National Aeronautics and Space Administration

Educators' Corner

Activity #6d
Weighing a Galaxy

Using Newton's Law for gravity, we can determine the mass of an object by measuring the motion of other bodies around it. We can show this by applying Newton's Law of motion to bodies orbiting around another body. We start with Newton's Second Law

where F is the force exerted on the orbiting body, m is its mass, and a is its acceleration. The force is the gravitational force exerted by the central object, and the acceleration is due to circular motion. So we now have

where G is the gravitational constant, M is the mass of the central object, r is the distance of mass m from M, and v is the velocity of m. Simplifying gives

Solving for M gives
Note that G = 6.67 x 10^-11m³/kg-s².

1. Apply this equation fo three of the planets in our solar system, given in the table below.

   Planet	Distance from Sun (km)	Velocity (km/s)	Mass (kg)
   Earth	1.5 x 108	29.8
   Jupiter	7.8 x 108	13.1
   Neptune	4.5 x 109	5.4

   What do you notice about the values of the Mass?__________________

   Wat would you conclude the mass of the Sun to be?___________________

   Now apply this equation to the galaxy F563-1. Determine the mass M using the equation and the velocity at various distances rom the center of the galaxy given in the table below. Each of these resulting mass values gives mass enclosed within that distance. [Note that 1 kiloparsec (kpc) = 3.1 x 10¹⁹ meters]

   Distance (kpc)	Velocity (km/s)	Mass (kg)
   5.0	95.0
   10.0	110.0
   15.0	110.0

   What do you notice about the values of the mass as the distance increases?

   _____________________________________________________________

   _____________________________________________________________

   What would you conclude the mass of the galaxy to be? _____________________________________________________________

   How much more massive is this galaxy than our Sun? _____________________________________________________________