Question

what would happen to the gravitational force between the sun and earth if the mass of the earth was double?

Answer 1

By Newton's law of gravitation,
F = G (M_sun M_earth) / r^2.
If M_earth is doubled (M_earth -> 2M_earth) and the Sun–Earth distance r stays the same, the gravitational force becomes twice as large: F -> 2F.

A few important consequences and notes:
- Earth’s acceleration toward the Sun is a = F/M_earth = G M_sun / r^2, which does not depend on Earth’s mass. So the orbital acceleration (and, for the same velocity, the orbit) would be essentially unchanged.
- The Sun feels the equal-and-opposite force, so the Sun’s acceleration toward Earth would double (a_sun = F/M_sun increases).
- The system’s barycenter (Sun–Earth center of mass) would move a bit farther from the Sun’s center. Currently it’s ≈450 km from the Sun’s center; doubling Earth’s mass would make it ≈900 km.
- The orbital period formula is T = 2π sqrt(a^3 / G(M_sun+M_earth)). Doubling Earth’s mass has a negligibly small effect on T because M_earth << M_sun.
- If Earth’s radius stayed the same, surface gravity would double (g = G M_earth / R^2), with large effects on life and structures.

Numerical example: current gravitational force ≈ 3.54×10^22 N, so doubling Earth’s mass would give ≈7.08×10^22 N.