Q1. If assumed as perfectly circular, the Earth’s orbit has a radius of 1.495978
ID: 2304498 • Letter: Q
Question
Q1. If assumed as perfectly circular, the Earth’s orbit has a radius of 1.495978707×108 km (or exactly 1 AU). This is on par with the size of the Schwarzschild radius of some supermassive black holes.
Q2. An accurate clock far, far from any black hole keeps track of the passage of time. Most of the Universe agrees that after 1 year, 365.25 Earth-days have passed. An identical clock sits on the surface of a planet orbiting around the supermassive black hole from Q1 data provided above with a distance from the singularity of 2.0 AU. How much time has passed according to the black hole orbiting clock?
Explanation / Answer
1. for perfectly circular orbit
R = 1.495978707×10^11 m = 1 AU
from the definition of swarchild radius
rs = 2MG/c^2
hence
M = c^2*rs/2G = 1.00928*10^38 kg
2. now for t = 365.25 earth days
tr/t = sqrt(1 - rs/r)
r = 2 AU
rs = 1 AU
hence
tr = t*sqrt(1 - 0.5) = t/sqrt(2) = 258.27075182838 days
hence according to the blackhole orbitting clock only 258.270751 days have passed where as for a clock far away from gravitational effects, 365.25 days have passed
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