational phase in other period search methods (e.g.: Schleicher et al.
(2013), Eisner et al. (2017)). Each phase is between zero and one, and each
cycle increases as our time baseline increases, with $JD_{0}=$ 28 August 2018
at 05:12 UTC. As an example, the first image was taken at 28 August 2018 at
08:58 UTC, which was 3.77 hours after $JD_{\circ}$. Taking the modulus of this
with respect to a 10.66 hour period yields a remainder of 3.77 hours. Dividing
this by the 10.66 hour period, the phase is found to be 0.35. Thus, the
nucleus has a phase of 0.35, or similarly, was 35% through a 10.66 hour
rotation at the time the image was taken.

##  3 Results

Comet 21P/GZ has two persistent CN jet features: a Northern feature and a
Southern feature (see Figure 1). For both features, we examine the projected
radial peak brightness distance as a function of time, which is used to
determine the periodicity and the projected CN outflow velocity (see §2 for
methodology).

###  3.1 Projected Radial Peak Brightness distance of CN Features

Figure 5 shows the projected radial peak brightness distance measured for each
CN jet feature for the four days of data that we have obtained, with the blue
circles representing the Northern jet and the orange circles representing the
Southern Jet. Apart from the scatter in measurements, the brightest regions of
both jets appear to be moving radially outward.

Figure 5: Projected radial distances of peak brightnesses for the Northern and
Southern Jets for the four nights of measurements. The blue circles represent
the Northern Jet and the orange circles represent the Southern Jet.

###  3.2 Periodicity

As explained in §2.4, we obtain the periodicity of the Northern and Southern
Jets by creating a Lomb-Scargle Periodogram of their peak brightness location
data versus time. As shown in Figure 6, the power spectrum for the Northern
Jet yields a maximum power at a frequency of 3.25 oscillations per day,
corresponding to a period of 7.39 $\pm$ 0.01 hours, with secondary peaks at
10.66 $\pm$ 0.01 hours and 5.7 $\pm$ 0.1 hours corresponding to frequencies of
2.25 and 4.24 oscillations per day, respectively. The power spectrum for the
Southern Jet yields a maximum power at a frequency of 2.85 oscillations per
day, corresponding to a period of 8.41 $\pm$ 0.01 hours, with secondary peaks
at 6.24 $\pm$ 0.01 hours and 13.0 $\pm$ 0.4 hours corresponding to frequencies
of 3.85 and 1.85 oscillations per day, respectively. The peaks seen in the
power spectrum for the Northern Jet can be seen as tertiary peaks in the power
spectrum for the Southern Jet, and vice versa. This indicates that the peak
frequencies for each power spectrum are significant and are not artifacts of
the periodogram. In addition, mixing the power spectra for the Northern and
Southern Jets by multiplying them by each other results in the same six most
powerful frequencies, although it is unclear why the most prominent peaks have
different strengths for the two jets.

Figure 6: Upper Left: Power density spectrum for the Northern jet. Prominent
peaks located at 2.25, 3.25, and 4.24 oscillations per day are labeled,
corresponding to periods of 10.66, 7.39, and 5.7 hours. Upper Right: Power
density spectrum for the Southern Jet. Prominent peaks located at 1.85, 2.85,
and 3.85 oscillations per day are labeled, corresponding to periods of 13.0,
8.41, and 6.24 hours. Potentially significant peaks for both jets are labeled,
although the peaks at 4.24 and 1.85 oscillations per day for the Northern and
Southern Jets, respectively, result in less compelling phase diagrams and
image divisions than the others. Note that in both power spectra, the most
prominent peaks are spaced by 1 oscillation per day. This is a common result
of periodograms due to daily aliasing, and performing a window clean did not
eliminate the primary frequencies observed due to the limited number of data
points. In addition, the most prominent peaks in either power spectrum appear
as weaker frequencies in the other. Lower Left: Mixed power spectrum created
by multiplying the power spectra of the Northern and Southern Jets. The six
most powerful frequencies in the mixed spectrum correspond to the three most
powerful frequencies in each of the Northern and Southern Jet power spectra.

These peak frequencies are then used to phase the peak radial brightness of
each feature as explained in §2.4 to begin eliminating weaker periodicities
from consideration for 21P/GZ’s rotational period. Phase diagrams using the
four frequencies resulting in the best phase-location correlations are shown
in Figure 7, with the greatest correlations corresponding to periodicities of
7.39 $\pm$ 0.01 and 10.66 $\pm$ 0.01 hours for the Northern Jet. The 5.7 $\pm$
0.1 and 13.0 $\pm$ 0.4 hour periods had the weakest phase-location
correlations are were eliminated from consideration.

Figure 7: Phase diagrams of the Northern and Southern Jets plotted for 6.24
$\pm$ 0.01 hours, 7.39 $\pm$ 0.01 hours, 8.41 $\pm$ 0.01 hours, and 10.66
$\pm$ 0.01 hours, which are the more prominent peaks of the power density
spectra in Figure 6. From these phase diagrams, the 7.39 hour and 10.66 hour
figures have the strongest correlation, as seen by a clear overlap of data
from all cycles.

Figure 8: Same phase divisions for the four periods, with the divisions with
the least residual features in the top row, and divisions with the most
residual features in the bottom row. Columns a, b, c, and d show divisions for
the 6.24 $\pm$ 0.01, 7.39 $\pm$ 0.01, 8.41 $\pm$ 0.01, and 10.66 $\pm$ 0.01
hour periods, respectively.

Figure 9: Divisions of image pairs offset by half rotations for each period.
Columns A, B, C, and D show divisions for the 6.24 $\pm$ 0.01, 7.39 $\pm$
0.01, 8.41 $\pm$ 0.01, and 10.66 $\pm$ 0.01 hour periods, respectively.

To continue testing the different periodicities, we divide different sets of
images, one set with images of the same phase and one set with images half a
phase apart, and compare the results as described in Samarasinha & Larson
(2014). Dividing two images of the same phase —a perfect period match— with no
significant viewing geometry change nor other abnormality, would result in a
flat image with no residual. For a half phase division, the results would be a
maximum residual. In our study, we test 6.24 $\pm$ 0.01 hours, 7.39 $\pm$ 0.01
hours, 8.41 $\pm$ 0.01 hours and 10.66 $\pm$ 0.01 hours as shown in Figure 8
for both the full phase division and half phase division, noting that the
viewing geometry changed less than approximately 15∘ for each division. Figure
8 shows the best and worst same phase divisions for each period. All same
phase divisions yielded few residual features, indicating each period is
plausible. Furthermore, Figure 9 shows the half phase divisions for the four
periods, all showing significant residual features. Because the same phase and
half phase divisions for all four periods show the expected residual features,
none can be ruled out by Figures 8 and 9. As a result, the 7.39 $\pm$ 0.01 and
10.66 $\pm$ 0.01 hour periods are chosen as the most likely periods for Comet
21P/GZ because they result in the strongest phase plots for both jets in
Figure 7, although the 6.24 $\pm$ 0.01 and 8.41 $\pm$ 0.01 hour periods remain
plausible.

Full phase divisions also allow for the reduction of the errors for the four
periods tested. This is done using a reference image (called image A) to
divide by another image of the same phase (called image B). Images of
increasingly different phase (image C) are then divided from the reference
image until there is a noticeable increase in residual features. Next, the
time elapsed between images B and C is calculated. This time represents the
magnitude of the error accrued over the integer number of rotations, N,
elapsed between images A and B. To find the error in the period per rotation,
the time elapsed between B and C is divided by N. This was rounded to the
nearest hundredth for the four periods used, and had a rounded value of 0.01
hours per rotation for each.

###  3.3 Projected Outflow Velocity of CN Gas

Table 2: CN projected outflow velocity measurements.  Date |  Northern Jet Velocity (m s-1) |  Southern Jet Velocity (m s-1)
---|---|---
2018-08-28 | 730 $\pm$ 30 | 590 $\pm$ 30
2018-09-04 | 610 $\pm$ 30 | 580 $\pm$ 20
2018-09-07 | 350 $\pm$ 40 | 400 $\pm$ 20
2018-09-12 | 590 $\pm$ 40 | 740 $\pm$ 30

Figure 5 shows that both the Northern and Southern Jets are moving radially
away from the nucleus as projected onto the plane of the sky for each night.
As explained in §2.3, we measure the CN projected outflow velocity, summarized
in Table 2, for both the Northern and Southern Jets. The velocities measured
represent the projected outflow velocities onto the plane of the sky and are
thus lower limits on the outflow velocities for each CN jet. However, as
explained in Samarasinha (2000), if the CN jets are broad, these may be
representing direct outflow measurements.

##  4 Discussion

As stated in §3, we find that comet 21P/GZ has a rotation period of either
7.39 $\pm$ 0.01 or 10.66 $\pm$ 0.01 hours in the August-September 2018 time-
frame. Additionally, we find that the maximum projected outflow velocity of
the CN gas is 730 $\pm$ 30 m s-1 for the Northern Jet, and 740 $\pm$ 30 m s-1
for the Southern Jet. We can further analyze the morphology of the CN features
to constrain the rotational state of the nucleus and gain valuable information
about the nature of the jet features themselves.

###  4.1 CN Jet Feature Morphology

We analytically determine the azimuthal location of each jet feature as a
function of projected distance for both the Northern and Southern Jets as
explained in §2.2. Figures 10–13 and Figures 14–17 show the morphology of the
Northern and Southern Jets, respectively, spaced over each night for 28 August
2018, 4 September 2018, 7 September 2018, and 12 September 2018. For the
Northern Jet, there is a clear corkscrew pattern shown in several of these
figures while the Southern Jet does not show as clear a pattern. We measure
the behavior of the corkscrew pattern in the Northern Jet by determining the
position angle of its central location, and its width. These results are
summarized in Table 3. The small width of the corkscrew pattern suggests that
the jet originates at a location near the pole ranging from $\sim$4∘ to
$\sim$6∘ away, indicating a cometocentric latitude of approximately $\sim$86∘
to $\sim$84∘. Additionally, considering the center of the corkscrew, we can
determine the axis along which the comet must rotate for each night of
observations.

Table 3: Measured position angle of the Northern Jet’s corkscrew center and the width of the pattern. | Position | Width of
---|---|---
Date |  Angle (deg) |  Corkscrew (deg)
2018-08-28 | 20 $\pm$ 4 | 11 $\pm$ 4
2018-09-04 | 29 $\pm$ 3 | 11 $\pm$ 3
2018-09-07 | 26 $\pm$ 4 | 8 $\pm$ 4
2018-09-12 | 15 $\pm$ 3 | 9 $\pm$ 3

Figure 10: Northern Jet morphology for 28 August 2018 at five different times
spread out across the night, showing outward movement and a clear corkscrew
pattern.

Figure 11: Northern Jet morphology for 4 September 2018 at five different
times spread out across the night, showing outward movement and a clear
corkscrew pattern.

Figure 12: Northern Jet morphology for 7 September 2018 at five different
times spread out across the night; this night does not clearly exemplify the
same corkscrew pattern seen in previous nights, nor does it show as drastic of
outward movements.

Figure 13: Northern Jet morphology for 12 September 2018 at four different
times spread out across the night, showing a definite change in the curvature.

Figure 14: Southern Jet morphology for 28 August 2018 at five different times
spread out across the night, showing a curved jet most likely due to radiation
pressure, with some near-nucleus curvature.

Figure 15: Southern Jet morphology for 4 September 2018 at five different
times spread out across the night, showing a potential corkscrew pattern more
visible at less than 20,000 km.

Figure 16: Southern Jet morphology for 7 September 2018 at five different
times spread out across the night, showing a strong curvature near the nucleus
that is most prominent in the images taken at 11:21 and 11:46 UTC, potentially
representing a corkscrew pattern.

Figure 17: Southern Jet morphology for 12 September 2018 at four different
times spread out across the night, showing a potential corkscrew pattern more
visible at less than 20,000 km and a strong curvature at greater distances
most likely due to solar radiation pressure.

###  4.2 Constraining the Pole Vector

On a given night, the sky-plane projected rotational pole direction coincides
with the measured position angle of the corkcrew’s center, as shown in Table
3. This, however, does not provide us with the non-projected information,
which can be obtained by having a sufficient change in observing geometry of
the comet. Utilizing similar methods as described in Samarasinha & Mueller
(2002) and Farnham & Cochran (2002), we are able to constrain the pole vector
using the Northern Jet.

We can determine the direction, in Right Ascension, $\alpha$, and Declination,
$\delta$, of the pole vector of 21P/GZ as described below. First, we define
the Earth’s direction from the frame of reference of the comet as
$\alpha_{\earth}$ and $\delta_{\earth}$ for the Right Ascension and
Declination, respectively. Next, using a spherical to Cartesian
transformation, we can describe the Cartesian coordinates of the pole vector
of 21P/GZ, $x_{p}$, $y_{p}$, and $z_{p}$, as:

$\displaystyle x_{p}$ $\displaystyle=\cos{\delta}\cos{\alpha}$ (2)
$\displaystyle y_{p}$ $\displaystyle=\cos{\delta}\sin{\alpha}$ (3)
$\displaystyle z_{p}$ $\displaystyle=\sin{\delta}.$ (4)

The next step is to perform a transformation between the comet’s frame of
reference and the Earth’s frame of reference. In this step, we are projecting
the comet’s frame of reference into the plane of the sky where the “Northern”
component – labeled as $x_{N}$, $y_{N}$, and $z_{N}$ – will be the component
projected on the skyplane’s north direction while the “Eastern” component –
labeled as $x_{E}$, $y_{E}$, and $z_{E}$ – will be the component projected
onto the skyplane’s eastern direction. Thus:

$\displaystyle x_{N}$
$\displaystyle=-\sin{\delta_{\earth}}\cos{\alpha_{\earth}}$ (5) $\displaystyle
y_{N}$ $\displaystyle=\sin{\delta_{\earth}}\sin{\alpha_{\earth}}$ (6)
$\displaystyle z_{N}$ $\displaystyle=\cos{\delta_{\earth}},$ (7)

and

$\displaystyle x_{E}$ $\displaystyle=\sin{\alpha_{\earth}}$ (8) $\displaystyle
y_{E}$ $\displaystyle=-\cos{\alpha_{\earth}}$ (9) $\displaystyle z_{E}$
$\displaystyle=0.$ (10)

The final step is then to project the comet’s pole vector onto the skyplane –
$Pole_{N}$ and $Pole_{E}$ for the northern and eastern components of the pole
vector, respectively – to be converted to a measurable position angle, and
compared to our results. The pole vector projected onto the skyplane is given
by:

$\displaystyle Pole_{N}$ $\displaystyle=x_{p}\cdot x_{N}+y_{p}\cdot
y_{N}+z_{p}\cdot z_{N}$ (11) $\displaystyle Pole_{E}$
$\displaystyle=x_{p}\cdot x_{E}+y_{p}\cdot y_{E}+z_{p}\cdot z_{E},$ (12)

with the projected position angle of the pole position, $PA$, being given by:

$PA=atan2(Pole_{N},Pole_{E}).$ (13)

Figure 18: Top: Potential pole vector solutions for 28 August, and 4, 7 and 12
September 2018 as projected onto the celestial sphere. Intersections between
pairs of nights are marked with $\times$, and the parabolic fit to the
intersection points is shown in black. The $\odot$ and $\oplus$ indicate the
Solar and Earth directions from the comet’s frame of reference on the days
corresponding to their color in the legend. The pole vector solution is marked
at a Right Ascension of ${169^{+28}_{-23}}^{\circ}$ and a Declination of
${73^{+5}_{-11}}^{\circ}$. The error bars are computed by taking the standard
deviations of the locations of the intersections on either side of the
solution in both RA and DEC. Bottom: The same results zoomed in at the
intersection point show that solutions for all four nights nearly overlap the
pole vector solution and reveal the resolution of our grid search.

Where $atan2(Pole_{N},Pole_{E})$ is the angle between the North direction on
the sky plane and the vector created by PoleN and Pole${}_{E}.$ To determine
the possible pole positions that correspond to our measurements in Table 3, we
conduct a grid search over all Right Ascensions and Declinations of potential
pole vectors and reject all values which are not within the uncertainty of our
measured projected pole position angles. Figure 18 shows the acceptable pole
positions for all four nights of data.

To find the pole solution, all intersection points between pairs of nights are
mapped, with six intersection points in total, as shown in Figure 18. This
produces a family of intersections for the pole position. The family of
intersections is then fitted with a parabola, which closely matches the
distribution of intersection points. The Right Ascension and Declination at
the halfway point of the parabola’s arc length are taken as the center of the
family of solutions and the pole position of 21P/GZ. The position of 21P/GZ’s
pole vector is found to be at a Right Ascension of ${169^{+28}_{-23}}^{\circ}$
and a Declination of ${73^{+5}_{-11}}^{\circ}$. While the Right Ascension
error is large, it is offset by the pole vector’s high declination. The
opposite pole vector would logically point to a Right Ascension of
${349^{+28}_{-23}}^{\circ}$ and a Declination of ${-73^{+11}_{-5}}^{\circ}$.
Due to the lack of information surrounding the spin direction, we cannot
conclude which pole corresponds to the conventionally defined positive pole.
The pole vector of 169∘ and 73∘ in Right Ascension and Declination,
respectively, is equivalent to an orbital longitude of 199∘ and obliquity of
10∘ in orbital coordinates, where orbital longitude is measured from the Sun-
perihelion direction in a prograde sense in the orbital plane of comet 21P/GZ.

###  4.3 Using our Determined Period to Calculate the $X$ parameter

When a comet approaches the sun, it releases gas that creates torques on its
nucleus; these torques affect the rotation of the comet over time. Samarasinha
& Mueller (2013) introduce the $X$ parameter, calculated by:

$X=\frac{|\Delta P|R^{2}}{P^{2}\zeta},$ (14)

where $\Delta P$ is the change in rotational period per orbit, $R$ is the
radius of the nucleus, $P$ is the period of the comet, and $\zeta$ is the
amount of water lost each orbit per unit surface area. The $X$ parameters of
comets are expressed as the ratio between their $X$ parameter and that of
comet 2P/Encke. Most comets have an $X/X_{Encke}$ within a factor of five
(Mueller & Samarasinha, 2018). We use our measured period, along with a period
of 9.5 hours measured in the 1985 21P/GZ apparition by Leibowitz & Brosch
(1986b) to determine the change in period per orbit. Table 4 provides $\Delta
P$ obtained from a combination of our measurements and those of Leibowitz &
Brosch (1986b) along with $R$ and $\zeta$ values obtained from Mueller &
Samarasinha (2018). Using Equation 14 we calculate that
$X_{GZ}/X_{Encke}\approx$ 3.4 $\pm$ 0.1 for a period of 7.39 $\pm$ 0.01 hours
or $X_{GZ}/X_{Encke}\approx$ 1.9 $\pm$ 0.1 for a period of 10.66 $\pm$ 0.01
hours. This further supports the conclusion found in Mueller & Samarasinha
(2018) that the $X$ parameter is a constant to within a factor of a few unity
and suggests that 21P/GZ has not undergone abnormal torque effects in the
recent past.

Table 4: Values used to calculate $X_{GZ}/X_{Encke}$. $\Delta P$ is calculated from the change in period from Leibowitz & Brosch (1986b) and our determined period, which was measured 5 orbits later. The other values are obtained from Mueller & Samarasinha (2018). Comet |  $\Delta P$ (minutes) |  $R$ (km) | P (hours) | $\zeta/\zeta_{Encke}$
---|---|---|---|---
21P/GZ (7.39 hour period) | 25.3 | 1.0 | 9.5 | 0.43
21P/GZ (10.66 hour period) | 14.4 | 1.0 | 9.5 | 0.43
2P/Encke | 4.0 | 2.4 | 11.0 | 1.0

###  4.4 Unanswered Questions about Jet Morphology

The morphology of the Southern Jet, as shown in Figures 14 to 17, does not
follow a clear and consistent pattern. However, it is possible to observe that
at large cometocentric distances, the Southern Jet appears to be curving away
from the sunward direction ($PA_{\odot}\approx 90^{\circ}$). This can be
explained easily through solar radiation pressure. The relativistic momentum
of a photon, $p$, is given by:

$p=\frac{E}{c},$ (15)

where $E$ is the energy of the photon and $c$ is the speed of light in a
vacuum. The average energy that the CN molecules will then receive can be
calculated using the fluorescence efficiency, $g$ as follows:

$E=gt,$ (16)

where $t$ represents the amount of time a molecule spends in the coma.
Combining Equations 15 and 16 provides:

$p=\frac{gt}{c}.$ (17)

Knowing that $F=dp/dt$, we can calculate the force due to radiation pressure
using Equation 17 to obtain $F_{rad}=g/c$. Finally, using Newton’s second law,
$a=\frac{F}{m}$, we can solve for the acceleration due to radiation pressure,
$a_{rad}$ in the direction opposite of the solar direction to obtain:

$a_{rad}=\frac{g}{cm_{CN}},$ (18)

where $m_{CN}$ is the mass of a CN molecule. Using the fluorescence
efficiencies from Schleicher (2010) for 21P/GZ on 28 August, 2018, we find the
acceleration due to radiation is $\sim$ 0.25 cm s-2. This decreases to 0.21 cm
s-2 by 12 September, 2018 due to a decreased fluorescence efficiency, $g$.

If we assume all the acceleration to be perpendicular to the motion of the
jet, and we assume a measured outflow velocity, v=730 $\pm$ 30 m s-1, at
50,000 km from the nucleus, the jet would have moved 5,900 $\pm$ 500 km in the
anti-sunward direction, the equivalent of $6\pm 1^{\circ}$. This is consistent
with the curvature at higher distances as seen, for example, in Figure 14.

##  5 Conclusion

We observed comet 21P/GZ over four nights from 28 August 2018 to 12 September
2018. We azimuthally enhanced our data to reveal the coma morphology of CN jet
features. We used a newly developed method that measured the projected radial
distance of the peak flux of a jet versus time, then using a Lomb-Scargle
Periodogram, we obtained a rotational periodicity of either 7.39 $\pm$ 0.01 or
10.66 $\pm$ 0.01 hours. From the rotational periodicity of 21P/GZ, and
previously obtained measurements dating from 1985 (Leibowitz & Brosch, 1986b),
we were able to measure the $X$ parameter, as defined in Samarasinha & Mueller
(2013), and obtain $X_{GZ}/X_{Encke}\approx$ 3.4 $\pm$ 0.1 for a periodicity
7.39 $\pm$ 0.01 hours or $X_{GZ}/X_{Encke}\approx$ 1.9 $\pm$ 0.1 for a
periodicity 10.66 $\pm$ 0.01 hours, which is consistent with contemporary
literature.

Additionally, we were able to obtain a maximum projected CN gas outflow
velocity for the Northern Jet of 730 $\pm$ 30 m s-1 and for the Southern Jet
of 740 $\pm$ 30 m s-1.

We further analyzed both jets’ morphologies to constrain rotational states. We
determined that the Northern Jet possessed a corkscrew pattern, approximately
10∘ in width, suggesting a proximity to the pole position. From that analysis,
we were able to constrain a pole position at a Right Ascension of
${169^{+28}_{-23}}^{\circ}$ and a Declination of ${73^{+5}_{-11}}^{\circ}$,
with undetermined sense of rotation. From the analysis of the Southern Jet, we
concluded that radiation pressures had a significant effect on the curvature
at large cometocentric distances.

##  6 Acknowledgements

We would like to thank the invaluable time and effort given by Dr. Walt Harris
in the mentoring necessary for the completion of this work. We would like to
thank Dr. Beatrice Mueller for her contribution to the data analysis process.
We would like to thank everyone who participated in the data gathering
process, including all students from the Lunar and Planetary Laboratory course
PTYS 495B/595B (Fall 2018). We would like to thank the Steward Observatory
technical staff for the amount of time they dedicated to this project.
Finally, we would like to thank the SSO grant NNX16A670G awarded to Walt
Harris and the NESSF grant 80NSSC18K1241 awarded to Cassandra Lejoly (PI:
Harris) for allowing this work to be completed.

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