Period of Titan

Research project by Winfree Spears and Ji-Seon Park

Cape Fear Natural Science Academy
Fayetteville, North Carolina

Goal 1: To determine the period and orbital radius of Titan by fitting image data to a sine curve.

Goal 2: To calculate the mass and density of Saturn using orbital data from Titan.

Materials:

Software:

HOU Image Processing

Astro Photo (freeware)

TheSky or Starry Night

Interactive Excel Spreadsheet

Click HERE to download

Saturn images Jan 9-10, 2001

19 images, .FIT format (845 KB zip)

Click HERE to download.

• Find and record coordinates for the centers of Saturn and Titan using HOU-IP Axes tool.
  
  
• From Image Info, record date and time of image in a log .

• Repeat this process for all 19 images.

HOU Home	Mass of Saturn	Orbital Period of Rhea	Correcting for Orbital Tilt of Rhea Method #1	Correcting for Orbital Tilt of Rhea Method #2	Links & Contacts	CFHS Home

	Measured Value	Accepted Value
Orbital Period of Titan	284.7 hrs	284 hrs
Orbital Radius of Titan	1.29 x 10^6 km	1.22 x 10^6 km

• Use the distance formula to calculate the orbital distances (in pixels) from Saturn to Titan for each image.
• We created a distance formula calculator in Excel.
• Record calculated distance in log for each image.

• From log, enter image number, date, time, and orbital distance for each image into Excel spreadsheet.
• Excel formulas will automatically calculate values for other columns.
  

• Using the graph as a reference, adjust phase angle, period, and amplitude (pink boxes) for the sine curve (pink line) until it matches Titan curve.
• As a first guess, set amplitude value to the largest distance value measured for Titan.
• Phase value ranges from 0 to 6.28 and is used to shift sine curve left or right. Start at 0. Increase phase angle value by .5's until it approximates the slope of the left side of Titan's curve.
• Period values widen / narrow the distance between peaks in the sine curve. Increase or decrease the period value for the cosine curve so that it passes through Titan curve data points for both days.
• Move back and forth between the phase angle, period, and amplitude boxes, adjustmenting values to minimize differences between the two curves.
• The closer S(r-rcalc)^2 is to zero, the better the fit of the curves. (S(r-rcalc)^2 is the sum of the squares of the differences between the curves)
• When no changes can be made to phase angle, period, or amplitude that will decrease S(r-rcalc)^2 you have determined the period and orbital radius (in pixels) for Titan. Record your values for amplitude and period.

• Open Saturn images in HOU-IP, adjust zoom, pallet, and image contrast as desired. Record the image location, date, and time from Image Info.
  
  
• Use a planetarium program such as TheSky or Starry Night to identify Titan. Reset the planetarium program to the site location, date, and time the image was taken, zoom in on Saturn, and match the moon pattern in the planetarium program to the image pattern to identify Titan.
  
  

• Use Geometric geocentric eliptical coordinates (r :) from TheSky or distance from Starry Night to find the distance from Earth to Saturn for Jan 10, 2001. Convert from AU to km.
  d = 8.535964AU x 1.496 x 10^8 km/AU = 1.277 x 10^9 km
  
• Use small angle formula to find the orbital radius of Titan by using angular width qrad and the distance from Earth to Saturn (d) in km:
  D = d x qrad =1.22 x 10^6 km

• Using the Astro Photo program enter CCD type, method of focus, type telescope, focal reducer size, and type binning to determine plate scale (2.9 '/pxl).
  
  
• Using the plate scale, convert amplitude (A) from pixels to arc seconds, then to radians to get the angular width of Titan's orbit: qrad = A x 2.9'/pxl x 1deg / 3600' x prad / 180deg