THE MOONS OF SATURN

 (This observing exercise is geared for high school students.)

 

The invention of the telescope is credited to a Dutch lens maker named Hans Lippershey in 1608. After hearing of this new device, Galileo Galilei built his own. Eventually he became famous as a telescope builder, and his improved designs increased the initial magnification of about three to his best of around thirty.

 

It was six months after the construction of his telescope that Galileo began to look at celestial objects. Some of his first observations showed that unseen (faint) stars became visible and that nebulous blurs (such as the Pleiades and Praesepe) were resolved into stars. His most important discoveries, though, dealt with Solar System objects. Galileo saw that the lunar surface showed craters, mountains, and seas, and that the Sun had dark spots. In addition, he witnessed Venus going through all phases like the Moon. (This observation supported the Sun-centered Copernican Model.) Galileo viewed Saturn, but his telescope did not have a high enough resolution to see clearly its rings. He commented that it “looked like Saturn has ears”.

 

Of particular relevance, Galileo discovered that Jupiter has four (large) moons with periods ranging from 2 to 17 days. He made an extensive study of the changing positions of these moons, and today these four are referred to as the Galilean moons. Their names are Io, Europa, Ganymede, and Callisto, and they are remembered with the pneumonic I Eat Green Carrots. Their orbital motion around Jupiter was convincing evidence that not all objects revolved around the Earth.

 

These observations and more were published by Galileo in his book The Sidereal Messenger. Many years later in 1632 he wrote a second book, Dialogue on the Great World System, in which he described the old Earth-centered and the new Sun-centered theories. However, instead of remaining neutral, he vigorously supported the Copernican view. The Pope in particular was extremely upset that Galileo had not followed his instructions. Via an Inquistion-like trial, Galileo was forced to denounce the Copernican theory and was confined to his house for the rest of his life.

 

Goal

From current and previous observations, one can determine the revolutionary periods and orbital distances of each moon around Saturn. The mass of Saturn can be then computed with Kepler's Third Law. This law relates the sum of the masses of two mutually gravitating objects with (a) the distance between them and (b) the period of the orbit.

 

Kepler's Third Law is (M + m) = a³ / P², where

M  =  the mass of the primary body in units of the mass of the Sun.

m  =  the mass of the orbiting body in units of the mass of the Sun.

a   =  the orbital distance (radius if circular; semi-major axis if elliptical).

P  =  the period of the orbit in years.

 

Procedure

The primary tasks of this exercise are (a) to measure each moon's distance from Saturn and (b) to determine each period of revolution. An excellent assumption is that the orbits are all circular. Unfortunately, because the orbits of Saturn and the Earth are nearly in the same plane, our perspective is more-or-less edge-on, rather than top-down.

Once these parameters are known, then Kepler's Third Law will be used to calculate the mass of Saturn. (Technically, it is the mass of Saturn and the moon, but the mass of the planet is more than a hundred thousand times larger.)

One will make a careful measurement of each moon's distance from the center of Saturn's disk. In order to convert this value, which is in millimeters or centimeters, to kilometers, one will need to measure the diameter of Saturn with the same ruler. Its true diameter has been determined to be 120,000 km.

 

The Scale of the telescopic image is equal to

Scale (km/cm)  =  120,000 (km) / Saturn diameter (cm)

To convert the moon's distance to km, multiply by the scale:

Distance (km)  =  Distance (cm) x Scale (km/cm)

Remember to accurately record your observing date and time.

 

Reduction of Data

Step 1: Convert the dates and times of the observations to decimal days. The horizontal axis is time (days).

 

Step 2: Convert the distance measurements to appropriate units for the graphs. The vertical axis is the observed distance from Saturn. If the moon was observed on the left (West) side of the planet, plot it as a negative value. Right-side (East) observations are considered positive.

 

Step 3: Draw (or compute) a sinusoidal curve through the data points.

a) Do not connect the dots through the data. Remember, the physical orbits of the moons are smooth and continous. These are just a few selected, discrete observations. Consequently, each curve must have a smooth, sinusoidal shape. On each graph, each cycle must have the same amplitude and period.

b) It is usually best to start with Titan, for it has the longest orbital period and this makes it easier to “see” the sine curve. Then proceed inward to the other moons.

 

Step 4: Determine the Period and the Distance from these graphs.

a) The period is just the time it takes for one orbit of the moon, and on the graph this is one cycle. More than one cycle can be used to determine the period, but remember to divide by the number of cycles covered. Kepler's Third Law requires that the period is given in years, so divide the computed period by 365.25 days.

b) The second needed parameter is the orbital distance, and this is found from the amplitude of the cycle. (Again, remember that our point-of-view is edge-on, and this amplitude is the farthest observed separation between Jupiter and the moon.) For Kepler's Third Law, the appropriate unit for the distance is the Astronomical Unit. [The average distance from the Earth to the Sun is defined as 1 AU, and it is equal to 150,000,000 km.]

 

Step 5: Calculate the Mass of Saturn.

Because the mass of Saturn (M) is much larger than the mass of the moon (m), we will simplify Kepler's Third Law to M = a³ / P².

 

Moon                     Period (day)            Period (yr)        Distance (cm)      Distance (AU)         Mass (Sun)          Mass (Earth)

Dione                ¦                              ¦                              ¦                              ¦                              ¦                              ¦                             

Tethys               ¦                              ¦                              ¦                              ¦                              ¦                              ¦                             

Rhea                 ¦                              ¦                              ¦                              ¦                              ¦                              ¦                             

Titan                  ¦                              ¦                              ¦                              ¦                              ¦                              ¦                             

  

Average Mass of Saturn  =  ____________________________

 

Compare this with the actual mass of Saturn, which 0.000286 mass of the Sun.

 

Percent Error  =  (Actual Value – Computed Value) / Actual Value x 100

 

Percent Error  =  ___________________________

  

Questions

1. Is the result close to that of the actual value? What are the largest sources of error in this exercise?

 

2. What would cause a larger error in the mass calculation – a ten percent error in the orbital period or a ten percent error in the distance? Why?

 

3. Because the Sun's mass is so much larger than that of the other objects in the Solar System, astronomers compare the masses of these objects to that of the Earth. To do so, divide the computed mass of Saturn by 0.000003, which is the mass of the Earth in terms of the Sun.

 

 

James Sowell, 2013