"Surface" gravity of gas giants

[nmsurobert]

I just read a couple articles discussing the surface gravity of Saturn and Jupiter. I would expect the "surface" gravity of these planets to be much higher than that of Earth. I understand how the low densities of these planets has influence on that, but I thought mass was related to gravity in where more mass = more gravity. Why is Jupiter able to capture so many moons? What am I missing here?

Thanks.

 

[gneill]

Surface gravity varies directly with the mass, but also inversely with the square of the radius.
$$g_{surf} = G \frac{M}{R^2}$$

 

[nmsurobert]

ok. but with gravity being so low, why does Jupiter and saturn catch so many moons?

 

[gneill]

Compare the gravities at the same distance from Jupiter and Earth, Say at the average distance separating our Moon from Earth. Here, a little Mathcad:

So Jupiter has ~318 times the pull of Earth for objects at the same distance.

Edit: Note that the distance $a_m$ actually cancels out when you form the ratio. The $G$ also cancels, and you end up with the simple ratio of the masses.

 

[russ_watters]

ok. but with gravity being so low, why does Jupiter and saturn catch so many moons?

Surface gravity isn't what pulls in distant objects. As gneill says, it's gravity at the same distance.

 

[Janus]

More mass = more gravity at the same distance. So, for Jupiter, which is some 333 times as massive as the Earth, it's gravitational hold on IO, which is ~10% further away from Jupiter's center than our Moon is from the Earth's center, is some 276 times stronger than the Earth's hold on the Moon. For the same pull, you would have to move out to ~18 times further away.
There is another factor involved. The Earth orbits closer to the Sun, which also exerts a gravitational pull on the Moon. If the Moon got too far away, the Sun would win this tug of war and pull the Moon into an independent orbit. Jupiter is 5 times further away and thus experiences even less of a pull of from the Sun

How far a moon can orbit from a planet is called the "Hill sphere" and its radius is found by:
$$ a \sqrt[3]{ \frac{m}{3M} }$$

where a is the distance between sun and planet, m the mass of the planet and M the mass of the Sun
With a being 5 times larger and M being 333 times greater for Jupiter, its Hill sphere is ~34.7 times greater in radius than the Earth's. ( if Jupiter were the same distance from the Sun, its Hill sphere would only be ~7 times larger than the Earth. )

Neptune, because is ~5.8 times further away than Jupiter, has a larger Hill sphere even though its mass is less than 1/18th that of Jupiter

 

[gneill]

How far a moon can orbit from a planet is called the "Hill sphere" and its radius is found by:
$$ a \sqrt[2]{ \frac{m}{3M} }$$

Should be a cube root I believe. Typo?

Edit:
Also, the ratio of Jupiter to Earth mass is more like 318:

But your point is taken, and the Hill Sphere is a great way to look at the relative volume of space that planets can use to hold onto satellites.

 

[nmsurobert]

awesome. thank you guys.

now that i look at the math you posted, it makes sense. that's not intuitive, is it? when i think of surface gravity i think of the astronauts bouncing around on the moon. and how we're told "they do that because the moon is smaller and you'll weigh less... blah blah blah". especially with saturn, i would expect me to weigh quite a bit more.

 

[Janus]

awesome. thank you guys.

now that i look at the math you posted, it makes sense. that's not intuitive, is it? when i think of surface gravity i think of the astronauts bouncing around on the moon. and how we're told "they do that because the moon is smaller and you'll weigh less... blah blah blah". especially with saturn, i would expect me to weigh quite a bit more.

Take the Moon for an example:
It is ~1/81 the mass of the Earth, yet the surface gravity is ~1/6 that of the Earth. It's smaller size not only means that it has a smaller mass, but it also gives it a smaller radius which puts the surface closer to the center of the Moon than the Surface of the Earth is to its center. And the lower density of the Moon is also a factor. With the same radius, at the same density as the Earth, surface gravity would be better than 1/4 that of the Earth's.
With Saturn, it low density mean that its volume has to be larger to contain the same mass. Saturn has 1/8 the density of the Earth. If its density were the same as the Earth, it volume would only have to be 1/8 as large as it is. 1/8 the volume means 1/2 the radius, and 1/2 the radius means 4 times the gravity at the surface, or 3.664 times that of the Earth's.
Going the other way, if the Earth had the same density as Saturn, you would have to double its radius to fit all its mass in, and thus reduce its surface gravity by a factor of 4.

 

[nikkkom]

ok. but with gravity being so low, why does Jupiter and saturn catch so many moons?

Note: the number of captured objects depends on many other factors as well, such as the density of asteroids/planetesimals crossing this planet's orbit, the size, density and longevity of protoplanet cloud when this planet was forming (this cloud was probably the most efficient capture mechanism).

A Jupiter orbiting at Mercury's distance would probably have very few to no moons.

 

[Janus]

Should be a cube root I believe. Typo?

yes, a typo. It's now been corrected.

Edit:
Also, the ratio of Jupiter to Earth mass is more like 318:

View attachment 233886

I just did a quick mental estimate based on the Sun being roughly 333,000 times more massive than the Earth and Jupiter being roughly 1/1000 the mass of the Sun.

But your point is taken, and the Hill Sphere is a great way to look at the relative volume of space that planets can use to hold onto satellites.