Hill Spheres and apparent radii

Suppose you are a passenger in a spaceship, you look outside, and you notice that, from your vantage point, the Sun and the Earth appear to be almost exactly the same size. Question: Is it possible that you are currently in orbit around the Earth?

Suppose instead that it’s the Earth and Moon that appear the same size. Is it possible that you are orbiting the Moon (i.e. the less massive of the two objects)?

What if the two celestial bodies you see are the Sun and Mars? Could you be orbiting Mars?

What about the Sun and Jupiter?

By my calculations and understanding, the answers to these questions are “yes”, “no”, “no”, and “no”.

The test is simple: You can orbit the less massive body if and only if it is at least three times as dense as the other body.

I hope I’m not wrong about this. I’ve never seen it explained in quite this way, and I could be misinterpreting the formulas.

Here’s a table of the density ratios of selected objects in the Solar System:

hillspheres1

The Earth is 3.9 times as dense as the Sun, so the answer to the first question is yes. Mars, though, is only 2.8 times as dense as the Sun, so in that case it doesn’t work.

I’m making some simplifying assumptions, including:

  • The smaller body’s orbit around the larger one is reasonably circular.
  • You and the smaller body are about the same distance from the larger body. So, either the larger body is much farther away than the smaller one, or they are around 90 degrees apart in your field of view.

In order to orbit an object that is itself in orbit, you must be inside the object’s Hill Sphere. The size of a Hill Sphere can be calculated fairly easily, in terms of various parameters. It turns out that if you calculate it in terms of the density of the objects, a lot of the math cancel out. And if the density ratio is precisely 3, even more of the math cancels out.

An issue to be aware of: Even in the Earth-Sun case where you could be orbiting the Earth, your orbit would not be stable in the long term. You have to be well inside the Hill Sphere in order to get long term stable orbits. You won’t find any natural moons orbiting that far out.

In my post on eclipses on other planets, I examined whether you could be standing on another planet, and have a moon of that planet appear the same size as the Sun. Now I ask the opposite question: Could you be standing on a moon, and have that moon’s planet appear the same size as the Sun?

We pretty much know the answer now: It is only possible if the planet is at least three times as dense as the Sun. Only the three innermost planets meet this criterion, and they have just one moon between them: Earth’s Moon. From Earth’s Moon, Earth appears much larger than the Sun, so our only candidate fails, and we know that this situation does not happen in the Solar System.

Realistically, for your vantage-point moon to be in a long term stable orbit, its planet’s density ratio would have to be considerably larger than 3. That seems unlikely in the stellar system of any star that is similar to the Sun. It could easily happen around red giant stars, though, due to their low density. So, although the situation cannot happen now, all you have to do is wait 5 to 7 billion years, for the Sun to become a red giant.

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