Are You Stronger Than Jupiter?

A comment on my earlier piece comparing the acceleration due to gravity that we feel from the Moon compared to what we feel from someone else in the room challenged me: how strong is the gravitational pull from Jupiter, compared to that of someone else in the room? Jupiter has a great edge in mass: someone else in the room weighs in at somewhere around 75 kilograms, while the planet comes to around 1,898,600,000,000,000,000,000,000,000 kilograms. On the other hand, your neighbor is somewhere around one meter away, while Jupiter will be something like 816,520,800,000 meters away. Maybe farther: that’s the farthest Jupiter gets from the Sun, and the Earth can be on the opposite side of the Sun from Jupiter, so add up to another 152,098,232,000 meters on top of that.

That distance is going to be a little bit of a nuisance. The acceleration we feel towards any planet will be stronger the nearer it gets, and while, say, Neptune is always about the same distance from the Earth, there are times that Venus and Mars are surprisingly close. Usually these are announced by clouds of forwarded e-mails announcing that Mars will be closer to the Earth than it’s been in 33,000 years, and will appear to be as large as the Empire State Building. Before you have even had the chance to delete these e-mails unread the spoilsport in your group of e-mail friends will pass along the report that none of this is true and the e-mail has been going around unaltered since 1997 anyway. But there is still a smallest distance and a greatest distance any planet gets from the Earth.

If we want to give planets the best shot, let’s look at the smallest distance any planet gets from the Earth. For Mercury and Venus, this happens when the planet is at aphelion, the farthest it gets from the Sun, and the Earth at perihelion, the nearest it gets. For the outer planets, it happens with the Earth at aphelion and the other planet at perihelion. (Some might say ‘apogee’ and ‘perigee’, although these are properly speaking only the words to use when something orbits the Earth. Some might say ‘apoapsis’ and ‘periapsis’, which talk about the nearest and farthest points of an orbit without being particular about what is being orbited, but no one actually does.) Here I’m making the assumption that there’s no weird orbital locks where, say, the Earth can’t be at perihelion while Venus is at aphelion, which might even be true. It’s probably close enough.

Once we figure out how far we want to get from a planet, to figure out the relative strengths, we can use the formula figured out back in the original post about this. Remember what the subscripts mean: we used “1” to represent the person being attracted to both the other person, denoted by “2”, and the subscript “3” to represent the Moon. So F1, 2 was the acceleration our test person, let’s say you, feels toward someone else in the room, and d1, 2 is how far that person is from you. Then the acceleration towards the other person in the room, divided by the acceleration towards the Moon, was:

F1, 2 / F1, 3 = (m2 / m3) * (d1, 32 / d1, 22)

The smaller this number, the weaker the pull of the person is compared to the pull of the planet, or, the stronger the planet is relative to the person. The reciprocal, or F1, 3/F1, 2, would tell us how much stronger the planet is.

If I’ve worked all this out correctly, and you might wish to check, then, for the major non-Earth and non-Moon bodies in the Solar System we have this data:

Body Mass (kg) Nearest Distance (m) Person / Planet Attraction
(F1, 2 / F1, 3)
Planet / Person Attraction
(F1, 3 / F1, 1)
Sun 1.99 * 1030 1.47 * 1011 8.16 * 10-7 1,226,000
Moon 7.35 * 1022 3.63 * 108 1.34 * 10-4 7,450
Mercury 3.3 * 1023 7.73 * 1010 1.36 0.737
Venus 4.87 * 1024 3.82 * 1010 0.0224 44.6
Mars 6.42 * 1023 5.46 * 1010 0.348 2.87
Jupiter 1.9 * 1027 5.88 * 1011 0.0137 73.1
Saturn 5.68 * 1026 1.2 * 1012 0.19 5.25
Uranus 8.68 * 1025 2.6 * 1012 5.83 0.172
Neptune 1.02 * 1026 4.3 * 1012 13.5 0.0738
Pluto 1.3 * 1022 4.28 * 1012 106,000 9.48 * 10-6

The numbers for the Moon are a little different from the earlier entry, since I used the Moon’s perigee distance today, rather than the average distance. At its nearest, the Moon pulls on you with about 7400 times the acceleration of someone else in the room, but even that’s slight compared to the Sun’s pull.

But the numbers seem fairly clear. Even at their nearest, Mercury, Uranus, Neptune, and Pluto can’t attract you nearly as well as someone beside you. And we may think of Mars as Earth’s nearest neighbor, but we’re falling towards Venus something like fifteen times as quickly, at least at their fastest.

It’s not much harder to work out these numbers for the greatest distance between Earth and these other bodies, and therefore get at the weakest the pulls are, and to figure out whether there’s ever times that Venus or Mars are weaker than your neighbor. I’ll leave that for the interested.