681: Gravity Wells
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.
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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 would be giving up energy, while an object traveling along a downward one would be gaining energy.
Escaping a planet's or moon's orbit requires enough energy to reach the top of either peak that defines the edge of the well. The peak to the left indicates the minimum energy to exit orbit by traveling toward the center of the solar system. The peek to the right indicates the maximum energy to exit orbit by traveling directly away from 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 earths surface gravity. 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. 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 planets 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 distance between the planets are condensed the gravitational potential, from the gravity pulling toward the sun, accumulates quicker. This is the reason for the large peeks between the planet. 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.
- Mars - The listed depth of the gravity well of Mars is 1286 km
- The Mars cutout shows how weak the gravity of Mars' moons really are. Deimos is so weak that a bike jump would be sufficient to escape its gravity. 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. It combines "Yo Mama is so fat" and "Yo Mama is so horny". The joke implies that she has a huge gravitational pull and has sex with the entire football team by demonstrating a football team being falling into her very deep gravity well. A "Yo Mama" joke also appears in comic 89: Gravitational Mass.
- The Earth/Moon cut out shows the significant difference in strength between the gravity well of the Earth and the Moon. Cueball comments that the Apollo Lunar Module was small and the Saturn V rocket was much larger because escaping the Earth's well takes much more energy than escaping the Moon's. The cut out also shows that objects like the International Space Station, the space shuttle, GPS satellites and geo-stationary satellites at their respective positions within Earth's gravity well.
- Jupiter — 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. Had a few dozen times the mass of gasses contained in Jupiter condensed in that location, the gravitational pull would cause the pressure and temperature to increase to a level that is sufficient to ignite nuclear fusion. Had that happen during creation of our solar system, we would have two Suns and our solar system would be a Binary system.
- Jupiter's moons
- Saturn & its rings — The diagram shows the position of Saturn's rings in its 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 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.
- Moons of Saturn
- 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
How to calculate gravity wells
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)
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 to gravity assist. This is know as a performing a slingshot or a gravity assist, and is done to gain speed or to break when needed. On earth the same principle is used when launching rockets. Rockets are always launched in a 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. Also, the use of rocket engines are more effective when used at a high speed. This is know as the Oberth effect. The use of engines are therefore more effective when used as part of a slingshot maneuver.
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.
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
|the Milky Way's gravity:||525||14,000|
- 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
- Earth - 5,478 km
- Moon - 288 km
- Mars - 1,286 km
- [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.
- Two figures: Weeoooeeoooeeooo
- 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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