What about Mr. Oberth?

On a space forum I was singing the virtues of EML1 and EML2, the earth-moon Lagrange regions closest to the moon. "What about Mr. Oberth?" asked a fellow who calls himself Rune.

This is a common complaint from Zubrin fans who prefer to depart for Mars from Low Earth Orbit (LEO). Zubrinistas point out there's a greater Oberth benefit doing a burn deep in a gravity well.

What is the Oberth benefit? Why is there a bigger Oberth benefit deep in a gravity well?

The Oberth benefit gives a lot of extra kinetic energy for a small change in speed.

Kinetic energy is equal to 1/2 * mass * velocity². A way to visualize the product of three factors (mass * velocity * velocity) is as a rectangular solid:

To get 1/2 mv², just cut the square diagonally from corner to corner as shown above.

What happens if you're already going fast and speed up a litte more? Say you increase your speed v by v_b, velocity from a rocket burn. Here's a picture:

Take 1/2 of m (v + v_b)², and you get 1/2 mv² as well as 1/2 mv_b², the kinetic energy you might expect from adding these two speeds. On top of that, you also get m(v * v_b). The pink rectangle above is Oberth gravy.

For example a kilogram going 10 meters/second has kinetic energy of 50 joules; a kilogram going 2 meters/second has kinetic energy of 2 joules. But a kilogram moving (10 + 2) meters/second doesn't have a kinetic energy of 52 joules, rather (50 + 2 + (10 * 2)) joules. Starting with a 10 meter/second speed and speeding up another 2 meters a second gives you a 20 joule Oberth benefit.

So accelerating a mass already moving fast gives you more kinetic energy for your buck. But what does that have to do with doing a burn deep in a gravity well?

A fair model of a gravity well is a vortex wishing well. You've probably seen this in a shopping mall or science museum:

Photo used permission of Michael Hanna of Online Vending.

If you've played with one of these things, you know the coin starts rolling slowly around the outer portion of the funnel. As it moves inward it rolls ever faster until it's spinning furiously at the center. In a similar fashion, satellites far from the earth orbit sedately, but sats in low earth orbit zoom along at about 8 km/s. But unlike the vortex wishing well coin, satellites aren't slowed by friction so they stay in more or less circular orbits instead of spiraling inward as the coin does.

So now Rune's question makes more sense. Here's a picture of Rune zooming around deep in earth's gravity well while I'm just barely moving at the well's edge:

Vinfinity for a Mars Hohmann is about 3 km/s. From low earth orbit it takes a little more than 3 km/s to achieve escape velocity. You'd think to get escape plus 3 more km/s for Vinfinity would take around 6 km/s. But due to the Oberth benefit it only takes 3.6 km/s for Trans Mars Injection (TMI) from LEO.

From an orbit in the moon's neighborhood, it only takes about .5 km/s to escape from earth's gravity well. But there is less Oberth benefit when you're only moving 1 km/s. For TMI, I would need to stomp on the gas and speed up 2.5 km/s.

But I don't stomp on the gas.

Rather, I tap the on the brake.

Recall high earth orbits are slow. It takes only a small deceleration to kill most of your orbital speed. With almost no orbital speed holding me aloft, I drop like a stone towards the earth.

When I'm approaching perigee, I've already fallen from a great height. I am traveling just a hair under earth's escape velocity.

Rune is still traveling about 8 km/s, orbital speed for a circular low earth orbit. But I'm traveling almost 11 km/s.

I zoom past Rune like he's standing still.

To give some numbers, a .7 km/s deceleration suffices to drop from EML1 to a perigee deep in earth's gravity well. At perigee the ship is traveling 3.1 km/s faster than a circular low earth orbit. So the net delta V advantage over LEO is 2.4 km/s.

EML2 is similar but a little more complicated. .2 km/s suffices to drop from EML2 to a perilune deep in the the moon's gravity. A little .2 km/s tap on the gas at perilune enjoys an Oberth benefit from the moon's gravity well to send the ship earthward. So it only takes .4 km/s to reach a low perigee. This perigee is also moving about 3.1 km/s faster than LEO, so the advantage is 2.7 km/s.

More on the Oberth Effect can be found at Winchell Chung's Atomic Rockets.

Moreover, earth propellant most climb a much steeper gravity well before reaching space. EML1 and 2 are only 2.5 km/s from potential propellant in the lunar cold traps.

So to answer Rune's question, It's largely because of the Oberth effect that EML1 and especially EML2 are so attractive. Those who believe circular LEOs have an Oberth advantage forget that high earth orbits can easily reach a deep perigee with a small tap of the brakes.