Gravitational Potential Energy, Escape Speed and Black Holes

Why is mgh not enough?

In SPH3U you used Eg = mgh for a ball on a shelf. That works because g barely changes over a few metres. But a rocket climbing 400 km to the ISS moves through a field that weakens as it goes, and a probe heading to Jupiter leaves Earth's field almost entirely. We need an expression for gravitational potential energy that works at any distance.

Remember from the unit Work, Energy and Simple Harmonic Motion: work done by a force changes energy, and when only conservative forces like gravity act, total mechanical energy (Ek + Ep) is conserved.

Eg = −GMm/r

The gravitational potential energy of a mass m at distance r from the centre of a mass M is

Eg = −GMm/r

Two features need explaining.

1. Where is zero? Only changes in potential energy matter, so we may choose the zero anywhere. For mgh we chose the floor. For planets and stars, the natural choice is r = ∞, where the masses no longer affect each other.

2. Why negative? Gravity attracts. To pull m away from M to infinity, you must do positive work against gravity, which raises Eg up to 0. So at every finite distance Eg is less than 0. The closer you are, the more negative Eg is: you are deeper in the "gravity well".

(Getting this formula needs the area under the F = GMm/r² curve from r to infinity, a calculus result you may see in MCV4U or first-year university. The answer is the GMm/r above.)

Change in potential energy: moving from r₁ to r₂,

ΔEg = GMm(1/r₁ − 1/r₂)

which is positive when r₂ > r₁ (moving away).

Energy of a satellite

For a circular orbit, mv²/r = GMm/r², so mv² = GMm/r and

• Ek = ½mv² = GMm/(2r)
• Eg = −GMm/r
• E_total = Ek + Eg = −GMm/(2r)

The total energy is negative (the satellite is bound) and exactly half of Eg. Also Ek = −E_total. To move to a higher orbit, the total energy must increase (become less negative), even though the satellite ends up moving slower.

Escape speed

To escape completely, an object must reach infinity (where Eg = 0) with Ek ≥ 0. So its total energy at launch must be at least zero:

½mv² + (−GMm/r) = 0 → v_esc = √(2GM/r)

The object's own mass cancels again. From Earth's surface, v_esc = √(2(3.982 × 10¹⁴)/6.37 × 10⁶) = 11.2 km/s. Notice that v_esc = √2 × (circular orbital speed at the same r).

Escape speed explains a lot. The Moon's escape speed is only 2.37 km/s, and over billions of years it could not hold an atmosphere. Jupiter's escape speed is about 60 km/s, so it has kept even its lightest gas, hydrogen.

Black holes

What if a mass were squeezed so small that its escape speed reached the speed of light, c = 3.00 × 10⁸ m/s? Setting √(2GM/r) = c gives

r_s = 2GM/c²

This is the Schwarzschild radius. A correct treatment needs Einstein's general relativity (Karl Schwarzschild found this solution in 1916), but it gives the same formula. If all of an object's mass is inside r_s, nothing, not even light, can get out. The sphere of radius r_s is the event horizon of a black hole.

• Stellar black holes form when very massive stars collapse at the end of their lives. They are a few to tens of solar masses.
• Supermassive black holes sit at the centres of galaxies: Sagittarius A\* has about 4 million solar masses (previous lesson).
• In 2015 the LIGO detectors first detected gravitational waves from two merging black holes, confirming another prediction of general relativity. You will compare Newton's and Einstein's views of gravity in Comparing Fields and the Fundamental Forces.

Away from the horizon, a black hole's gravity is no stronger than any other object of the same mass. If the Sun were replaced by a 1 solar-mass black hole, Earth's orbit would not change (though it would get very cold and dark).