National Aeronautics and Space Administration

Educators' Corner

Activity #6b - Extension: Weighing a Galaxy

Objectives:

• Students will be able to use Newton’s Laws of Motion to determine the mass of the Sun from the motions of the planets.
• Students will be able to estimate the mass of a galaxy by applying Newton’s Laws to the motion of stars in the galaxy.
• Students will be able to convert among different units used in astronomy.

Directions:

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

F = ma,

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

GMm/r² = mv²/r,

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

GM/r = v².

Solving for M gives

M = v²r/G.

Note that G = 6.67 x 10^-11 m³/kg-s².

1. Apply this equation to 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 ? ___________________

What would you conclude the mass of the Sun to be ? _________________

2. Now apply this equation to the galaxy F563-1. Determine the mass M using the equation and the velocity at various distances from 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 ? _______________________.