# Effect of a magical increase in Earth's mass on Moon's orbit

Some technobabble handwavium process increases Earth’s mass by a significant amount - say 5% - by, I dunno, teleporting basalt here from the basalt dimension. The process doesn’t affect the Moon, other than by changing its orbit due to the increase in Earth’s gravitational pull. What’s the easiest way to determine period of the new lunar orbit? Can it be claimed that the total energy of the orbit before and after are equal?

increases in Earth’s mass means an increase in Earth"s gravitational pull on the Moon.
So the Moon will increases it’s speed toward Earth up to a new equilibrium. Energy will not be equal, since you added potential energy to the Moon by increasing Earth"s mass.
The 3rd law of Kepler says T²/a3 = 4Pi²/GM
Problem is that the period T and the rayon a are unknown…
you can postulate that one or the other will maintain and calculate the other.

This sounds like a job for Kepler’s laws. At least for a starting crude approximation. The equation relating the relevant quantities is

v^2 = GM\left(\frac{2}{r}-\frac{1}{a}\right),

where v is the orbital speed, GM the gravitational parameter of the Earth, r the distance between the bodies, and a the length of the semi-major axis. So if you (somehow???) suddenly increase M (which also messes with the energy), the moon will be shot into an orbit with higher a and therefore a longer period.

Er, I mean a decreases if the mass increases (which is intuitively obvious). Sorry about the confusion!

I suspect the moons orbit will become more eccentric (less circular) with apogee close to where it is when the earth mass suddenly increases.

Depends on whether or not Flash Gordon is able to defeat Ming the Merciless before the 14-hour timer reaches zero.

Cool orbit simulator:

If the Moon stayed at the same orbital distance but the Earth suddenly became more massive, the new orbital period would be 26.6 days, down from 27.3 days.

But if the Moon retains its current orbital speed, the Moon would not be travelling fast enough to maintain a circular orbit at that distance. As Saffer suggests, this would be an eccentric orbit. The new orbit would have the same orbital speed at the apogee as the old lunar orbit, but would be faster everywhere else. I don’t know what the new semimajor axis would be; 95% of the old one, perhaps?

This is a fantastic tool, thanks for posting.

I just realized that in addition to tweaking gravity and time, you can actually click-and-drag the sun/earth/moon, putting them in each other’s way or saving them. Things quickly get out of control, though: you can slingshot a satellite off into deep space, and the influence of gravity fades surprisingly quickly with distance. Once a satellite disappears off screen with any significant velocity, it takes a very long time to circle back…

The way to solve this is to start with total energy and momentum, and keeping those constant.

So I googled “energy and momentum of orbits”, and came to this wiki page: Elliptic Orbits

The relavent equations are:

e = GMm/(2*a). Where e is total energy, G is a constant, M and m are the two masses, and a is the major axis. Here e is kept constant and M increases, thus the major axis must decrease accordingly.

Next,
T^2 = 4 pi^2 a^3 / (G M)
Where T is the period (time for one orbit), rest is above. Since we already know a increases in proportion to M, you can cancel one a for the M. Thus, the period will decrease in proportion with the increase of the mass. (as discussed by a poster above)

EDIT:
Looking further down the set of equations, you discover the minor axis only depends on Angular momentum, total energy, and mass of the moon. Thus, in the scenario of changing earths’s mass, the minor axis will not change… which seems odd to me?

e is not kept constant. The Moon keeps its instant velocity and its height. So kinetic energy in the instance of magical change stays the same, while potential energy increases. So the major axis would stay the same.

Why would the major axis stay the same? It definitely depends on the gravitational parameter. If the Earth lost enough mass, the moon would continue off to infinity…

You are correct in that the total energy is not actually constant. At least the gravitational energy must change.

For that matter the angular momentum probably isn’t constant either (if you assume the speed of the Earth stays the same, though I’d think it’s reasonable to assume L is constant and speed changes but that probably isn’t what the OP envisions). But… if either the speed or angular momentum is kept the same, then the kinetic energy must also change in some way.

Note, which I forgot about in previous post, that you cannot keep both momentum and kinetic energy constant if you change the mass.

Thinking about this more deeply. It matters WHEN in the orbit the Earth’s mass changes. Simplest way to see this is if you imagine speed stays constant, then it matters what that speed is. Similar result if you hold either kinetic energy or angular moment constant…

Huh, turns out to be surprisingly difficult to be precise.

Note istm that the change in the major axis (in this elliptical orbit approximation) is not exactly a linear effect: imagine the Earth got really really massive so that the Moon fell down into a highly eccentric orbit; it still cannot reduce the major axis to less than the Moon’s current distance.

I think it can. If the Earth suddenly became 5% more massive, that would be the same as accelerating the Moon towards the Earth; the Moon would fall into a new orbit, where the perigee was closer to the Earth but the apogee was still at the same distance (because the Moon’s orbital velocity at that point would remain the same).

Because the perigee is now closer to the Earth, the major axis (and of course the semi-major axis which is half that value) would be smaller.

If the apogee remains the same, but the perigee becomes zero, then the major axis would be half its original value. You cannot reduce it further without a second change.

With a perigee of less than 8000 km, the Moon would hit the Earth, so that’s kind of a moot point.