Paper on the orbital determination of near-earth asteroid 68950

1. Orbital Determination of Asteroid 68950 (2002QF15)
Team 3: Galileo Al Dente
E. Higgsa, K. Kimb, A. Rupertoc
Summer Science Program, Etscorn Observatory, New Mexico Institue of Mining and Technology, 801 Leroy Pl, Socorro, NM
87801
21 July 2019
a
North Kitsap High School, Washington, United States of America
b
Leigh High School, California, United States of America
c
The Bolles School, Florida, United States of America
Abstract
Tens of thousands of near-Earth asteroids exist, with roughly 95% of them currently known and monitored.
To track near-Earth asteroids and determine their orbits however, careful measurements of the asteroid’s
celestial coordinates at several points of orbit must be taken in order to calculate the asteroid’s orbital
elements. Through several observations of our asteroid 68950 (2002 QF15) at Etscorn Observatory, we
perform astrometry on images of our asteroid taken to determine the respective right ascensions and decli-
nations of our asteroid at each observation. The orbital elements, as well as the position and velocity vectors
of the asteroid were then calculated using the Method of Gauss, a mathematical approach for preliminary
orbit determination that requires celestial information from at least three observations from three different
days over the span of two weeks for an accurate determination. After gathering the orbital elements for
our asteroid, we compared them with those calculated by JPL Horizons, a highly accurate online ephemeris
system.
1. Introduction
The solar system contains a vast array of celestial objects, including, but not limited to, planets, comets,
asteroids, and debris. While many comets are found in the Kuiper Belt, a region of non-planetary objects
found just beyond the orbit of Neptune, most asteroids are found in the asteroid belt between the orbits of
Mars and Jupiter. Consequently, slight gravitational pushes against these asteroids can result in their orbits
coinciding with that of Earths, thus posing a hazardous threat.
Near Earth Asteroids (alternatively called Near Earth Objects) mostly originate in the asteroid belt and
orbit at a relatively close distance of less than 1.3 AU perihelion distance from the sun. Monitoring the orbits
of these near-Earth asteroids and other celestial objects are one of the many goals of modern astronomy;
the use of telescopes and observatories both on the ground and in space help to accomplish this goal.
Among Near Earth Asteroids, there are certain asteroids that are more threatening than others, collectively
referred to as Potentially Hazardous Asteroids, which are defined as asteroids which come within 0.05 AU
of Earth’s orbit. While this still a very large distance, these asteroids have a small chance of going off-course
and crashing into the Earth. As such, it is particularly important to monitor the orbits of these asteroids.
Since 2002 QF15 is a Potentially Hazardous Asteroid, our research will be valuable in helping surmise how
much of a risk it poses to the Earth. We intend to predict the orbit of our asteroid using the Method of
Gauss, which relies on 3 successful observations of asteroid spread out over time and the right ascension,
Preprint submitted to Elsevier July 31, 2019

2. declination, and exact time of those observations. We then plan to use various programs programmed in
Python3 to predict the path of our asteroid.
Fig. 1. Visualization of the orbital elements [5]
Table 1. Orbital Elements
a Semi-Major Axis The distance of the longest radius of the ellipse. Measured
in AU.
e Eccentricity Describes the elongation of the ellipse. Has no dimensions.
i Inclination Angle between ecliptic plane and the asteroid’s orbital
plane. Measured in degrees.
Ω Longitude of Ascending
Node
Angle between the Vernal Equinox and the ascending node.
Measured in degrees.
ω Argument of Perihelion Angle between the Vernal Equinox and the perihelion.
Measured Eastward in the ecliptic plane in degrees.
M Mean Anomaly The angular position of the asteroid projected onto a cir-
cular orbit, precessed to July 21, 6:00 UTC. Measured in
degrees.
2

3. Table 2. Definitions
α, δ The celestial coordinates, right ascension and declination, respectively. Right as-
cension is an angular measure between the object and the Vernal Equinox, and
declination is an angular measure between the object and the equator.
The six orbital elements are mathematically determined numbers and constants that are used in describ-
ing a celestial objects orbit. For instance, the eccentricity, e, of an orbit describes the elongation of ellipse,
which can be determined through the semiminor and semimajor axis of the orbit. Each orbital element
can be obtained mathematically through the Method of Gauss, which relies on 3 successful observations
of asteroid spread out over time and the right ascension, declination, and exact time of those observations.
Through processing, and performing calculations on data gathered in several observations of an object, its
orbit can be obtained. Through several observations spread out over 5 weeks at Etscorn observatory, we will
use the Method of Gauss for preliminary orbit determination to calculate each of the six orbital elements
(table 1) for asteroid 68950 (2002 QF15), as well as its position and velocity vectors.
2. Observations and image processing
2.1. Methods
2.1.1. Instruments and Software
Our team used a 0.36m diameter C-14 telescope with an f/11 reflector and a CCD chip at the Etscorn
Observatory in Socorro, New Mexico. In addition, we used the TheSkyX software to control the telescope
as well as DS9 software to create star charts for finding our asteroid, locate focus stars, and find the po-
sition of stars for doing astrometry and photometry on our images. We used AstroImageJ to reduce our
images. Lastly, we created Python programs to process our data and perform astrometry, photometry, orbit
determination, orbit visualization, and two different types of statistical analysis.
2.1.2. Observations
For our observations, we spent seven shifts of three hours at Etscorn Observatory taking images of our
asteroid. Out of these seven, three were hindered by weather conditions preventing us from observing our
asteroid. After starting up the telescope and computer, we focused the camera using the focuser, bahtinov
mask, and a focus star, acquired using DS9. Then, we slewed the telescope to our asteroid’s predicted
position and began taking visual images. Generally, we took one set of visual images amounting to roughly
15 minutes, then a set of darks and a set of biases, before taking another few sets until our shift was over.
2.2. Observation sessions
Table 3. Observation session data
Date UT Time Images Filter Exposure Time Conditions
20 June 2019 3:00-5:00 18 Visual 60 s Clear sky
23 June 2019 3:00-5:00 20 Visual 60 s Clear sky
28 June 2019 5:00-7:00 n.a. n.a. n.a. Cloudy
2 July 2019 5:00-7:00 n.a. n.a. n.a. Cloudy
5 July 2019 5:00-7:00 23 Visual 120 s Mostly Clear
10 July 2019 5:00-7:00 40 Visual 60 s Clear sky
13 July 2019 5:00-7:00 n.a. n.a. n.a. Cloudy
3

4. While there were no clouds during our second observation, our asteroid was very high in the sky, at
around 86◦ altitude, and the dome partially blocked our field of view, no matter which of our three focus
stars we slewed to. Thus, we tried to jog the telescope down until it no longer saw the dome and proceeded
to take sets of visuals from that location; however, our asteroid was no longer in the telescopes field of view
and we did not obtain any useful data from that observation.
2.3. Image Reduction and Asteroid Location
Once we’ve taken our science images, we can begin to analyze them. Unfortunately, bias and dark
current in the raw images, as well as slight misalignments between images, prevent us from extracting any
useful information. In order to make these images usable, we have to correct for these errors. We did this
using AstroImageJ, as well as dark and bias images. Dark exposures are long exposures with the shutter
closed, which collects signal from internal charge accumulation. Bias frames are images with zero exposure
time, which collect signal from existing charge interference. With these images, we can generate a master
dark and master bias image for each observing session, generated by averaging all images in each set. We
also have to remove bias from the master dark. To remove any source of error in an image, we do what is
referred to as ”subtraction,” where the software takes each image and, pixel by pixel, subtracts the signal
of one from the other. Using AstroImageJ’s data reduction software, we reduced our science images by
subtracting the bias, and then subtracting the bias-corrected dark. Then, we must deal with interference
from the instrument itself, such as variance in pixel sensitivity, or dust on the telescope lenses. To do this,
we correct using flat images. Flat images are images of a uniform illumination source. For our purposes,
flat images for each week, as well as biases and darks to correct the flats, were taken for all teams to use.
Before we can correct our science images with the flats, we have to bias- and dark-correct the flats, which
we do using the biases and darks taken at the same time as the flats, using the same process as we did for the
science images. We can then divide out the flat signal to correct the science images.Once we’ve removed
dark and bias current, we use AstroImageJ to align the images, using manually selected stars as reference.
Fig. 2. Reduced Image from July 5, with 2002 QF15 labelled
4

5. With our reduced images, we can use AstroImageJ to display all of the images from one night as a
sequence, which we can blink through and see moving objects. With a star chart and reference of expected
asteroid position, we can zoom in on the correct region and locate the asteroid by finding the spot that moves
from image to image. Once we’ve located our asteroid, we can use AstroImageJ to label the asteroid with
an annotation (for future reference), as well as finding the asteroid’s X and Y position on the image to use
in our data analysis.
3. Orbit determination
3.1. Centroiding
After reducing the images obtained from observation, it was necessary to use centroiding to determine
the positions of several celestial objects based on their brightness. Centroiding is the process of determining
the fractional pixel-location center of an object based on an array of values passed in to the program (in this
case, it was an array of brightness values from reducing the CCD outputs). Centroiding requires a rough
estimate of the coordinates of the celestial object in the array, as well as a numerical size (or radius) of an
aperture that will set the bounds for calculating the position of the object in the array. As most of the celestial
objects used in orbit determination were circular in nature, we used a circular aperture that excluded border
pixels to centroid our objects. After obtaining an array of pixels inside the aperture, we get the weighted
average of the x and y rows and columns respectively.
¯x =
xiwi
wi
¯y =
yiwi
wi
In this,x is the x-position, y is the y-position, and w is the pixel count of the position referenced. Then,
we proceed to find both the standard deviation of the centroid and the standard deviation of the mean,
σ¯x = σ√
N−1
, where σ is the standard deviation and N is the number of x values in the centroid calculation,
which we use to show the error on the centroid calculation. [3]
3.2. Least Squares Plate Reduction (LSPR) Astrometry
The next step in the analysis of our images involves mapping our images onto celestial coordinates, in
order to find our asteroid’s observed position in the sky. To do this, we used a Least Squares Plate Reduction
(LSPR) model, which takes the known position of a number of reference stars (typically around 12), as well
as their X-Y coordinates in the image. Right Ascension and Declination can be related to X-Y coordinates
using 6 plate constants, unique to each image, which describe how the coordinates are scaled, rotated, and
translated on the image as follows:
α
δ
=
b1
b2
+
a11 a12
a21 a22
x
y
where b1, b2, a11, a12, a21, a22 are the plate constants. The LSPR model follows a χ2 minimization model,
which works by minimizing χ2 where χ represents the sum of the squares of the difference between the
actual data and the model. For LSPR, the data is the actual celestial coordinates of the reference stars, and
the model is the coordinates produced using the plate constants. Upon relating the plate constants to χ using
the equation above and minimizing χ2, we find that:
5

6. 

Σαi
Σαixi
Σαiyi


=


N Σxi Σyi
Σxi Σx2
i Σxiyi
Σyi Σxiyi Σyi




b1
a11
a12


and


Σδ
Σδxi
Σδyi


=


N Σxi Σyi
Σxi Σx2
i Σxiyi
Σyi Σxiyi Σyi




b2
a21
a22


where N refers to the number of reference stars used, and αi, δi, xi, and i refer to the celestial and image
coordinates of each reference star. Using these equations, we can find the plate constants for a given image,
which we can then use to produce the celestial coordinates of the asteroid in the image using the X-Y
centroid and the above equation. From there, we determine uncertainty using the standard deviation of the
mean, defined as:
σα =
1
(N − 3)
N
i=1
(αactual − αfit)2
and
σδ =
1
(N − 3)
N
i=1
(δactual − δfit)2
where σα and σδ refer to the uncertainties, αactual and δactual refer to the actual celestial coordinates for a
given star, and αfit and δfit refer to the celestial coordinates produced by applying the plate constants to the
X-Y coordinates of a given star in the image. [3]
3.3. Astrometry
Astrometry describes the process of determining the J2000 coordinates of our asteroid based on a set of
image obtained from the CCD chip after an observation. To do astrometry, we use our centroid code to get
the x-y centers of all the references stars we have chosen and to find the center of our asteroid in the image.
Then, we plug that information into the LSPR code in order to find the right ascension and declination of
our asteroid based on its x-y position in the image. Additionally, we can find the error in our calculations
by comparing our calculated values of right ascension and declination of stars to the stars’ known position.
6

7. 3.4. Photometry
Photometry describes using the known location of the asteroid in the image and trying to find its visual
magnitude (signal) and the error on that (noise). While this is unimportant for orbital determination, it is
required to submit our data to the Minor Planet Center. In our images, we can find the signal on an object
by taking the average signal inside a circular aperture, and subtracting the average sky background (found
by taking the average signal in a circular annulus around the object). From there, we find the instrumental
magnitude Vinst and signal to noise ratio (SNR) with:
−2.5log(signal)
and
S NR =
√
S
1 + nap(1 +
nap
nan
)(S kye+De+ρ2
S
where
ρ = r2
+ (
g
√
12
)2
and S e refers to the object’s signal in electrons, na p refers to the number of pixels in the aperture, nan refers
to the number of pixels in the annulus, S kye refers to the average sky background in electrons, De refers to
the dark current, r is the read noise of the image, and g is the gain of the CCD. For the C-14 at Etscorn,
the gain is 0.8. However, instrumental and visual magnitudes are not identical- they are instead related by
a zero-point offset. To determine this offset, we can use data from reference stars located in the image and
perform a linear regression. After that, we can use the modelled linear relationship to calculate an observed
visual magnitude for our asteroid.[3]
3.5. Method of Gauss
3.5.1. Preliminaries
The method of Gauss solves for the position vector of the asteroid r and velocity vector of the asteroid ˙r
for the central observation given three sets of data ti, αi, δi assuming that the asteroid is in Keplerian motion
about the sun. Subsequently, given r and ˙r, and t2 we can solve for the six orbital elements. Thus, we
focus on using the Method of Gauss to obtain our position and velocity vectors. With the asteroid’s position
vector from the earth p calculable, and the earth sun vector R known, we can use the fundamental triangle
and vector addition to calculate r and ˙r.
7

8. Fig. 3. The Fundamental Triangle
where
r = ρ − R = ρˆρ − R (1)
3.5.2. Terms
Some terms appear throughout the calculation, and are defined as follows:
k = 0.01720209895
µ = 1
C = 173.145
τi are the Gaussian time intervals calculated from time of observations
τ1 = k(t1 − t2)
τ3 = k(t3 − t2)
τ = k(t3 − t1) = τ3 − τ1
Using the celestial coordinates right ascension (α) and declination (δ) obtained from observation, we obtain
the position vectors from all 3 or i number of observations expressed in equatorial Cartesian basis between
the Earth and the asteroid using the following equation below.
ˆρi = (cos αi cos δi)ˆi + (sin αi cos δi)ˆj + (sin δi)ˆk
For our determination, we assume that the sun vector R is known. However, we only know the unit vectors
ˆρ1, ˆρ2, and ˆρ3, instead of the position vectors ρ1, ρ1, and ρ1 needed for the fundamental triangle calculation.
Employing vector math, we use the scalar equation of range which gives us the magnitudes of the rho
vectors, pi
ρ1 =
c1D11 + c2D12 + c3D13
c1D0
ρ2 =
c1D21 + c2D12 + c3D23
c2D0
ρ3 =
c1D31 + c2D32 + c3D33
c3D0
(2)
8

9. in which
D0 = ˆρ1 · ( ˆρ2 × ˆρ3)
D1j = (Rj × ˆρ2) · ˆρ3
D2j = ( ˆρ1 × Rj) · ˆρ3
D3j = ˆρ1 · ( ˆρ2 × Rj)
(3)
j = 1, 2, 3
Using the concept of conservation of angular momentum, all three position vectors of the asteroid are in the
same plane, implying that r2 is a linear combination of r1 and r2:
r2 = c2r1 + c2r3
where
c1 =
g3
f1g3 − g1 f3
c2 ≡ −1
c3 =
−g1
f1g3 − g1 f3
(4)
3.5.3. Method of Gauss First Pass
However, to find the scalars ci, we must find the time dependent functions fi and gi, which we solve as
the f and g series derived from a Taylor expansion of the position vector ri about the central value r2.
fi = 1 −
µ
2r3
2
τ2
i +
µ(r2 · ˙r2)
2r5
2
τ3
i ...
gi = τi −
µ
6r3
2
τ3
i ...
Because scalar quantity r2 is unknown and vectors we are trying to solve for r2
ˆr2 are not known, we solve
the truncated Taylor series expansion to the second order:
fi = 1 −
µ
2r3
2
τ2
i
gi = τi −
µ
6r3
2
τ3
i
we can now solve for the f and g scalars with only scalar quantity r2 known, which can be found using the
scalar equation of Lagrange:
r8
2 + ar6
2 + br3
2 + c = 0
where a, b, and c are constants defined by operations on known τ and ˆρi. From the scalar equation of
Lagrange, we can compute up to three r2 using built in Python functions, thus up to three different sets of
orbital elements.
9

10. Now that f and g to the second order are found, using eqs.(4) and (3) and (2), we can solve for r2, r1, and
r3 using eq.(1). Moreover, the velocity vectors ˙ri can be found with the following equations
˙r2 = d1r1 + d3r3
where
d1 =
−f3
f1g3 − f3g1
d3 =
f1
f1g3 − f3g1
3.5.4. Subsequent Iterations
By iterating the Method of Gauss several times until the r2 converges, we will have to repeat the process
of finding the f and g series which will lead to different r2, r2, and ˙r2 quantities, thus up to three different
sets of orbital elements. We repeat the process for all subsequent iterations, however we not proceed to
calculate r2 using the scalar equation of Lagrange, instead substitute the r2, r2 and ˙r2 from the previous
iteration into a fourth order f and g series:
fi = 1 −
u
2
τ2
i +
uz
2
τ3
i +
3uq − 15uz2 + u2
24
τ4
i
gi = τi −
u
6
τ3
i +
uz
4
τ4
i
where
u =
µ
r3
2
z =
r2 · ˙r2
r2
2
q =
˙r2 · ˙r2
r2
2
− u
Moreover, between subsequent iterations, we must correct for light travel time, in which
ti = t0,i −
ρi
C
where C is the speed of light in AU per Day. These calculations carry on for the respective τi values as
well. To find a convergent r2, we compared the newly calculated r2 and returned the vectors r2 and ˙r2 if the
difference between the r2 magnitudes between iterations were less than around 10−20 [2]
3.6. Orbit Visualization
Furthermore, we used VPython to create a code that showed the 3D orbit of our asteroid around the
Sun with the additional reference of the Earth revolving around the Sun. By using our calculated orbital
elements and the orbital elements of the Earth from JPL Horizons, we can write a while loop that constantly
updates the position of the asteroid and Earth and leaves a trail that shows their orbits. Thus, we have a
10

11. 3D model that accurately depicts the relative orbits and inclinations of the Earth and asteroid, although the
sizes of the celestial objects are not to scale for visualization purposes.
Fig. 4. Orbit Visualization, showing the orbital paths
Fig. 5. Orbit Visualization, showing the inclined planes of the orbits
4. Results and Data Analysis
4.1. Calculated Orbital Elements
The following shows the outputs of each author’s Method of Gauss python code.
11

12. Table 4. Calculated Orbital Elements
Orbital element E. Higgs K. Kim A. Ruperto
a 1.064416 1.059485 1.064416
e 0.347751 0.345131 0.347751
i (◦) 25.436460 25.22866 25.436460
Ω (◦) 236.096518 236.170181 236.096517
ω (◦) 256.292440 255.80387 256.292441
M (◦) 215.412966 117.569562 116.427122
4.2. Differential Correction
After gathering the six orbital elements from the calculated r2 and
˙
r2, we attempted to further the
accuracy of our orbital elements using Differential Correction. By changing each cartesian component of
the r2 and
˙
r2 vectors, we can calculate the partial derivative of each celestial coordinate αi (which includes
the δi) in respect to each cartesian component that is being modified (one of x, y, z, ˙x, ˙y, ˙z). Only one
component is being changed at a time, which totals 6 partial derivatives for each n = 2N celestial coordinate
of every n observation:
∂αi
∂x
≈
αi(x + ∆, y, z, ˙x, ˙y, ˙z) − αi(x − ∆, y, z, ˙x, ˙y, ˙z
2∆
We are able to determine the modified celestial coordinates by obtaining the orbital elements from the mod-
ified position and velocity vectors, and then running those modified orbital elements through an ephemeris
generator we programmed to obtain the changes in αi and δi. After gathering all the partials, they were run
in a simultaneous least squares regression:


∆αi
∂αi
∂x
∆αi
∂αi
∂y
∆αi
∂αi
∂z
∆αi
∂αi
∂˙x
∆αi
∂αi
∂˙y
∆αi
∂αi
∂˙z


=


(∂α
∂x )2 ∂α
∂x
∂α
∂y
∂α
∂x
∂α
∂z
∂α
∂x
∂α
∂˙x
∂α
∂x
∂α
∂˙y
∂α
∂x
∂α
∂˙z
∂α
∂x
∂α
∂y (∂α
∂y )2 ∂α
∂y
∂α
∂z
∂α
∂y
∂α
∂˙x
∂α
∂y
∂α
∂˙y
∂α
∂y
∂α
∂˙z
∂α
∂x
∂α
∂z
∂α
∂y
∂α
∂z (∂α
∂z )2 ∂α
∂z
∂α
∂˙x
∂α
∂z
∂α
∂˙y
∂α
∂z
∂α
∂˙z
∂α
∂x
∂α
∂˙x
∂α
∂y
∂α
∂˙x
∂α
∂z
∂α
∂˙x (∂α
∂˙x )2 ∂α
∂˙x
∂α
∂˙y
∂α
∂˙x
∂α
∂˙z
∂α
∂x
∂α
∂˙y
∂α
∂y
∂α
∂˙y
∂α
∂z
∂α
∂˙y
∂α
∂˙x
∂α
∂˙y (∂α
∂˙y )2 ∂α
∂˙y
∂α
∂˙z
∂α
∂x
∂α
∂˙z
∂α
∂y
∂α
∂˙z
∂α
∂z
∂α
∂˙z
∂α
∂˙x
∂α
∂˙z
∂α
∂˙y
∂α
∂˙z (∂α
∂˙z )2




∆x
∆y
∆z
∆˙x
∆˙y
∆˙z


or in the format:
a = Jx
where summations of i, = 1, n are implied for all elements in matrices a and J. As the changes in components
exist in matrix x, we invert matrix J and solve for x:
x = J−1
a
After making the changes to the new r2 and ˙r2 vectors by adding the x matrix, we calculate a new set
of orbital elements. To determine whether the correction improved the accuracy of the orbit, the devitation
values before and after the corrections are compared through the respective RMS values:
RMS orbit =
(αobs − αfit)2
n − 6
12

13. After running Differential Correction on our data with ∆ = 10−4, we obtained the following results:
Table 5. Orbital Elements after Differential Correction
Orbital element K. Kim
a 1.059747
e 0.345329
i (◦) 25.2512407
Ω (◦) 236.164753
ω (◦) 255.810268
M (◦) 117.527041
With the following RMS values:
Before differential correction: RMSorbit = 0.0014551
After differential correction: RMSorbit = 0.00994774
As our RMS values were higher, we conclude that running differential correction, contrary to theory, did
not improve our orbit.
4.3. Monte Carlo
After calculating and correcting our orbit, we used the Monte Carlo method to calculate uncertainties
and mean values on our orbital elements. The Monte Carlo method is a method of statistical analysis which
relies on randomly sampled inputs to produce numerically generated results describing the inputs. For our
Monte Carlo model, we used the uncertainties on our observed RA and DEC, which we had calculated in
our LSPR routine, to model each input with a Gaussian distribution. The mean of each distribution was
the observed value, and the standard deviation was the uncertainty on that value. The model sampled from
each distribution to generate a certain number of slightly different, randomized RA and DEC values for
each of the three observations. The number of generated values was user-inputted.Then, the simulation
ran the Method of Gauss on each of these inputs and records each orbital element before plotting them all
in histograms and fitting the outputs to Gaussian distributions. The fitted distributions were characterized
by mean and standard deviation values, which referred to the final value and uncertainty on each orbital
element after running the Monte Carlo sequence.
For our Monte Carlo sequence, randomized inputs would sometimes result in issues such as diverging
Methods of Gauss or math domain errors. To deal with this easily, we had the code find any inputs which
caused these errors and disregard them. As a result, a Monte Carlo result with a given number of iterations
may have, in actuality, executed slightly fewer iterations.
Iterations a
(Au)
e i
(◦)
Ω
(◦)
ω
(◦)
M
(◦)
1,000 1.403026
±0.77
0.431107
±0.10
29.009285
±4.17
233.211899
±3.44
269.916718
±15.31
194.887804
±103.34
10,000 1.619307
±16.62
0.429184
±0.11
28.868984
±4.30
233.276566
±3.45
269.526571
±15.48
192.570008
±102.47
100,000 1.445180
±3.30
0.430293
±0.11
28.908163
±4.27
233.228376
±3.47
269.713540
±15.52
192.698433
±101.98
13

14. Fig. 6. Generated histogram of inclination angle. Note that the Gaussian Fit is not clearly visible, as it is along the x-axis.
Unfortunately, because of the relatively high error values given, and the noticeable difference between
the generated mean values and the actual computed orbital elements, as well as the poor quality of the his-
tograms generated, we must conclude that the Monte Carlo sequence did not produce significantly accurate
or useful information.
5. Conclusions
The process of determining the orbit of Asteroid 68950 (2002QF15) included gathering the celestial
coordinates and observing times from three observations at Etscorn observatory over the span of several
weeks. The data obtained was then ran through LSPR Astrometry, leading to the calculation of the six
orbital elements through the Method of Gauss for preliminary orbit determination. While seven observation
sessions were conducted, only three sets of data were obtained due to weather conditions that hindered a
proper observation of our asteroid. However, to increase the accuracy of our data, one set of data from
another team observing the same asteroid were used.
Some aspects that may have affected the accuracy and integrity of our collected data may be to the short
span of observation data collected. The method of Gauss becomes more accurate with the input of more
data over longer periods of time, as well as the methods of differential correction, and with a relatively short
span of data of 3 observations, inaccuracies may have been created. Using a truncated Taylor series in the
method of Gauss also may have lead to inaccuracies in our results. Moreover, inaccuracies and errors in
the data may have resulted from the imperfections of the equipment used in observation, as well as minor
errors in all the programs we coded to determine the orbit of our asteroid.
To assess the accuracy of our calculated orbit, we’ve compared our average orbital elements with those
of JPL [6]:
14

15. Table 6. Orbital elements obtained by averaging collected data
Orbital element Averages JPL Horizons Percent Error (%)
a 1.06277 1.05708725 0.53780675321
e 0.346878 0.34424804 0.7639030051
i (◦) 25.36719611 25.1530773 0.8512628988
Ω (◦) 236.1210721 236.235017 0.0482336910
ω (◦) 256.1295857 255.540013 0.2307163988
M (◦) 149.8032166 149.80321 0.0000043894
To determine the uncertainty, we used the percent error equation:
Error = 100 ×
|Vapprox − Vexact|
Vexact
As our respective errors for each orbital element were relatively small, we conclude that the method of
Gauss for preliminary orbit determination is indeed effective for the set of data we have collected.
6. Acknowledgements
We would like to thank the Academic Director Adam Rengstorf, Assistant Academic Director William
Andersen for teaching us the material required to write this paper, and dealing with us for five weeks,and
TA’s Emma Louden, Anthony Flores, Cyndia Cao, and Descartes Holland for taking time out of their
summers to critique our work and help us learn the material. We would also like to thank Dr. Aaron Bauer
for coming to teach us Python.Finally, we would like to thank SSP as a program for allowing to do all of
this, and creating such an inspirational and motivating environment for us this past few weeks; it has truly
been the best experience of our lives.
Appendix A. Cloudy Night Experiments
We collected our data over nights of June 28, July 2, and July 13, when we happened to be clouded out
and, therefore, unable to observe our asteroid. On those days, we performed three different experiments to
look deeper at dark and bias current.
Appendix A.1. Experiment 1: Dark Current dependence on exposure time
For Cloudy Night Procedure 1, we collected six sets of five dark images, taken at 22.2◦C and varying
exposure times. We started with an exposure time of 10 seconds and doubled the exposure time with every
set until we reached 160 seconds. We also took a set of 11 biases for reducing our dark images with bias
subtraction. Given that one must take dark images with the same exposure time as their longest images,
it is a reasonable assumption to say that dark current increases and builds up over time. Since 120 second
exposure dark images do not seem to have significantly more noise than 60 second exposure dark images,
we hypothesize that dark current increases linearly over time.
We reduced our images by subtracting bias current using AstroImageJ. Using a Python code, we calcu-
lated the mean signal for each exposure time, then created a scatterplot to show signal in pixel count plotted
against exposure time. (Fig. A.1.) Using a linear regression function, we found a line of best fit for the data
15

16. and the error on both the slope and intercept of the line. According to our data, we can conclude that dark
current varies with exposure time by y = 1.41x − 0.07.
Table A.7. Uncertainties on linear fit
Value Uncertainty
Slope 1.41 0.000
Intercept .07 0.00003
Fig. A.7. Scatterplot with linear fit
Given that the line of best fit is linear and matches the data very well, as we can see from the minimal
uncertainties on the fit line (Table A.5.), we conclude that dark current increases linearly with time. Thus, it
seems that our initial conclusions were justified in that dark current does increase linearly over time, which
we surmised from our experience in taking dark exposures to reduce our visual images.
Appendix A.2. Experiment 2: Dark Current dependence on temperature
For Cloudy Night Procedure 2, we took five sets of dark images with an exposure time of 120 seconds,
varying the temperature with each set. We started at 22◦C, and decreased the temperature by 5◦C for each set
until we reached -3◦C. We hypothesize that dark current varies exponentially with temperature, following
the Arrhenius law.
After bias-correcting our images in AstroImageJ, we calculated the mean signal for each temperature
by averaging across each image, then across all images in each set. We then created a scatterplot such
that x = log( 1
Temperature) and y = log(dark current) in order to use a linear regression to find the function.
According to the linear regression, our data fit the equation y = −1.65x + 1.06. From how we’ve defined x
and y, we can determine that dark current varies with temperature by y = 11.482x1.65.
Table A.8. Uncertainties on linear fit
Value Uncertainty
Slope -1.65 0.00357
Intercept 1.06 0.00957
16

17. Fig. A.8. Scatterplot with linear fit
This result, however, is not an exponential relationship. As a result, we can conclude that our dark
current does not agree with the Arrhenius Law.
Appendix A.3. Experiment 3: Bias Current dependence on temperature
For Cloudy Night Procedure 3, we took 19 sets of biases at 24◦C, varying the temperature each time.
We started at a temperature of 21◦C and decreased by 2◦C every set until we reached -15◦C. We hypothesize
that the signal and noise in biases should not vary with temperature, as we only took one set of biases at a
single temperature during the Cloudy Night Procedure 2, where we varied dark current versus temperature.
Thus, we should end up with a straight line with a slope of zero for a plot of bias versus temperature, as
they have should have no relation to each other.
We did not reduce our images, as we can not subtract anything from biases using AstroImageJ and it
would introduce more elements, as well as being unnecessary, if we combined our biases with any other sort
of exposure. Using a Python code, we created a mean signal and noise from each set of biases, to gain a data
point at every temperature. We used these to create a scatterplot of signal in the biases versus temperature of
the exposure. Using a polynomial fitting function, we found a curve of best fit for the data and the general
error of the plot. Through this method, we determined that bias current depends on temperature by the
equation y = 3.28423x2 + 6.62689x + 656.56356.
17

18. Fig. A.9. Scatterplot with quadratic fit
Given that we were able to find a quadratic line of fit for the biases, they must rely quadratically on
temperature. However, the high error on our curve of best fit shows that it may not be a direct correlation.
Therefore, our initial assumptions that bias signal is independent of temperature seems to be incorrect, as
we do not have a line with zero slope. Thus, we conclude that biases are temperature dependent, not unlike
darks.
Appendix B. Referenced Data
Because we only had three successful observations, and since differential correction requires at least four
inputs, we had to borrow data from Team 9, the other team observing 2002 QF15 [1]. Of their successful
observations, we borrowed data from July 12, 2019.
Table B.9. Observation session data for Team 9
Date UT Time Images Filter Exposure Time Conditions
12 July 2019 3:00-5:00 5 Clear 120 s Somewhat
clear
Appendix C. MPC Report
This is the plain text that was submitted to the Minor Planet Center (MPC) to give them more data to
calculate a future orbit for our asteroid, 2002 QF15.
18

19. COD 719
CON A. W. Rengstorf
CON adamwr@pnw.edu
OBS E. Higgs, K. Kim, A. Ruperto
MEA E. Higgs, K. Kim, A. Ruperto
TEL 0.36-m f/11 reflector + CCD
NET GAIA-DR2
BND V
NUM 3
ACK Team 3 - 2002 QF15
68950 C 2019 06 19.16985 14 22 29.87 +23 33 44.4 16.2 V 719
68950 C 2019 06 19.19821 14 22 34.44 +23 33 34.6 15.1 V 719
68950 C 2019 07 05.25601 14 58 37.55 +21 20 33.9 17.3 V 719
68950 C 2019 07 05.28086 14 58 40.23 +21 20 16.0 17.9 V 719
68950 C 2019 07 10.23117 15 07 34.34 +20 28 12.1 16.7 V 719
68950 C 2019 07 10.27331 15 07 38.67 +20 27 41.9 16.6 V 719
References
[1] Data courtesy of Beard, Hong, Xin.
[2] A. W. Rengstorf, W. L. Andersen, Orbit Determination Packet.
[3] A. Rengstorf, lecture notes
[4] W. L. Andersen, lecture notes
[5] SSP Handbook
[6] JPL HORIZONS Web-Interface
19