Visualising Gravity Wells

ps200306

Another poster prompted me to think about why it's so hard to get off planet Earth, and whether the problem could be represented visually. I've often come across "rubber sheet" pictures of curved space like this:



The idea is that the gravity of a planet or star warps the space around it, like a heavy object making a dimple in a rubber sheet. Other massive objects follow orbits in the curved space, while making their own dimples. The analogy is imprecise, since in the real 3D world the dimples are warps in higher dimensions.

But another imprecision is that the picture is qualitative, not quantitative. We get no true idea of the scale of the dimples. To do that we could try to take an edge-on look through the rubber sheet, so that we get a one-dimensional line warped into two dimensions:



This gives us an approach to graphing a gravity well (dimple) in familiar X-Y coordinates. How do we now pick a scale for our graph? We can't show curves in higher dimensions, but we can pick a proxy for the curvature, such as escape velocity or escape energy. The escape energy is proportional to the square of the escape velocity, so we'll use the velocity merely because the numbers are smaller and easier to show to scale.

You'll remember that the escape velocity is the velocity you'd have to be traveling away from the gravitating object so that it's gravity would slow you down but never quite stop you. As you'd expect, escape velocity gets smaller as you travel away, but only reaches zero at an infinite distance. It's conventional to show the velocity as a negative number (both because it represents traveling away, and because it makes our picture look like a well instead of a Mexican hat).

So we're going to try to show the gravity wells (or dimples) of the Sun and planets, but this time to proper scale. In the following graphs, the horizontal axis will represent distance from the Sun in astronomical units (one AU = distance from Sun to Earth = 150 million km). The vertical axis will represent escape velocity as a negative number in kilometres per second. It turns out the gravity wells are much, much steeper and deeper than the popular pictures tend to show.

So here are the gravity wells of the first six planets -- Mercury, Venus, Earth, Mars, Jupiter, Saturn -- embedded in the gravity well of the Sun:



Note that the circles just serve to mark distances from the Sun. On this scale the planets themselves are invisible dots sitting at the bottom of their gravity wells. And here we see our first of several problems. The gravity of our Solar System is completely dominated by the Sun. Our scale above can't accommodate the Sun, whose vertical coordinate is at -618. That is, the Sun's escape velocity at its surface is a massive 618 kilometres per second!

Our next problem is that the gravity wells of the inner planets are needle-like lines on this scale. This reflects the inverse-square nature of gravity. It's effect is very strong at close range, but falls off very rapidly at distance. And that really is the nub of the problem with getting off planet earth. Wherever you are going, you have to do almost all the work just to travel the first short distance.

This hard work is also characteristic of living on a dense, rocky planet. The depth of our gravity well is not just dependent on the mass of a planet, but also on "how close you can get to it". If all the mass of the planet was concentrated in an infinitely dense point, the well would be infinitely deep (ignoring General Relativity and black holes for simplicity). But the surface of the planet gets in the way. The bottom of our well represents the surface gravity, and the escape velocity there.

That explains why Saturn's gravity well is only three times as deep as the Earth's, when it is nearly a hundred times as massive. Saturn is much less dense than the Earth -- so much so that it would float on water! (Indeed, Saturn would even float on petrol ... it's barely half the density of the other gas giants and one eighth of the Earth).

Let's take a zoom in on the inner rocky planets:



Even on this scale we can't see the planets themselves. (The spacing between grid lines is 0.1 AU or about 15 million km). But at least we can start reading off escape velocities. The escape velocity of the Earth looks to be about 11 km/sec. That's the depth of the gravity well on top of the more than 40 km/sec escape velocity of the Sun at our distance from it.

Wait a minute though. Doesn't the Earth's gravitational influence extend out to infinity? How can we read the value reliably just by looking at the skinny part of our gravity well? We're using a nifty mathematical graphing tool so we can switch off the Sun to find the answer!

Here are the same four inner planets' gravity wells as if each one existed in isolation:



The Earth's escape velocity is indeed just a hair over 11 km/sec. The thing is: we use up almost the entire 11 km/sec travelling just a very short distance from Earth. Let's switch the Sun back on and zoom in even closer on Earth:



Note the broken Y-axis required to show the bottom of Earth's gravity well at this scale. Our horizontal grid lines are now only 0.01 AU, or 1.5 million km, apart. Even so, our lines are too thick to be able to discern the Earth itself on this scale. I've marked a point with a red X. Intuitively it looks like a kind of balancing point, where a carefully placed marble might not roll down into either the Earth's or the Sun's gravity well. And so it is -- this is the Earth-Sun L1 Lagrange point. It's just about 1.5 million km on the Sunward side of the Earth and just over 10 km/sec of launch speed gets us there. After that we can can go to any point closer to the Sun for free.

If we want to head out to Mars, though, we have to pay a price in additional launch speed, not so much because of Earth's gravity -- remember we've already paid almost all of that price just to travel the first million km, leaving less than 1 km/sec required to go anywhere in the universe if only Earth were involved -- but because now we're climbing higher out of the Sun's gravity well. If you look at earlier pictures, you can see that will cost us an additional 8 km/sec or so.

In our Solar System, the Sun is the big cahuna. Of course, the Sun's dimple is itself embedded in an even bigger dimple due to the Milky Way galaxy. At our distance from the galactic centre, the galaxy's escape velocity is more than 500 km/sec! The galaxy is an even bigger sucker!

Anyway, hope this hasn't been too long and boring. The take away point is that going anywhere in space starting from a dense body like the Earth costs a lot in energy terms. And most of that is expended early on. If you want to play around with gravity well simulations, here's the one I modeled with desmos.com, a simple but nifty graphing calculator -- feel free to pan and zoom, tear it apart and build your own solar system, or even change the laws of physics:

https://www.desmos.com/calculator/rhoardynmd

greedygoblin

Great post ps200306!

Reminded me of the xkcd explanation:

ps200306

greedygoblin wrote: »

Reminded me of the xkcd explanation

Brilliant! Way better than my explanation (obviously). I hadn't seen that before. The little side-comments like the one about getting back from the Moon are gems.