Determining the Mass of Saturn using Newton
by
Tamzeed Ahmed
PHYS 440 Final Project
Saturn is the sixth planet from the Sun and it is the second largest planet in the
solar system after Jupiter. It has a prominent system of rings, consisting
mostly of ice particles with a smaller amount of rocky debris and dust.
[Wikipedia]
Titan
is the largest moon of Saturn and the second largest moon in the solar system
after Jupiter’s moon Ganymede. It was the first satellite in the Solar system
to be discovered after the Galilean moons of Jupiter. It is the only moon in
the solar system to have a dense atmosphere. [Wikipedia]
Rhea
is the second largest moon of Saturn. [Wikipedia]
My
project is to determine the mass of Saturn using
T is
the period of the orbit, in seconds,
a is
the length of the semi-major axis, in centimeters,
G is
the gravitational constant and is equal to 6.672*(10^-8) cm3 g-1
s-2 and
M and
m are the masses, in grams, of the celestial bodies under consideration.
In my
project, I will try to determine the mass of Saturn using two of its moons,
Titan and Rhea. As a result, I will have two equations for each of the moons. T1
and T2 are the orbital periods of Titan and Rhea respectively. a1
and a2 are the lengths of the semi-major axis of Titan
and Rhea respectively. M and m are the masses of the two celestial bodies. In
my project, M is the mass of Saturn and m is the mass of the moon. When I am
trying to determine the mass of Saturn using Titan’s orbital period and length
of semi-major axis, I will use the mass of Titan as m. When I am trying to
determine the mass of Saturn using Rhea’s orbital period and length of
semi-major axis, I will use the mass of Rhea as m. However, since the masses of
the both Titan and Rhea are very insignificant, when compared to that of
Saturn, I will use m=0 in both cases.
I made
nine observations of Saturn and its moons Titan and Rhea from the Macalester
Observatory. Of the nine images I took using the CCD Camera, only one of the
images (taken on 4/4/2006) appeared out of focus. It was most probably a
problem with the CCD but the final image was good enough to be included in my
project. I took no sky flats or dome flats for any of the images since my
project did not require me to. I did however take dark frames to get rid off
any dark current and bias from the final image. The dark frame exposure time
was the same as the scientific image exposure time and the bias frame was a
zero second exposure.
1st
Observation taken on 3/27/2006 at UT 3.56.26.
Exposure
time: 1.45 seconds
Exposure
time: 1 second.
Exposure
time: 1 second.
Exposure
time: 1 second.
Exposure
time: 1 second.
Exposure
time: 1 second.
Exposure
time: 1 second.
Exposure
time: 1 second.
Exposure
time: 1.3 seconds.
I also
measured the diameter of Saturn in pixels in each of the nine images and developed
a conversion equation from pixels to cm using the information on the diameter
of Saturn, in cm, that I obtained from Wikipedia. The average diameter of
Saturn, in pixels, from the nine images is 25.78. The diameter of Saturn, in
cm, is 12053600000 cm. So, the conversion factor is 1 pixel = 467556245.2 cm.
The Julian days were converted into seconds using the conversion equation 1
Julian day = 86400 seconds.
The following is the plot of Titan-Saturn distance (cm) against time (seconds):
In the
above plot, when the observation is below the x-axis, that indicates that Titan
was on the left side of Saturn and vice versa. The distance marked T1
is the orbital period of Titan around Saturn and the distance marked a1
is the length of the semi-major axis of Titan’s orbital period. From my
calculations, T1 is 1249344 seconds and a1 is
approximately 16*(10^10) cm.
Using these values for T1 and a1 in
The following is the plot of Rhea-Saturn distance (cm) against time (seconds):
In the
above plot, when the observation is below the x-axis, that indicates that Rhea
was on the left side of Saturn and vice versa. The distance marked T2
is the orbital period of Rhea around Saturn and the distance marked a2
is the length of the semi-major axis of Rhea’s orbital period. From my
calculations, T2 is 398304 seconds and a2 is
approximately 7*(10^10) cm.
Using these values for T2 and a2 in
The mass of Saturn is 5.6846×1026. So, the values of M
that we found using Titan and Rhea are good approximations for the mass of
Saturn. The value of M found using Titan is a better approximation that the one
found using Rhea.
One of
the reasons why my results are not so close to the actual mass of Saturn is
because I did not have enough data points due cloudy skies that made observing impossible.
I also had to use CCDOPS program to measure the distances and not the IRAF in
the Linux machine, which broke down recently. The distances measured using
CCDOPS were very crude since I had to find the centers of Saturn, Titan and
Rhea on my own using my best guess while IRAF would find it for me
automatically.
I
would like to thank, Prof. Barton Pritzl, Jim Johnson and my classmates for
helping me with this project.
I used
the following websites to get information for different parts of my project:
http://www.skyandtelescope.com/
http://scienceworld.wolfram.com/