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Lesson 9:
Neutron Stars, Black Holes,
Galaxies, and Dark Matter
Math Notes

Schwarzschild Radius
 vesc = escape speed from surface of object
 M = mass of object
 R = radius of object
 Rs = Schwarzschild radius of object
 G = Newton’s gravitational constant
 c = speed of light
 Recall the escape speed equation from Lesson 2:
 vesc = (2GM / R)1/2
 Solving for R yields:
 R = 2GM / vesc2
 Setting vesc = c yields the Schwazschild radius of the object:
 Rs = 2GM / c2
 If an object of mass M is compressed to the point that its radius R is less than its
Schwarzschild radius Rs, then it would require a speed greater than the speed of
light to escape its surface. Since nothing travels faster than light, nothing, not
even light, would be able to escape its surface, making a “black hole”.
 For stars, M is usually measured in solar masses (Msun). Hence, you might find
this, equivalent, form easier to use:
 Rs = 3 km (M / Msun)
Object
Earth
Sun
Neutron star
Neutron star


M
3 x 10-6 Msun
1 Msun
1.4 Msun
3 Msun
R
6,400 km
7 x 105 km
10 km
9 km
Rs
0.9 cm
3 km
4.2 km
9 km
Hence, Earth would have to be compressed to a radius of 0.9 cm, the sun to a
radius of 3 km, and a 1.4 solar-mass neutron star to a radius 4 km. However, >3
solar-mass neutron stars would be smaller than their Schwarzschild radii and
consequently collapse to form black holes.
Galactic Mass
 vc = speed of object in circular orbit around Galaxy
 M<r = mass of Galaxy within radius r
 r = radius of orbit




G = Newton’s gravitational constant
Recall the equation for the speed of an object in a circular orbit from Lesson 2:
 vc = (GM<r / r)1/2
 Note: The gravitational effects of the mass exterior to the object’s orbit
cancel out and consequently do not affect the speed at which the object
orbits. Only mass interior to the object’s orbit matters.
Solving for M<r yields:
 M<r = rvc2 / G
Since the Galactic rotation curve is approximately flat, vc is approximately the
same for all radii. In this case, M<r is proportional to r.
Homework #9
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