explain xkcd - User contributions [en]

3181: Jumping Frog Radius

2026-02-27T10:44:22Z

2601:1C0:8080:3C70:10F8:D748:A1E9:ECB3: /* Explanation */

{{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==

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''s = (2*''G*M'') / ''c''2
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 cm in diameter, it would become a black hole.)

The comic suggests a "more useful" radius: the ''Jumping Frog radius'' ''r''jf, which is the size of a "planet" such that its gravity keeps a champion {{w|Frog jumping contest|jumping}} {{w|The Celebrated Jumping Frog of Calaveras County|frog}} from being able to achieve {{w|escape velocity}}. Thus [[Randall]] has instead of ''c'', the 299,792,458 m/s speed of light, used a much smaller value of 4.5 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 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''jf. 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 m/s needed to escape.

The title text points out that the ''r''jf 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''jf 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. Furthermore, five Solar System bodies (the Sun and the four giant planets) have gravity wells greater than Earth's, and therefore larger ''r''jf and greater ability to collect any frogs that might be gravitationally up for grabs. 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. Earth's ''r''jf exceeding its physical radius does accurately explain why, after evolving on Earth, no frogs have jumped to other celestial objects.

If you were to take a frog off the earth and put it in a tiny frog space suit, which somehow did not unduly inhibit its movement, it could jump off any number of the smaller bodies in the solar system. However, few of these bodies are small/low-mass enough for a frog to escape them, ''and'' large enough and close enough for us to observe them and accurately estimate their escape velocities. (The diameter of asteroid {{w|4942 Munroe}} is known to be about 3.45 km, but its shape and mass are unknown. Its surface has an [https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=2004942 exceptionally high albedo of 0.936], which suggests that the surface is mostly some kind of ice. If we assume that asteroid Munroe is spherical and entirely composed of water ice, with a density close to 1 g/cm3, its mass is 2.16 × 1010 kg, and its escape velocity is 0.041 m/s. If instead it's a solid sphere of meteoric iron/nickel with a density of about 8 g/cm3, its mass is 1.72 × 1011 kg, and its escape velocity is 0.115 m/s. In either case, Space Frog would have no trouble jumping away from Munroe.) Some examples:

{| class="wikitable"
!Celestial Body!!Escape Velocity (m/s)!!Frog Escape?!!Notes
|-
|Deimos||5.6||X||The smaller of Mars's two moons
|-
|Ersa||ca. 1||✓||Minor moon of Jupiter
|-
|Halley's Comet||ca. 2||✓||Notable comet, orbiting the sun every 76 years
|}

==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''jf = 2''GM'' / (4.5 m/s)2

:[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''jf
: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]]
[[Category:Geometry]]
[[Category:Math]]
[[Category:Pages with broken file links]]

3208: SNEWS

2026-02-27T03:43:10Z

2601:1C0:8080:3C70:10F8:D748:A1E9:ECB3: /* Explanation */

{{comic
| number = 3208
| date = February 16, 2026
| title = SNEWS
| image = snews_2x.png
| imagesize = 740x321px
| noexpand = true
| titletext = People say setting of fireworks indoors is dangerous, but I looked at their energy release and it's like 10^-40 foe; totally negligible.
}}

==Explanation==
{{incomplete|This page was created by a Type Ia firework display that Ponytail set off. Don't remove this notice too soon.}}
[[Ponytail]] is 'showing [[Hairy]] her bedroom'. Hairy asks about the large device on the ceiling, and Ponytail explains that it is part of the {{w|SNEWS}} (SuperNova Early Warning System). This provides advance notice of {{w|supernova}}e by detecting {{w|neutrino}}s (tiny particles that travel near the speed of light, rarely interacting with matter). Neutrinos are produced in large quantities during the collapse of the star core, which occurs hours before the brightness of the star surface starts to increase (drastically). Neutrinos from a supernova can be distinguished from those generated by the Sun: the latter are relatively steady in their flux (1010–1011 cm-2s-1) with energies < 20 MeV, while the former come in a much higher flux for a few seconds and have energies of 10–50 MeV. She explains this gives {{w|astronomer}}s warning, allowing them to observe the event with telescopes and other instruments.

Hairy reasonably assumes that the device is either a detector, forming part of the SNEWS, or some kind of telescope to be used in the event the SNEWS goes off. However, Ponytail explains that it is a {{w|fireworks}} launcher — presumably linked into the detection network and triggered if it registers an observation — for the purposes of waking her up so she can witness the supernova herself. This is a '''very''' bad idea, for a multitude of reasons. Reckless use of fireworks is known for causing significant property damage and personal injury, even when used outdoors; launching fireworks inside the house means causing an explosion in a confined area, guaranteeing that it will hit the building, maximizing the opportunity to ignite something flammable on the structure, and containing, and therefore amplifying, the sound of the burst (which can already deafen people who are too close). Understandably, Hairy {{tvtropes|ScrewThisImOuttaHere|leaves to sleep at his own house}}.

Some people aren't easily woken up by a simple {{w|alarm clock}}, especially if it is in reach and has a "snooze" function where a button will silence the alarm for several minutes before it beeps again. The similar sounds of "snooze" and SNEWS may be part of the joke.

Ponytail is being {{w|hyperbole|hyperbolic}}, because even if all astronomers were interested in supernovae, not every individual or observatory will be immediately situated to view a particular point in the sky. For example, they may need to wait for the Earth's rotation, causing the phenomenon to "rise" in the east. Others may be located at unfavorable latitudes where the object will never appear above the Earth's horizon. It may also take some time before the supernova reaches an apparent magnitude that is visible during the daytime. Which would be particularly disappointing for ''everyone'' with an interest (on Earth) if it all happens to a star currently too close to conjunction with the Sun to see, in spite of the advanced neutrino warning.

Since historical supernovae have been visible from 6 months to nearly 2 years, it would be unlikely that Ponytail sleeps through a new one in its entirety, although there would still be significant cachet for any astronomer lucky enough to be able to legitimately say that they had seen the 'first light' at the earliest opportunity. It would have been difficult for her to ''not'' sleep through part of the supernova, for the same reason if she had not set up the fireworks; ironically, she has made that a more likely possibility, because injuries from the fireworks may leave her in a coma or under medical sedation.

The title text is a play on the tremendous amount of energy released by a supernova. The {{w|Foe (unit)|foe}} is an unofficial unit of energy equal to 10^44 Joule (but named directly from initials in the original quantity of "ten to the fifty-one ergs", involving a {{w|Erg|pre-SI}} measurement of energy), which is approximately on the order of the usual amount of energy released by a supernova. In comparison, human-scale amounts of energy — even relatively significant ones such as firework detonations — are negligible. This ignores the fact that energy releases that are "negligible in comparison to a supernova" can still be easily fatal to humans; even the largest man-made nuclear explosion is approximately ''twenty-seven'' orders of magnitude less than the baseline 'foe' value. The described "10-40 foe" is equal to 10 kJ, the energy content of approximately 3.3 grams of pyrotechnic gunpowder (for instance, a string of sixty or so 50-mg firecrackers).

==Transcript==
{{incomplete transcript|Don't remove this notice too soon.}}
:[Ponytail is walking into her room. The room has a bed, a set of drawers and a large sci-fi device mounted on the ceiling. Hairy is standing in the room, pointing up at the device.]
:Hairy: What's that device?
:Ponytail: Part of the supernova early warning system.
:Ponytail: There hasn't been a Milky Way supernova in over a century.
:Ponytail: Astronomers don't want to miss the next one.

:[Close up of Ponytail, now sitting on the end of the bed]
:Ponytail: 20 years ago, we set up a supernova alert system using neutrino detectors.
:Ponytail: It should give us a few hours' advance notice.

:[In a frame-less panel the view zooms back out, showing Ponytail and Hairy.]
:Ponytail: If it ever goes off, every astronomer on earth will scramble to point their equipment at the sky.
:Hairy: Oh, OK. So is that a detector? Or some kind of telescope?

:[The panel moves to the right, showing Hairy walking away. Ponytail is still on the end of the bed, raising a clenched fist for dramatic effect.]
:Ponytail: Fireworks launcher.
:Ponytail: I '''''refuse''''' to sleep through a supernova.
:Hairy: I think I'll spend the night at my place instead.

==Trivia==
The word "off" is misspelled as "of" in the title text.

This comic was published on the eve of the {{w|Chinese New Year}} or Lunar New Year, which is reckoned by the new moon appearing at this time of year. Celebrations throughout Asia and communities worldwide include setting off firecrackers and launching fireworks.

{{comic discussion}}<noinclude>

[[Category:Comics featuring Ponytail]]
[[Category:Comics featuring Hairy]]
[[Category:Astronomy]]

3181: Jumping Frog Radius

2026-02-27T03:34:31Z

2601:1C0:8080:3C70:10F8:D748:A1E9:ECB3: /* Explanation */

{{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==

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''s = (2*''G*M'') / ''c''2
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 cm in diameter, it would become a black hole.)

The comic suggests a more useful radius: the ''Jumping Frog radius'' ''r''jf, which is the size of a "planet" such that its gravity keeps a champion {{w|Frog jumping contest|jumping}} {{w|The Celebrated Jumping Frog of Calaveras County|frog}} from being able to achieve {{w|escape velocity}}. Thus [[Randall]] has instead of ''c'', the 299,792,458 m/s speed of light, used a much smaller value of 4.5 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 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''jf. 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 m/s needed to escape.

The title text points out that the ''r''jf 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''jf 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. Furthermore, five Solar System bodies (the Sun and the four giant planets) have gravity wells greater than Earth's, and therefore larger ''r''jf and greater ability to collect any frogs that might be floating around in interplanetary space. 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. Earth's ''r''jf exceeding its physical radius does accurately explain why, after evolving on Earth, no frogs have jumped to other celestial objects.

If you were to take a frog off the earth and put it in a tiny frog space suit, which somehow did not unduly inhibit its movement, it could jump off any number of the smaller bodies in the solar system. However, few of these bodies are small/low-mass enough for a frog to escape them, ''and'' large enough and close enough for us to observe them and accurately estimate their escape velocities. (The diameter of asteroid {{w|4942 Munroe}} is known to be about 3.45 km, but its shape and mass are unknown. Its surface has an [https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=2004942 exceptionally high albedo of 0.936], which suggests that the surface is mostly some kind of ice. If we assume that asteroid Munroe is spherical and entirely composed of water ice, with a density close to 1 g/cm3, its mass is 2.16 × 1010 kg, and its escape velocity is 0.041 m/s. If instead it's a solid sphere of meteoric iron/nickel with a density of about 8 g/cm3, its mass is 1.72 × 1011 kg, and its escape velocity is 0.115 m/s. In either case, Space Frog would have no trouble jumping away from Munroe.) Some examples:

{| class="wikitable"
!Celestial Body!!Escape Velocity (m/s)!!Frog Escape?!!Notes
|-
|Deimos||5.6||X||The smaller of Mars's two moons
|-
|Ersa||ca. 1||✓||Minor moon of Jupiter
|-
|Halley's Comet||ca. 2||✓||Notable comet, orbiting the sun every 76 years
|}

==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''jf = 2''GM'' / (4.5 m/s)2

:[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''jf
: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]]
[[Category:Geometry]]
[[Category:Math]]
[[Category:Pages with broken file links]]

3181: Jumping Frog Radius

2026-02-27T01:52:30Z

2601:1C0:8080:3C70:10F8:D748:A1E9:ECB3: /* Explanation */

{{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==

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''s = (2*''G*M'') / ''c''2
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 cm in diameter, it would become a black hole.)

The comic suggests a more useful radius: the ''Jumping Frog radius'' ''r''jf, which is the size of a "planet" such that its gravity keeps a champion {{w|Frog jumping contest|jumping}} {{w|The Celebrated Jumping Frog of Calaveras County|frog}} from being able to achieve {{w|escape velocity}}. Thus [[Randall]] has instead of ''c'', the 299,792,458 m/s speed of light, used a much smaller value of 4.5 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 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''jf. 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 m/s needed to escape.

The title text points out that the ''r''jf 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''jf greater than their physical radius, and five of them (the Sun and the four giant planets) have ''r''jf greater than Earth's. 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. Earth's ''r''jf exceeding its physical radius does factually explain why, after evolving on Earth, no frogs have jumped to other celestial objects.

If you were to take a frog off the earth and put it in a tiny frog space suit, which somehow did not unduly inhibit its movement, it could jump off any number of the smaller bodies in the solar system. However, few of these bodies are small/low-mass enough for a frog to escape them, ''and'' large enough and close enough for us to observe them and accurately estimate their escape velocities. (The diameter of asteroid {{w|4942 Munroe}} is known to be about 3.45 km, but its shape and mass are unknown. Its surface has an [https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=2004942 exceptionally high albedo of 0.936], which suggests that the surface is mostly some kind of ice. If we assume that asteroid Munroe is spherical and entirely composed of water ice, with a density close to 1 g/cm3, its mass is 2.16 × 1010 kg, and its escape velocity is 0.041 m/s. If instead it's a solid sphere of meteoric iron/nickel with a density of about 8 g/cm3, its mass is 1.72 × 1011 kg, and its escape velocity is 0.115 m/s. In either case, Space Frog would have no trouble jumping away from Munroe.) Some examples:

{| class="wikitable"
!Celestial Body!!Escape Velocity (m/s)!!Frog Escape?!!Notes
|-
|Deimos||5.6||X||The smaller of Mars's two moons
|-
|Ersa||ca. 1||✓||Minor moon of Jupiter
|-
|Halley's Comet||ca. 2||✓||Notable comet, orbiting the sun every 76 years
|}

==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''jf = 2''GM'' / (4.5 m/s)2

:[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''jf
: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]]
[[Category:Geometry]]
[[Category:Math]]
[[Category:Pages with broken file links]]