HW 2 Solutions - Schwarzschild Radii
To find the Schwarzschild radius, multiply the mass (in solar masses) by 3 km. That is, find the mass of the object you're worried about, divide it by the mass of the sun (in the same units)
and multiply the result by 3000 meters.
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Saturn's mass is 5.68 x 1029 grams, or 5.68/2 x 1029-33Msun.
That's 2.84 x 10-4 solar masses.
so Rs = 3000 meters x 2.84 x 10-4 = 0.85 meters
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Via The Empire State Building has a mass of about 3.2 x 108 kg
Dividing that by the mass of the sun (2 x 1030 kg) gives its mass as 1.6 x 10-22Msun.
so Rs = 3000 meters x 1.6 x 10-22 = 4.8 x 10-19 meters.
- Mount Everest... lots of different ways to find this mass.
- Here's an upper limit - the mass in Mt. Everest from sea level to the summit. It is about 8840 meters above sea level. Assuming its base is
about 10x bigger than its height, then it is about 80 km across at the base. Assuming the shape is rougly conical with a radius of 40,000 m and height of 8800m, the volume is 1/3 pi r2 h,
or 1.4 x 1013 m3. The density of rock is about 3500 kg/m3, so the upper limit to its mass is
3x1015 kg = 1.5x10-15Msun
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On the other hand, it sits in a massive, and high, mountain range. So we can ask how much of that range is Mt. Everest alone? Then for the height
we should use the elevation above the base camp, which is about 3500 km. The base has a diameter of about 8 km, so the exercise as above yields a
smaller volume of 5.9x1010m3. Multiplying that by the mean density of rock yields a mass of
2x1014kg = 1.03x10-16Msun
The corresponding values of RSch are 4.5x10-12 and 3x10-13 m
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Masahiro Tanaka weighed in at 205 pounds when he reported to Yankees Spring Training camp. That's 93.2 kg, or 4.66x10-29Msun.
So his Schwarzschild radius is RSch = 1.4x10-25m
For perspective, the diameter of a hydrogen atom is about 10-12m, and the radius of a proton is 10-15 meters. So Mt. Everest
would need to shrink to less than the size of a single atom, and the Empire State Building to the size of a proton!