681: Gravity Wells

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Gravity Wells
This doesn't take into account the energy imparted by orbital motion (or gravity assists or the Oberth effect), all of which can make it easier to reach outer planets.
Title text: This doesn't take into account the energy imparted by orbital motion (or gravity assists or the Oberth effect), all of which can make it easier to reach outer planets.

The xkcd page links to a much larger version.

[edit] Explanation

Ambox notice.png This explanation may be incomplete or incorrect: Fix Grammar and explain title text
If you can address this issue, please edit the page! Thanks.

The comic shows the relative strengths of the gravitational fields of each planet in the solar system as well as some of their satellites. Each well is scaled such that rising out of a physical well of that depth, in constant Earth surface gravity, would take the same energy as escaping from that planet's gravity in reality. Each planet is shown cut in half at the bottom of its well, with the depth of the well measured down to the planet's flat surface.

This is a simplified method of comparing the escape velocities of the various planets and satellites. In reality, the strength of gravity decreases the further you get from the planet. However, a comparison of energy expended to escape the gravitational pull allows for a simpler comparison between the objects.

Moons shown in the chart are at the appropriate distance from their respective planets' gravity wells for their orbits. Those moons featured are: Earth's moon, Mars' tiny moons Phobos and Deimos (in an inset panel), Jupiter's moons Ganymede, Io, and Europa, and Saturn's moon Titan.

The Sun's gravity well is not shown in its entirety, but is just indicated on the far left as "Very very far down". Had it been shown in its full extent it would have made the rest of the drawing so small in comparison that it would have been unreadable.

[edit] Inner Planets

  • Mercury -- no facts listed
  • Venus -- no facts listed
  • Earth & Moon -- listed with the depth of the gravity well of 5,478 km for Earth and 288 km for the Moon. Calculations of depth is explained in the Saturn insert.
  • Mars -- listed with the depth of gravity well of 1286 km

[edit] Cut outs

  • The Mars cutout shows how small Mars' moons really are, specifically in terms of gravity. Deimos is so small that a bike jump would be sufficient to escape its gravity well and on Phobos you could launch a baseball into space simply by throwing it.
  • The drawing next to Jupiter is playing on the classic "Yo Mama" joke, combining the aspects of your mom being very fat (having a huge gravitational pull) and very slutty (having sex with the entire football team), represented by the football team being attracted to, and falling into, her gravity well. A "Yo Mama" joke also appears in comic 89: Gravitational Mass.

[edit] Outer Planets

  • Jupiter -- is so massive and dense that it is comparable in mass to a Brown dwarf which is the smallest kind of star. Saturn, while similar in size, is composed of much lighter gas material. Hence Saturn's mass and therefore its gravitational pull are much smaller. Strong gravitational pull is a significant factor in igniting fusion. As gravity pulls matter together the pressure and temperature increase. With sufficient increase in pressure and temperature nuclear fusion begins. Had that happen during creation of our solar system, i.e. had a few dozen times the mass of gasses contained in Jupiter condensed in that location, then we would had two Suns and our solar system would have been a Binary system.
  • Saturn & its rings -- The "Rings" in Saturn's gravity well are Saturn's rings. The farther you get from a planet, the weaker the effect of its gravity on you, so, at some point, when climbing out of Saturn's gravity well, you've reached the point equivalent to starting your climb from the rings of Saturn, or, in fact, from specific rings of Saturn. Saturn's rings start fairly near the planet and extend out quite far, therefore multiple stripes are shown in the figure. The rings are also shown in multiple colors and roughly match the observed colors from photos take by the Cassini spacecraft expedition as it passed Saturn. All of the colors of the planets and moons represent the predominant color of that object as observed from earth.
  • Uranus -- notably absent is any "your-anus" jokes.
  • Neptune -- Megan's quote is a paraphrase of Carl Sagan's quote, "...but from a planet orbiting a star in a distant globular cluster, a still more glorious dawn awaits, not a sun-rise, but a galaxy rise." [Video here]

[edit] How to interpret gravity wells

To escape a planet's or moon's orbit, you need merely climb to the highest of the two peaks to either the left or right of the object. So, for example, to escape Venus, you need merely reach the peak left of it, the slope right of it is part of the Sun's gravity well, and shows how much further you'd need to climb out of the Sun's gravity well to get from Venus to Earth. Likewise, the peak left of Jupiter is slightly lower than the one to the right; this is because the rightmost peak includes the difficulty of moving away from the Sun to get to Saturn, the next planet along.

The text explains that the depth of the well is mass-of-planet over radius-of-planet with newtons constant and 9.81 m/s² as constants, where 9.81 m/s² is the acceleration of a free falling body at Earth's gravity.

The calculation for a gravity well is:

depth = (G * Planet-mass ) / (9.81 m/s2 * Planet-radius)
where G is Newton's gravitational constant, and
9.81 m/s2 is the acceleration rate of a free falling body on earth at sea level (g).

[edit] Escape Velocities

The following table was adapted from the table in Escape velocity, using h = V_e^2 / 2g:

Location with respect to Ve (km/s) Well depth (km) Location with respect to Ve (km/s) Solar well (Mm) Total depth (Mm)
on the Sun, the Sun's gravity: 617.5 19,435,000 19,435
on Mercury, Mercury's gravity: 4.3 942 at Mercury, the Sun's gravity: 67.7 233.6 235
on Venus, Venus' gravity: 10.3 5,407 at Venus, the Sun's gravity: 49.5 124.9 130
on Earth, the Earth's gravity: 11.2 6,393 at the Earth/Moon, the Sun's gravity: 42.1 90.3 97
on the Moon, the Moon's gravity: 2.4 294 at the Moon, the Earth's gravity: 1.4 91
on Mars, Mars' gravity: 5 1,274 at Mars, the Sun's gravity: 34.1 59.3 61
on Jupiter, Jupiter's gravity: 59.5 180,400 at Jupiter, the Sun's gravity: 18.5 17.4 198
on Ganymede, Ganymede's gravity: 2.7 372
on Saturn, Saturn's gravity: 35.6 64,600 at Saturn, the Sun's gravity: 13.6 9.43 74
on Uranus, Uranus' gravity: 21.2 22,907 at Uranus, the Sun's gravity: 9.6 4.7 28
on Neptune, Neptune's gravity: 23.6 28,400 at Neptune, the Sun's gravity: 7.7 3.02 31
on Pluto, Pluto's gravity: 1.2 73
at Solar System
galactic radius,
the Milky Way's gravity: 525 14,000

[edit] Transcript

Main Text
Gravity Wells scaled to Earth surface gravity
This chart shows the "depth" of various solar system gravity wells.
Each well is scaled such that rising out of a physical well of that depth — in constant Earth surface gravity — would take the same energy as escaping from that planet's gravity in reality.
Each planet is shown cut in half at the bottom of its well, with the depth of the well measured down to the planet's flat surface.
The planet sizes are to the same scale as the wells. Interplanetary distances are not to scale.
Depth = (G × PlanetMass) / (g × PlanetRadius)
G = Newton's constant
g = 9.81 m/s2


Planetary Descriptions
To Sun, very very far down
Mercury
Venus
Earth - 5,478 km
Moon - 288 km
Mars - 1,286 km
Ganymede
Io
Jupiter
[A drawing of a "very deep" gravity well, "Your mom" at the bottom, several member of "local football team" falling down towards her.]
Jupiter is not much larger than Saturn, but much more massive. At its size, adding more mass just makes it denser due to the extra squeezing of gravity.
If you dropped a few dozen more Jupiters into it, the pressure would ignite fusion and make it a star.
Europa
Titan
Two figures: Weeoooeeoooeeooo
Saturn
Rings
Uranus
Neptune
Megan: An even more glorious dawn awaits!


Mars Inset
[Mars gravity well, the Pathfinder probe on its surface, with its moons Deimos and Phobos as smaller gravity wells.]
[Figure of a man (to scale) in Deimos's gravity well.]
You could escape Deimos with a bike and a ramp.
[Figure of a man (to scale) in Phobos's gravity well.]
A thrown baseball could escape Phobos.


Earth Inset
[Zoomed-in view of Earth/moon gravity well, featuring the relative locations of the atmosphere, Low Earth Orbit, the International Space Station, the Space Shuttle, GPS satellites, and satellites in geosynchronous orbit.]
Cueball: This is why it took a huge rocket to get to the moon but only a small one to get back.
It takes the same amount of energy to launch something on an escape trajectory away from Earth as it would to launch it 6,000 km upward under constant 9.81 m/s2 Earth gravity.
Hence, Earth's well is 6,000 km deep.
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Discussion

Why is Earth's well's depth listed as 5478km but as 6379km in the inset? Compare with Mars which has 1286 in both places. 87.174.225.131 07:21, 12 April 2013 (UTC)

Best guess is either a goof, or that the lower number is just for Earth itself, while the greater number is for the Earth/Moon system as a whole. Proportionally speaking, we have the largest moon in the solar system, so maybe it wouldn't nicely fit in the Earth well as easily as Mars's and Jupiter's moons do.--Druid816 (talk) 08:28, 12 April 2013 (UTC)
It may be the height needed to go from one gravity well to another. You don't have to get all the way up to escape speed for that.
Randall wasn't kidding about the Sun being "very very far down"; its well is 100 times deeper than Jupiter's!
Wwoods (talk) 19:47, 12 April 2013 (UTC)
OTOH, from the table above i'm thinking that the 5.4 might be the Venus figure, and it was wrongly placed besides Earth...
Secondly, what i found interesting was that the Earth's 6.4 looks so much like its radius! I wonder if it's merely a coincidence, or there's a connection between the two... -- 141.101.99.233 21:25, 30 October 2013 (UTC)
The table is great, it must be included in the article; layout and time is just my problem right now. PRO TIP: Do not care about the x-axis.--Dgbrt (talk) 22:18, 30 October 2013 (UTC)
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