Editing 3181: Jumping Frog Radius

{{comic
| number    = 3181
| date      = December 15, 2025
| title     = Jumping Frog Radius
| image     = jumping_frog_radius_2x.png
| imagesize = 339x243px
| noexpand  = true
| titletext = Earth's r_jf is approximately 1.5 light-days, leading to general relativity's successful prediction that all the frogs in the Solar System should be found collected on the surface of the Earth.
}}

==Explanation==
{{incomplete|This page was created by an A frog stuck on mars. I have added a bit about the drawing. It is important I think that the planet with the frog has exactly the r<sub>jf</sub> radius. This means the frog cannot escape but just barely. Is there a physics relation behind the fact that the jumps height seems to be very close to the radius of the planet, i.e. r<sub>jf</sub>? Also Can someone calculate the size and mass of the largest object from which a champion frog can achieve escape velocity? Are there some named asteroids that are of sow low a mass that it would be possible for frog to jump of? (Of course there are some small enough... but do any of them have real names, like the one named after Randall)? Don't remove this notice too soon.}}
The {{w|Schwarzschild radius}} is essentially the size of a {{w|black hole}} -- the maximum distance from the center where gravity is so strong that light can't escape. It is part of a solution to {{w|Einstein's field equations}}. It is usually calculated as
:''r''<sub>s</sub> = (2*''G*M'') / ''c''<sup>2</sup>
where ''G'' is the {{w|gravitational constant}}, ''M'' is the mass of the object, and ''c'' is the {{w|speed of light}}. 
If ''M'' were the mass of the {{w|Earth}}, it would give the Schwarzschild radius for the Earth, which is about 9 mm. (If all of Earth's mass were compressed into a sphere of a bit less than 2&#8239;cm in diameter, it would become a black hole.)

The comic suggests a more useful radius: the ''Jumping Frog radius'' ''r''<sub>jf</sub>, which is the size of a "planet" such that its gravity keeps a champion {{w|Frog jumping contest|jumping frog}} from being able to achieve {{w|escape velocity}}. Thus [[Randall]] has instead of ''c'', the 299,792,458&#8239;m/s speed of light, used a much smaller value of 4.5&#8239;m/s, to represent the maximum speed of a jumping frog. It is possible that Randall got that value from [https://www.researchgate.net/publication/5661154_Explosive_Jumping_Extreme_Morphological_and_Physiological_Specializations_of_Australian_Rocket_Frogs_Litoria_nasuta this paper], which on page 179 puts an upper limit on the maximum velocity of adult Australian {{w|striped rocket frog}}s at 4.52&#8239;m/s. (The frog is shown making a "ribbit" sound, which is made by {{w|Pacific tree frog}}s and their relatives in North America and not by rocket frogs, but it's [https://www.imdb.com/list/ls052470723/ widely attributed to frogs all over the world].)

The drawing to the right of the formula shows a planet with exactly the radius ''r''<sub>jf</sub>. Thus the frog can jump really high compared to the planet's size (in this case about as high as the planet's radius), before it falls back down. This implies that the frog is jumping at somewhat less than the 4.5&#8239;m/s needed to escape.

The title text points out that the ''r''<sub>jf</sub> of the Earth is about 1.5 light days, which is about 7 times the distance to {{w|Pluto}} (compare to the 9 mm Schwarzschild radius). Since Earth's radius is much smaller than this, no frogs will be able to escape, so all frogs that stray into Earth's gravitational well would collect here on Earth. As far as we know, all the frogs in the Solar System are on Earth{{Citation needed}}, so the data apparently matches the theory. However, the reasoning is incorrect, as many other astronomical bodies in our solar system also have ''r''<sub>jf</sub> greater than their physical radius. If a frog were to be on any of those other bodies, it wouldn't be able to jump away to fall to Earth. A flawed argument neither supports nor refutes the conclusion, although it is true as far as we know that all frogs in the solar system do live on Earth.

However, if you were to take a frog off the earth and put them in a tiny frog space-suit. They could jump off any number of the smaller bodies in the solar system. As an example:

<table>
<tr><th>Celestial Body</th><th align="center">Escape Velocity (m/s)</th><th align="center">Frog Escape?</th><th>Notes</th></tr>
<tr><td>Sedna</td><td align="center">4.83</td><td align="center"><b>X</b></td><td>A distant trans-Neptunian object.</td></tr>
<tr><td>Haumea</td><td align="center">4.43</td><td align="center"><b>&#10003;</b></td><td>A dwarf planet in the Kuiper Belt.</td></tr>
<tr><td>Makemake</td><td align="center">4.36</td><td align="center"><b>&#10003;</b></td><td>Another dwarf planet located in the Kuiper Belt.</td></tr>
<tr><td>Eris</td><td align="center">4.43</td<td align="center"><b>&#10003;</b></td>><td>A dwarf planet that is slightly smaller than Pluto.</td></tr>
<tr><td>Trojan asteroids</td><td>Varies (generally < 5)</td><td align="center"><b>?</b></td><td>Many of these have low escape velocities due to their small sizes.</td></tr>
</table>

==Transcript==
:[The panel shows a large formula to the left and a small drawing to the right. The formula's right side is drawn above and below the division line:]
:''r''<sub>jf</sub> = 2''GM'' / (4.5<sup> m</sup><small>/</small><sub>s</sub>)<sup>2</sup>

:[The drawing to the right shows a very small planet with the radius indicated with a labeled dotted arrow pointing from the center straight up to the edge of the planet. A frog is shown jumping on the surface. This is indicated with a parabolic dotted line going from a frog sitting on the surface near the top of the planet, up to the frog shown soaring through the air with its limbs stretched out about as high above the surface as the planet's radius. At this point the frog is making a sound. Then the dotted line goes down to about a quarter of the way around the planet where the frog lands making a noise, with lines around the frog representing the impact.]
:Arrow label: ''r''<sub>jf</sub> 
:Frog: Ribbit
:Landing: Plop

:[Caption below the panel:]
:More practically useful than the Schwarzschild radius, the '''''Jumping Frog Radius''''' is the radius at which an object's gravitational pull is so strong that even a champion jumping frog can't escape.

{{comic discussion}}<noinclude>
[[Category:Physics]]
[[Category:Astronomy]]
[[Category:Animals]]