Editing 681: Gravity Wells

{{comic
| number    = 681
| date      = December 28, 2009
| title     = Gravity Wells
| image     = gravity_wells.png
| titletext = 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.
}}
{{TOC}}
* The xkcd page links to [http://xkcd.com/681_large/ a much larger version].

==Explanation==
The comic shows the gravitational potential (energy transferred per unit mass due to gravity) for the positions of each planet in the solar system — including some moons and Saturn's rings. An object traveling along an upward slope loses energy, while an object traveling along a downward slope gains energy. Escaping a planet or moon's orbit requires enough energy (e.g. by walking, jumping, or rocket) to reach the top of either peak that defines the edge of the well. The peak to the left indicates the minimum energy required to exit orbit. The peak to the right indicates the maximum energy required to exit orbit. In order to exit orbit with the minimum amount of energy, you would have to travel towards the center of the solar system; to exit orbit with the maximum amount of energy, you would have to travel away from the center of the solar system (the Sun). In reality, the strength of gravity decreases with distance from the planet. However, a comparison of energy expended to escape the gravitational pull allows for a simpler comparison between the objects.

The height of the graph is scaled to kilometers via the gravitational potential an object has at the given height assuming at a constant acceleration due to Earth's surface gravity. The {{w|Sun|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. As the gravitational potential increases with distance from the sun, the graph has a general upward slope. To rise out of each well on the diagram, and therefore escape the planet's gravity, it would require the same energy required to rise out of a physical well of that depth at Earth's surface gravity. 

The length of each gravity well is scaled to the diameter of the planet and the spacing between the planets is not to scale with distance from the sun. This is necessary to make the graph readable. Because the distances between the planets are condensed, the gravitational potential - from the gravity pulling toward the sun - accumulates quicker. This is the reason for the large peaks between the planets. The moons shown in the chart are at the appropriate distance from their respective planets' gravity wells for their orbits. 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 Earth's gravity well's depth in the inset and in the main part of the comic don't match. This is most likely a mistake by Randall.

==== Inner Planets ====
* {{w|Mercury (planet)|Mercury}} — no facts listed
* {{w|Venus (planet)|Venus}} — no facts listed
* {{w|Earth}} and {{w|Moon}}: The listed depth of the gravity well at Earth was originally listed at 5478&nbsp;km rather than the correct value of 6379&nbsp;km seen in the cutout.  Randall has since corrected it.  The Moon's is 288&nbsp;km.
* {{w|Mars}}: The listed depth of the gravity well of Mars is 1286&nbsp;km.

==== Outer Planets ====
* {{w|Jupiter}}: Jupiter is so massive and dense that it is one thirteenth the mass of a small {{w|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. If a few dozen times the mass of gasses contained in Jupiter had condensed in that location, the gravitational pull would cause the pressure and temperature to increase to a level that is sufficient to ignite {{w|Nuclear fusion|nuclear fusion}}. Had that happened during the creation of our solar system, we would have two {{w|Sun|Suns}} and our solar system would be a {{w|Binary system (astronomy)|Binary system}}.  Jupiter has {{w|Moons of Jupiter|67 moons}} of which 3 are shown;
** {{w|Ganymede (moon)|Ganymede}} — moon of Jupiter, no facts given
** {{w|Io (moon)|Io}} — moon of Jupiter, no facts given
** {{w|Europa (moon)|Europa}} — moon of Jupiter, no facts given

* {{w|Saturn}}: The diagram shows the position of the {{w|rings of Saturn}} in Saturn's gravity well. 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 {{w|Cassini–Huygens|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. Saturn has {{w|Moons of Saturn|62 moons}} of which one is shown;
** {{w|Titan (moon)|Titan}}, a moon of Saturn. The figures on Titan are sirens, a reference to Kurt Vonnegut's ''{{w|The Sirens of Titan}}''.

* {{w|Uranus}}: Notably absent is any "your-anus" jokes.
* {{w|Neptune}}: Megan's quote is a paraphrase of {{w|Carl Sagan|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." [http://www.youtube.com/watch?v=zSgiXGELjbc Video here]

==== Cut outs and sketches====
The following items are listed from top to bottom and left to right.

* Mars moons: The Mars cutout shows the Mars moon system, including the moons Deimos and Phobos. The depth of the Mars gravity well is listed at 1286&nbsp;km.
** {{w|Deimos (moon)|Deimos}}: The gravity on Deimos, a moon of Mars, is so weak that a bike jump would be sufficient to escape its gravity.
** {{w|Phobos (moon)|Phobos}}: The gravity on Phobos, a moon of Mars, is so weak that you could launch a baseball into space simply by throwing it.

* Your mom and a local football team: The sketch next to Jupiter is playing on the classic "Yo Mama" joke. It combines "Yo Mama is so fat" and "Yo Mama is so horny". The sketch implies that she has a huge gravitational pull because she is very fat, and has sex with an entire football team by demonstrating a football team falling into her very deep gravity well. A "Yo Mama" joke also appears in comic [[89: Gravitational Mass]].

* Earth's Moon: The cut out shows the significant difference in strength between the {{w|gravity well}} of the Earth and the Moon. Cueball comments that the {{w|Apollo Lunar Module|Apollo Lunar Module}} was very small and the {{w|Saturn V}} rocket was very large because escaping the Earth's gravity well takes much more energy than escaping the Moon's. The cut out also shows objects like the {{w|International_Space_Station|International Space Station}}, the {{w|Space shuttle|space shuttle}}, {{w|GPS satellite|GPS satellites}} and {{w|Geostationary orbit|geo-stationary satellites}} at their respective positions within Earth's gravity well. The depth of Earth's gravity well is listed correctly at 6 379&nbsp;km (note the difference from the non-cutout number). The depth of the Moon's gravity well is listed at 288&nbsp;km.

=== How to calculate gravity wells ===
The text near the bottom of Jupiter's gravity well explains that the depth of the well is mass-of-planet over radius-of-planet with Newton's constant and 9.81&nbsp;m/s² as constants, where 9.81&nbsp;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/s<sup>2</sup> * Planet-radius)
::where G is {{w|Isaac Newton|Newton}}'s {{w|gravitational constant}}, and
::9.81 m/s<sup>2</sup> is the {{w|acceleration}} rate of a {{w|Gravity of Earth|free falling body on earth}} at sea level (g).

====Title text====
The title text indicates that the planets motion can affect the amount of energy for escape velocity. It is possible to change speed by using the planets orbital speed and gravity. This is known as a performing a slingshot or a {{w|gravity assist}}, and is done to gain speed or to brake when needed. The use of rocket engines are more effective when used at a high speed slingshot maneuver, which is known as the {{w|Oberth effect}}, where most energy is going into moving the rocket as opposed to moving the exhaust — conserving the maximum useful energy.   On earth the same principle is used when launching rockets. Rockets are always launched in an eastward direction to make maximum use of the rotational energy of the earth. Launching rockets in a westward direction would require significant additional energy. Because of this most artificial satellites are flying east around the globe.

The size of the gravity-well as described in this comic is not accounting for these factors. Therefore, leaving the solar system (or any of the gravity wells of the planets) could require less energy than described by the graph, assuming that the launch and slingshots are properly designed and executed.

==== Escape Velocities ====
The following table was adapted from the table in {{w|Escape velocity#List of escape velocities|Escape velocity}}, using ''h'' = ''V_e''^2 / 2''g'':
{| class="wikitable" border="1"
| 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 <br/>galactic radius, || the Milky Way's gravity: || 525 || 14,000 
|}

==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 <u>surface</u> 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/s<sup>2</sup>


:'''Planetary Descriptions'''
:To Sun, very very far down
:Mercury
:Venus
:Earth - 6379 km [originally 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 alarms: Weeoooeeoooeeooo
:Saturn
::Rings
:Uranus
:Neptune
::Megan: An even more glorious dawn awaits!


:'''Mars Inset'''
:[Mars gravity well, with one of the Mars rovers 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/s<sup>2</sup> Earth gravity.

:Hence, Earth's well is 6,000 km deep.

==Trivia==
This comic used to be [https://web.archive.org/web/20211215032220/https://store.xkcd.com/products/gravity-wells-poster available as a poster] in the xkcd store before it was [[Store|shut down]].

{{comic discussion}}

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