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Solving for the Mass of the Sun

The Sun
Credit: NASA

Because the gravitational attraction of our Sun for the Earth is the centripetal force causing the Earth's circular motion around the Sun, we can use Netwon's law of universal gravitation to find the mass of the Sun without ever actually visiting the Sun. This is the same technique you would use to determine the mass of Cygnus X-1 with a probe. The Earth, orbiting the Sun, plays the same role as the probe sent to orbit Cygnus X-1.

Force of Gravity = Centripetal Force

From this, it follows that:

GMm/d^2 = mv^2/r

Because 'm' can be eliminated from both sides of the equation, and because 'd', the distance from the Earth to the Sun is also 'r', the orbital radius of Earth around the Sun, the equation can be rearranged to solve for the mass of the Sun:

M = (v^2)(r)/G

To actually compute the mass of the Sun, we need to know how far the Earth is from the Sun and how fast it is moving around the Sun.

Radius of Earth's orbit is 1.5x10^8 km

  • The value for G, the universal constant of gravitation, is 6.67 x 10-11 N m2 kg -2 (where N is Newtons).
  • The distance separating the Earth and the Sun (the orbital radius of the Earth around the Sun), r, is 1.5 x 108 km.
  • The Earth's velocity around the Sun is just the total distance travelled divided by the time required for the Earth to make one complete orbit around the Sun, T:
    v = (2)(pi)(r)/T

You need to be sure your distance measurement is in meters and your time measurement is in seconds in order for the units to cancel correctly.

x kg

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