Ask Miss Science

How do they measure the mass of Saturn?

The question is:

Hey, what a great idea! Here's a question for you. i came across this sentence when reading some basic information about Saturn (the planet, not the car brand): Saturn's mass is about 5.69 x 10(to the)26th kg. How do they know?

Good question, Nancy! This is really a physics and astronomy question, but it's one that I can answer.

Masses in the solar system (and, indeed, beyond) are measured by observing their gravitational pull (force) on each other. The equation describing the force, F, between two objects is:

F = G m1 m2 / r 2

Equation 1

In this equation:

• F is the force (the "pull", essentially) between two objects
  
• G is the universal gravitational constant (so far, only members of the Q, a fictional race of omnipotent beings have been known to be able to be able to alter G, and they do so intuitively, seemingly unable to explain the process to us)
  
• m₁ is the mass of one of the two objects
  
• m₂ is the mass of the other object
  
• r is the distance between the two objects
  

So, if we can somehow figure out F, m₁ and r (remember, G is a constant), we can easily calculate the size of m₂. The equation above can be rearranged to make this easier:

m2 = F r2 / (G m1)

Equation 1.1

This is, indeed, one way the Earth's mass is measured. The only problem with using this equation for things that are far away is that F is not easy to determine.
Fortunately, a man named Kepler had worked out a relationship that can be used to side-step the issue of determining F. Newton, the man who worked out
Equation 1 for us, combined Equation 1 with Kepler's third law to get the following equation describing the motion of one body circling another (say, the Earth around the Sun or Io around Jupiter):

p2 = 4 pi r3 / (G(m1 + m2))

Equation 2

The new letters in this equation are:

• pi is what it typically means: the ratio of a circle's circumference to its diameter
  
• p is the period of the revolution (for example, the period for the Earth's revolution is about 365.25 days)
  

One simple way the mass of Saturn can be determined using Equation 2 is to observe the motion of one of Saturn's satellites. It will be best to choose a small satellite.
If you choose a satellite small enough that m_satellite is much, much smaller than m_Saturn, then you can say that adding the mass of the satellite to the mass of Saturn doesn't make much difference in the final answer. As an analogy, consider adding the mass of a small dog to the mass of an automobile: the additional mass doesn't change things by very much. So, now we can say:

m_satellite + m_Saturn ~ m_Saturn

Now, Equation 2 is easy to solve! We can rearrange it like so:

mSaturn = 4 pi r3 / (G p2)

Equation 2.1

All we need is:

• The distance from one of Saturn's moons to Saturn's center (r)
  
• The period in which the moon revolves around Saturn (p)
  
• The universal gravitation constant, G (presuming, of course, that the Q haven't been tampering with it..)
  
• And, of course, 4 and pi
  

posted by Ask Miss Science @ 8:01 PM