Indonesian Review of Physics, Volume 1, Number 1, 2018 19

Measurement Eccentricity the Moon’s Orbit with Image Analysis
Technique by Using Tracker Software

Ricka Tanzilla1, Ishafit2 and Yudhiakto Pramudya3
1, 2, 3Physics Education,Universitas Ahmad Dahlan, Indonesia
Jl. Prof. Dr. Soepomo SH, Janturan Warungboto Yogyakarta
E-mail: tanzilla.ricka@gmail.com

One of the interesting physical phenomenon to be studied isabout eccentricity of the Moon's
orbit on the movement of the Earth and the Moon as explained in Kepler's Law. In order to
explain the phenomenon,several research has been done to determine themeasurement accuracy
of Moondiskdiameter using Trackers and to know the result of the eccentricity value in the
application of Kepler's Law. This research can produce eccentricity values accordance with the
application of Kepler’sLaw.The research method used is ImageAnalysis Techniqueassisted with
Tracker. The result of this research shows that the eccentricityvalue obtained is 0.07 ± 0.01.
This is in accordance withKepler's first Law which stated that orbits of each planet andsatellite
is an ellipse and has an eccentricity value of 0 <e <1. The value of eccentricity obtained has
accuracy of 22%,if it compared with the actual value.Whereas, if it compared with the value of
ephemeris then the value of accuracy is 8%.

Keywords: Kepler’s Law, Eccentricity, Moon, Tracker

I. Introduction

Physics is the study of nature and symptom of objects
in the universe with various physical phenomenon such as;
the movement of celestial objects, changing process of
moon phases, etc. [1]. Physics consists of various
branches of science. An interesting branch of science to
study is astrophysics. The aim of astrophysics is to find
out the physical nature of objects in the universe [2].
Physical phenomenon related to the universe can be
studied through physics. Even those that are far from the
reach of the eyes. As the basis of astrophysical science,
one thing to know is the movement of the Planet.
Planetary movements can be explained through Kepler’s
Law. Kepler’s Law was initiated by a classical physics
scientist, Johannes Kepler [3].Kepler explains the
movement pattern of planets surrounding the Sun and
other celestial objects. FromKepler’s observations it turns
out that it agrees with what Copernicus has stated is that
planets do not surround the Earth but surround the Sun
[4].Kepler’s and Copernicus' laws have difference. The
difference is, according to Kepler the orbit is in the form
of an ellipse, while according to Copernicus the orbit is in
the form of a circle [5].JohannesKepler and Tyco Brahe
made more precise measurements so that the results
obtained were more accurate. Therefore the results of the
observations and analysis were concluded in the form of 3

empirical laws regarding the movement of the Planet
calledKepler’s Law [6].What is expressed "Empirically"
shows that these laws are not based on deep theory, but
accurately describe the observed motion of the planet.
Kepler's first Law and Kepler's second Lawwere first
published in 1609 and Kepler'sthird Law was published
in 1618 [7]. Kepler’s first Law states all planets surround
the Sun in an elliptical orbits, having the Sun as one of
the foci.Kepler’s second law states a radius vector joining
any planet to the Sun sweeps out equal areas in equal
lengths of time. Kepler’s third Law, the squares of the
sidereal periods (of revolution) of the planets are directly
proportional to the cubes of their mean distances from
the Sun.Kepler uses Planet to prove his physical law [3].

Kepler’s law that will be studied in this study is
Kepler’s first Law. Most physics books are still stated that
all planets surround the Sun in an elliptical orbits, having
the Sun as one of the foci. Whereas, what has happened
at this time there are many studies showing that not only
planets surround the Sun in elliptical orbits. But this also
happens with the satellites that surround the Planet. For
example, the Moon surrounds the Earth in an elliptical
orbit. [3].

There are a lot of methods that show that the Moon's
orbit is an ellipse. Research by E. Jay Sarton under the
title Measuring the moon’s orbit by using projector screen
projection to measure the Moon's diameter so that it



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obtained the results of the Moon's orbit eccentricity that
is 0.0580 [8].

Seeing from the lack of the previous research, this
study will design eccentricity measurements of the
Moon's orbit by using Tracker so that it will be easier to
process the image produced from the Moon. Moon image
is then calculated the Moondisk diameter image in pixel
and then converted into the moon angle diameter. This
experiment was conducted because it knew the accuracy
of the diameter measurement of the Moon disk using
Tracker and knowing the results of eccentricity values in
the effort to implementKepler’s Law.

II. Literature Review
Research on the application of Moon diameter for

measuring the eccentricity of the Moon's orbit has been
be inspected by Banjamin Oostra  in an experiment
entitled Measuring the Moon’s orbit using a hand-held
camera the method used by taking pictures consistently
and measuring the visible diameter in the picture. In this
study the measurement of the diameter of the Moon's
appearance using the Iris program. In this study the
measurement of the diameter of the Moon's appearance
using the Iris program. After obtaining the diameter value,
it appears that the next step is analyzed so that the
eccentricity value of the Moon's orbit is 0.047.
Eccentricity of the Moon's orbit is obtained from
elliptical geometry analysis. The agreed value is that the
eccentricity value of the Moon's orbit is 0.0549 [9].
Although the results are inaccurate, this research is able to
indicate a simple method to measure the eccentricity of
the Moon's orbit [10].

Subsequent research was carried out by Jorge I.
Zuluaga et al. [11] entitled Thesimplest method to
measure the geocentric lunar distance: a case of citizen
science. This study describes the results of the Moon's
geocentric distance measurement which aims to explain
simple methods with the results of the prescription. This
measurement uses the Moon's geocentric distance
counting method only with images and measurements of
the precision of the moon's visible diameter
measurements. Measuring the diameter of the Moon's
appearance using the Python Matplotlib application. So
that the geocentric measurements of the Moon distance
are obtained by D / RE =60.51 [11].

Subsequent research was carried out by Benjamin
Oostra [10] with the title Introducing the Moon’s
Orbital Eccentricity. This research provides a way to
introduce the eccentricity of the Moon's orbit in the
introductory class of Astronomy. The method used is to
exercise direct observation with students to determine the
eccentricity of the Moon's orbit. Moon images taken
when Moon is in the position of Perigee and Apogee by

using a 12 megapixel camera phone with 12 times the
optical zoom shows variations in size due to different
distances. The eccentricity value obtained is 0.043 [12].

The next research was carried out by Kevin Krisciunas
with the title Determining the eccentririty of the Moon’s
orbit without a Telescope. This study was able to
measure the eccentricity of the Moon's orbit with a value
of e = 0.039 ± 0.006. This research uses a piece of
cardboard with a small hole inside which slides up and
down with a ruler that allows measuring eccentricity [13].

Subsequent research was carried out by Tanzilla et al.
[14] with the title Measurement of the Visible Moon
Diameter on the Position of Perigee and Apogee. This
study uses GIMP Software to help calculate the Moon's
Visible Diameter. The percentage comparison value when
lunar Apogee and Perigee is 15.77% [14].

a. Kepler's Law
From the study of observational data on the position

of the planets collected by Johannes Kepler with Tycho
Brahe's formulating three laws of planetary motion which
are always associated with their names. The three laws are:
1) Planet move around the sun in ellipses, with the Sun

at one focus
2) The line connection the Sun to a planet sweeps equal

areas in equal time.
3) The square of the orbital period of a planet is

proportional to the cup (3rd power) of the mean
distance from the Sun

The first law states that the orbit of Mars and
implicitly all other planets is an ellipse with the Sun in
one focus. An ellipse is a form obtained by slicing a cone
with a plane at an angle to its base. This is a symmetrical
and well-defined form with two focus and properties.
The length of a line or string from one focus to any point
in the ellipse and then to the other focus is constant [15].

The second law states that the speed of a planet varies
so that the line that connects to the Sun sweeps out the
same area at the same time. So, when the planet is closest
to the Sun (perihelion) it moves most rapidly and if it is
the furthest distance (aphelion), it moves the slowest.
Although Kepler takes nine years to find and publish its
third planetary movement law. This law concerns the
relationship between the period of the planet and their
distance from the Sun. The first is known precisely
because of the large number of revolutions that have been
observed since ancient times, but the distance is very
unknown so there is no basis for determining the period
varies with distance [16].

b. Moon
The moon is one of the natural satellites of the Earth

[17]. The distance of the Moon to Earth is only around



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384,400 Km [18], so that the Moon's radiance to Earth
only takes about one second less (the speed of light is
300,000 km/ second). The moon is the five largest
satellites in the solar system and only on the Moon is the
only natural satellite that has ever been footed by humans.
The diameter of the Moon is much smaller than the
Earth, which is only 3,474 km while the Earth is 12,742
km [19]. The moon has a surface that is never illuminated
by the Sun. This is due to the axial minimum slope of the
Moon, and has an impact on the crater area and other
topography near the Kurub. The moon remains protected
from the Sun directly throughout the year [20].

c. Ellipse Geometry

Kepler’s first law states that the planets revolve the sun
in the form of elliptical orbits with the sun at one focal
point of the ellipse. Ellipse geometry will provide an
explanation of Kepler’s first law shown infigure 1[2].

Figure 1. Ellipse geometry in polar coordinates

Subsequently, by remembering cos (π - θ) = - cos (θ), as
a result,

,
)cos(1

)1(

)cos(1()1(

)cos(

)cos(4444

)cos(4444

2

2

222

222

22222











e

ea
r

erea

reeara

raeeaara

raerearara














(1)

or

,
)cos(1 e

p
r




(2)
with

)1( 2eap  (3)
This is an ellipse in the polar coordinates,p referred as
ellipse parameters.

If θ = 00

)1( earp  (4)

It is termed as perifocal distance. If θ = 1800thus,
)1( eara  (5)

This is called as apfocus distance.
With:
a = one half of the major axis (semimajor axis)
r = radius
ae = focus distance to ellipse foci
The site of points with the distance number to the fixed-
focus pointis called as ellipse.
In the satellite (moon) movement around the earth, the
distance is referred as apogee and perigee.

The orbit of celestial body in Kepler orbit is not only
ellipse, but in general, the orbit has a conic section form
like in Figure 2which has an eccentricity value.
Eccentricity is the oval sizeofellipse[21].
The eccentricity values can be described as follows:
a. If the orbit has 0 eccentricity (e=0), it called as
circularorbit
b. If the orbit hasthe eccentricity between 0 and 1
(0<e<1), it called as elliptic orbit
c. If the orbit has 1 eccentricity (e=1), it called as
parabolic orbit
d. If the orbit has the eccentricity greater than 1 (e >1), it
called as hyperbola

Figure 2. Conic section[15]

III. Research/ Experiment Method
In this research, the data collection was conducted at

the 4th campus of Universitas Ahmad Dahlan, Ring Road
Selatan street, Tamanan, Banguntapan, Bantul Yogyakarta
at 7°50”Southern Latitude and 110°22”East Longitude.
The subject in this study is the depiction of the Moon.
The collection of Moon image is done by using the Prime
Focus Technique. The data that have been taken are
processed by Image analysis technique in the Tracker.
Research instrument can be seen in Figure 3.

θ
r2a-r

2ae

Π-θ



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Figure 3. Research instrument
The research instrument (Figure 3) was used to

capture the full Moon depiction.The Full Moon
Depictionwhich was obtained is process to get the value
of Moon disk diameter. Moon disk diameteris used to
find the eccentricity value of Moon orbit.
a. Determine the Moon diameter by using Tracker
1) Insert the Moon depiction on Tracker and begin the

push by using line profilein the Full Moon depiction.
2) In everyline profileonTrcaker, take the value intable x

and then processed in excel in figure 4.
3) The sky arena in the Moon depiction has a value

lower than the Moon disk.There is transition from
the low value to high value that occurred in the edge
of the Moon disk. Transition value is used as
threshold valueto determine the edge of the Moon
disk. To make it easier to get the threshold value,
then use themoving average. Luminosity value in
every area pixcel near the Moon disk and sky is
processed to obtain the difference value between
adjacent pixcel. The difference in value is up and
down randomly, then smoothed with a moving
average. There is a significant change. This value is
used as the threshold value for the determination of
the edge of Moon disk. After getting the transition
value, maka dapat ditetukan titik tepi piringan Bulan.
This point is referred asx1andx2.The x value has
pixcel unit.

Figure 4. Finding the x1andx2in Moon disk

4) After getting the x1 dan x2value, we can calculate the
value of the Moon disk diameter which is displayed

in each line of profile by counting the difference
betweenx1andx2 (Δx).

To determine the diameter of Moon disk, then graphs are
made for each line profile and difference valueof x (Δx),
To get the value of Moon disk diameter more accurately,
we can use the equation (6)

22
0 )

2

1
()( nnn xyyr  , (6)=2 (7)

rnis radius value of Moon disk in each line profile, ynis
theyvalue that obtained in every line profile, whileyois the
yvalue in the line profile which is considered as diameter
focal point of Moon disk.Δxnis the difference valueof xon
each line profile. Hence,the diameter of Moon disk is
obtained by averaging the rnvalue then multiplied by 2
following the equation of a Circle.

b. Calculating the eccentricity value
In order to obtain the value of angular diameter (θ) of

depiction, then look for nominal angular resolution of the
camera (NAR) with equation (8) in rad/pixcel. After the
NAR value is obtained, multiply it by the diameter of the
Moon disc (equation 7), so the angular diameter value is
obtained.

telescopelengthfocal
ensionimagewidth

ensionsensorweidht
NAR __

dim__

dim__










(8)
θis angular diameter. The Angular diameteris affected

by the distance of Earth and the Moon as shown in
Figure 5.If the distance of Moon is closer from Earth,
thus theangular diameteris bigger. It is called asangular
diameterperigee(θp).Whereas, if the distance of the
Moon is further from Earth, the angular diameter is
smaller, it is called asangular diameterapoge (θa). The
relation betweenangular diameterand distance can be
described by equation (9).

= (9)
Or= (10)

Then, the eccentricity value that obtained is

= , (11)



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Figure 5. The angular diameter of an object(source: [22])

IV. Research Result and Discussion
a.Determining the value of eccentricity and comparing it
with Actual Value.

The eccentricity value is obtained from the maximum
value of angular perigee diameter and minimum angular
diameter apogee by using equation (9). To determine the
angular value of perigee diameter and angular value of
apogee diameter can be seen in Figure 6. The highest or
maximum diameter angular value is the perigee diameter
angular value and the lowest or minimum value is the
angular value of apogee diameter.

Figure 6. Graphical the relation of angular diameter to time from
the results of the Image Analysis Technique

After obtaining the eccentricity value experimentally,
it can be concluded that the eccentricity value
experimentally and the actual value of 22% can be seen in
table 1. The relatively larger error due to the actual value
obtained from the observation results hundreds of years
or even thousands of years is then averaged so that the
actual value is obtained. The value of error is quite large
due to data collection done up to 10 months because
there are several obstacles. The constraints that often
occur are weather. The  weather greatly affects the quality
of shooting the full moon and the full moon is less than
once in a span of 27 or 28 days. In addition, the telescope
used also has drawbacks. The telescope used is a refractor
telescope or a lens telescope that will affect the image of
the Full Moon image received.

Table 1. Relative errors between experiment and Actual Value
Eksperimen Actual Value Ralat

0,07052779 0,0549 22%

Based on the experimental calculations the sexentricity
values obtained were 0.07 ± 0.01. The existence of
differences in eccentricity values can be seen from NASA
calculations over 5000 years of eccentricity values
included in the susceptible value of 0.0255 to 0.0775.
While averaged over 5000 years the eccentricity value is
0.0549 [23].

Figure 7. The moon eccentricity from 2008-2010 [23].

Figure 7 shows that eccentricity reaches the maximum
when the main axis of the moon's orbit points directly
towards or directly away from the Sun (0° and 180°).
This occurs at an average interval of 205.9 days, which is
longer than half a year because of the shift of the main
axis to the east. Eccentricity reaches a minimum point
when the main axis of the Moon's orbit is perpendicular
to the Sun (90° and 270°) [23].

b. Determining the Value of Eccentricity and Comparing
with Ephemeris Value

Ephemeris value is an accurate calculation that can
calculate the angular diameter of the Moon from year to
year so that it can be used as a reference value [24].

In Table 10 is the difference of the Angular Diameter
value Experimently and Emphemeris Values so that
obtained the average value of error is 0.653%.

1700

1750

1800

1850

1900

1950

2000

2050

A
ng

ul
ar

 d
ia

m
et

er
("

)

time



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Table 2. The differences of Angular Diameter Values Experimentally
and Emphemeris Values

Date
Angular Diameter
Value
Exsperimentally

Angular Diameter
Value
Emphemeris

Deviation

22/04/2016 1759.179128 1764 0.273%

14/11/2016 2026.150021 2009.2 0.844%

10/04/2017 1810.84386 1814.6 0.207%

11/05/2017 1772.639268 1769.3 0.189%

09/06/2017 1770.155642 1763.6 0.372%

10/07/2017 1795.741158 1788.5 0.405%

07/08/2017 1828.457749 1804.4 1.333%

06/09/2017 1869.714657 1859.1 0.571%

05/10/2017 1935.562327 1903.3 1.695%

01/11/2017 2007.655893 1994.8 0.644%

From Table 8 can be described in the form of diameter
angular graphs to the time with experiment values and
emphemeris values in Figure 8.

Figure 8. the graph of  angular diameter relation to the time with
Experimental and Emphemeris value data.

The eccentricity value which obtained from
emphemeris measurements is 0.065. So that the
eccentricity value is measured by the experimental value
and emphemeris value, the errors obtained by 8% can be
seen in Table 3.

Table 3. Relative error between experiment and ephemeris
Eksperimen Emphemeris Ralat

0,07052779 0,06509754 8%

V. Conclusion
Based on the results of the research and discussion, it

was concluded that the measuring accuracy of the Moon
disk diameter using the Tracker can be viewed from the
eccentricity value obtained.

The eccentricity value obtained has an error of 22% if
compared to the actual value, whereas if the eccentricity
value obtained compared to the ephemeris value, the error
value is 8%. The eccentricity value obtained was 0.07 ±
0.01. This is in accordance with first Kepler’s first law
which states that the orbit of each planet and satellite are
ellipses and has an eccentricity value of 0 <e <1.

VI. Acknowledgment

Thank you to PASTRON and ANDROMEDAthey
helpedto guide the continuity of this experimental
research and thank you to all friends who helped us in
conducting this experiment.

VII. Bibliography

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