This project is an investigation in finding the mass of Saturn using the orbital motion of only one of its moons, Titan. Titan's orbital period and orbital radius had to be found in order to calculate the mass of Saturn.
1. Calculator
2. Protractor
3. HOU Image Processing software
4. Astronomy Book
5. FITS format telescope images
6. Ruler
1. Open images taken from the internet [HOU Internet Site] source: (1) 020121N106; (2) 020121N0112; (3) 020122_0248; (4) 020202_040015; (5) 020202_040217; (8) 020209_C; (9) 020213_031601
2. Use "Image Info" to gather data. Record.
3. Rotate and flip two of the images.
A. Rotate 020122_248.fts 315°
B. Flip 020209_c.fts horizontally
4. Use AXES to determine the position of Saturn in each image.
5. Use 020121N0106.fts as the reference image.
6. Shift the remaining images, except 020213_031601 so that all are aligned.
7. To combine images: add or subtract from the 020121N0106 image. Try to always create the clearest possible image.
8. Print out new combined image.
9. Print out 020213_031601.
10. Align both print outs, and do a literal "cut and paste" to see the path of Titan. Titan will be the most prominent moon. [dot]
11. Determine the angular position of Titan in several of its identified locations along the orbit.
12. Find the orbital radius using a formula [see "calculations"]
A). The lengths of lines from 1 and 9 to the center of Saturn are compared with the measured diameter of Saturn. Use the known diameter of Saturn to calculate the average orbital radius of Titan.
13. Compare position of 1&9, and of 5&8 along with the time intervals between images to figure out the orbital period.
14. Use the period and radius of Titan to calculate the mass of Saturn.
15. Look up the accepted value for Saturn's mass and determine the percent error.
IMAGE INFO.
Image 1: #1
SAT 020121N0106
Center: 382.78, 282.96
Date: 2002-01-21
UT: 01:06:01'
Image 2: #2
SAT 020121N0112
Center: 384.59, 283.26
Date: 2002-02-02
UT: 01:12:15'
Image 3: #3
SAT 020122_0248
Center: 367.40, 264.28
Date: 2002-01-22
UT: 02:37:47'
Image 4: #4
SAT 020202_040015
Center: 242.35, 251.03
Date: 2002-02-02
UT: 04:00:15'
Image 5: #5
SAT 020202_040217
Center: 242.88, 251.48
Date: 2002-02-02
UT: 04:02:17'
Image 6: #8
SAT 020209_C
Center: 272.84'241.71
Date: 2002-02-09
UT: 03:23:27'
Image 7: #9
SAT 020213_031601
Center: 173.39,243.77
Date: 2002-02-13
UT: 03:16:01'
Image 8: #10 [Rotated Image @ 315°]
Center: 340.98, 182.87
Image 9: #12 [Flipped Horizontal]
Center: 145.62, 41.25
SHIFTED FROM IMAGE ONE:
#2: -1.81;-.30 [new image number #13]
#3: 41.8;100.09 [new image number #14]
#4: 140.43;31.93 [new mage number #15]
#5: 139.90;31.48 [new image number #16]
#8: 145.62;41.25 [new image number #17]
#9: 209.39;39.19 [new image number #18]
ADDING/SUBTRACTING
1. Subtract image number 1 from image number 13. [this will be image number #20]
2. Add image14 and image 20.
3. Subtract 15.
4. Add 16.
5. Subtract 17.
6. Add 18.
From the combined images, we determined the orbital radius and orbital period for the moon, Titan. We determined its direction from the color of the moons in the add/subtract sequence. The orbit is clockwise.
For the orbital radius of Titan:
Titan's orbital plane and Saturn's rings are tipped relative to the line of sight. If the orbital plane was further tipped so as to be viewed edge on, then positions 1 and 9 would both be nearly at the outer edges of Titan's orbital plane from our point of view. We found the distance from moon 1 and moon 9, to the center of Saturn and averaged them. We then put it into proportion with the diameter of Saturn. This is equivalent to the orbital radius of Titan.
For the orbital period of Titan:
We compared the time it took from 1 and 9 to 5 and 8. [We did this because, both were about 'halfway'.] From 5 to 8 took 7 days, [for slightly less than half an orbit] and 1 to 9 had an interval of 23 days; this meant that Titan went one and a half times around. We made angles to make it [Titan] perfectly halfway in position. We figured out that it takes Titan 14.1 days to orbit Saturn.
Proportion: x / 313 pixels = 120,660,000 m / 34 pixels
x = orbital radius of Titan = 1,110,781,765 m.
Mass of Saturn = 4(3.14)^2R^3 / GP^2
where R = orbital radius of Titan in meters,
P = orbital period of Titan in seconds,
G = 6.67 x 10^-11 Nm^2/kg^2
M = 4(3.14)^2 (1,110,781,765 m)^3 / G ((14.1 days)(24 hours/day)(3600 sec/hour))^2
= 5.47 x 10^26 kg
Our finding is that Saturn has a mass of 5.47 x 10^26 kg. The known mass of Saturn is 5.68 x 10^26 kg, so our experimental error is only 3.89%. The result is remarkably accurate, considering the averaging and approximations involved in analyzing the orbital motion. Maybe we were just lucky, but we prefer to think that we were very good!
Ivan Alcantar soccerballer684@aol.com
Stefania Maricchio stefy@adriacom.it
Jeff Sweet (project advisor) jsweet@redshift.com