How to evalute the mass of a planet

Newton's law of universal gravitation states that every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

The equation for universal gravitation thus takes the form:

F=G{\frac {m_{1}m_{2}}{r^{2}}}\

The constant of proportionality G is known as the universal gravitational constant.

The gravity formula can be used to predict the motion of moons and planets.

Cavendish first performed an experiment in 1797 which could and did indeed later establish the value of G.

The setup consisted of a torsion balance to attract lead balls together, measuring the torque on a wire and then equating it to the gravitational force between the balls.

Valueof G: 6.674*10^(−11) N-m^2/kg^2

where 10^(-11) is ten to the power minus 11 and m^2 is metres squared etc.

So to find values for the mass of the Earth, Sun and indeed other planets given their disatance from the sun.

Mass of The Earth

From the above equation if m1 is the mass of the earth, m1= F* Re*Re/(m2*G).  If m2 is 1KG, F = 9.8N (g=9.8m/sec^2)

where Re is the radius of the Earth. Re=6.4* 10^+6m (D=12,750Km),

m1= 9.8*Re*Re/G

m1=6*10^24 Kg

Mass of the Sun

If g reperesents gravitaitonal and c centripetal forces.

Fg = Fc

where Fg= G*M*m/r^2 and Fc=m * v^2/r (A body that moves in a circular motion (of radius r) at constant speed (v) is always being accelerated. The acceleration is at right angles to the direction of motion (towards the center of the circle) and of magnitude v2 / r.)

If M is the mass of the sun, m is the mass of the earth and v its orbital speed i.e.

As G*M*m/r^2 = m * v^2/r, m cancels and so:


In this equation, we know  the constant G.

However, we need to know how far the Earth is from the Sun and how fast it is moving around the Sun.

  • The value for G is 6.67 × 10-11 N m2/kg2 (where N is Newtons)
  • The Earth Sun distance one Astronomical unit (assuming circular orbit) r, is 1.5 × 1011 m.
  • The Earth's velocity around the Sun is the distance travelled divided by time taken i.e. 1 year, 365.25 days or 3.1557*10^7sec
  • Therefore the speed of the earth in orbit is 2*pi*r/T = 29,860 m/s  or 107,000 Km/hr, 1.07*10^5
  • Plug into M=v^2*r/G and get:

 Thus the sun's mass is 2 * 10^30 Kg.

Galileo Relativity

From the last section the speed of the earth in orbit is of the order 100,000 Km/hr. So why don't we feel it?

Galileo came up with an answer by considering being confined to a windowless cabin on a ship bleow deck. Observing all things in the room was the ship moving? There was no way of knowing.

This formed part of the argument for a sun centered solar system.