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Jumping Frog Radius

|< < Prev #3181 (December 15, 2025)

Title text: 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.

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Explanation

The Schwarzchild radius is essentially the size of a black hole -- the maximum distance from the center where gravity is so strong that light can't escape.

The comic suggests a more useful radius of large gravitational bodies. The "Jumping Frog radius" is the size of a planet so that its gravity keeps a champion jumping frog from being able to achieve escape velocity.

The title text points out that the r_jf of the Earth is about 1.5 light days, which is about 3 times the distance to Pluto. Since Earth's surface is much smaller than this^{[citation needed]}, no frogs will be able to escape, so they'll all collect here on Earth. As far as we know, all the frogs in the Solar System are on Earth, so the data apparently matches the theory.

The Schwartzchild radius is part of a solution to Einstein's field equations. It is usually calculated as the following:

r_{jf} = (2*G*M)/(c^2),

where G is the gravitational constant, M is the mass of the object, and c is the speed of light.

Instead of the 299,792,458 m/s speed of light, this comic uses a much smaller value of 4.5 m/s. It is possible that Randall got that value from this paper, which on page 179 puts an upper limit on the maximum velocity of adult Australian rocket frogs at 4.52 m/s. This much smaller denominator increases the calculated radius from the 9 mm Schwartzchild radius to almost 4*10^13 m, or just over 1.5 light-days. As Pluto is never more than about 0.3 light-days away from Earth, we can safely say that any frogs in the Solar System should be found on the surface of Earth, despite there likely not being a real frog prediction in general relativity.^{[citation needed]}

Transcript

[The panel shows a formula and a drawing to its right.]

r_jf = 2GM/(4.5 m/s)²

[A small planet with the radius marked as r_jf and a frog jumping on the surface. The frog jumps about as high as the planet's radius, saying "ribbit" in midair and "plop" as it lands.]

[Caption below the panel:]

More practically useful than the Schwartzchild 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.

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