<p>You ask about the larger body's magnetosphere providing protection from the solar rwind for the smaller body. Presumably you want the magnetosphere to prevent the solar wind from stripping away the atmosphere of the smaller body.</p>
<p>You describe the smaller body:</p>
<blockquote>
<p>The other body is 10.4M☽, or about .13M, slightly larger than Mars and lacking a large iron core.</p>
</blockquote>
<p>And possibly that world would be so small that it would lose atmopshere so fast that worrying about the solar wind would be pointless.</p>
<p>Wikipedia has an article about atmospheric escape which should be the first place to look to find out about different possible methods of atmospheric loss.</p>
<p><a href="https://en.wikipedia.org/wiki/Atmospheric_escape" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Atmospheric_escape</a></p>
<p>If you have any desire for either of your hypothetical planets to be habitable for oxygen brathing animals like humans in particular, or for liquid water using life in general, you should do research on planetary habitability.</p>
<p>There is one scientific study about the requirements for planets to be habitable for human beings from Earth (and thus for any lifeforms with the same requirements) <em>Habitable Planets for Man</em>, Stephen H. Dole, 1964.</p>
<p><a href="https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf" rel="nofollow noreferrer">https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf</a></p>
<p>On pages 34 to 35 Dole mentions complex equations for how fast a world's atmopshere will escape into space. And also mentions a simpler rule of thumb based on the ratio of the planet's escape velocity divided by the  of the particles of a gas in the exosphere of the atmosphere, the layer that gas particles escape into from.  The root-mean-square velocity of gas particles in the exosphere depends on their temperature in the exosphere, which seems to be higher than the surface temperature of a world.</p>
<p>According to table 5 on page 35, a comparatively minor change the ratio can make a comparatively big change in the length of time it takes for the amount of gas to be reduced to 1/<em>e</em>, or 0.368 of its original amount.</p>
<p>According to table 5. if the ratio is 1 or 2 the time to reduce the gas to 0.368 of hte original amount is zero, if the ratio is three a few weeks, if the ratio is four a few thousand years, if the ratio is five about 100 million years, if the ratio is six infinite time.</p>
<p>If an atmospheric gas is lost slowly enough, it might be replaced from various sources as fast or faster than it is lost.  And there are various pocesses which can speed up atmospheric loss.
So for each kilometer per second of escape velocity a world has, the mximum root-mean-square velocity of gas particles in its exosphere should be no more than  0.1666 to 0.2 kilometers per second, if the world is going to retain that gas for signifiantly long time periodss.</p>
<p>On pages 53 to 58 Dole discusses the mass range for a human habitable planet.  He decides that humans wouldn't settle a planet with a surface gravity higher than 1.5 <em>g</em>, 1.5 times the surface gravit of Earth.  Dole calculated a planet with a surface gravity of 1.5 <em>g</em> would have 2.35 the Mass of Earth, 1.25 the radius of Earth, and an escape velocity of 15.03 kilometers per second (which is 1.367 times Earth escape veloity of 11.186 kilometers per second).</p>
<p>The average surfce temperaure of Earth is about 288 degrees K (Kelvin). on page 54 Dole says the expshere temperatures of Earth are bout 1000 K to 2000K.  Dole says that if a planet can have an Earth like surface temperature of about 288 K and a maximum exopshere stemperature of 1000k, the root-mean-square velocity of oxygen atoms in the exosphere would be about 1.25 kilometers per second.  A world would need to have an escape velocity five times that, or 6.25 kilometers per second, to retain 0.368 of its original oxygen atmosphere for about 100 million years.</p>
<p>So Dole selects an escape velocity of 6.25 kilometers per second (0.558 tha tof Earth) that of Earth) as the minimum for a human habitable world.  According to Dole, that corresponds to a world with 0.195 the mass of Earth, 0.63 the radius of Earth, and a surface gravity of 0.49 <em>g</em>.</p>
<p>Dole didn't believe that a world that small could produce a dense oxygen atmosphere.  Two different calculations on page 56 gave lower mass limits of 0.25 and 0.57 Earth mass to produce a dense oxygen rich atmosphere, and Dole rather arbitaraily selected 0.4 the mass of Earth as the minimum mass to produce an oxygen atmosphere, correponding to 0.78 Earth radius and a surfce gravity of 0.68 <em>g</em>.</p>
<p>More recent discussions of planetary habitiablity discuss habitability of liquid water using life in general.  Any sort of atmosphere will do for such a planet, as long as it is dense enough to support liquid water on the surface and has the right influence on planetary temperature for water to be liquid on the surface.  So an atmosphere with a lot of oxygen is not a requiement for a world to be habitable in such discussions.</p>
<p>This article from 2013 <a href="https://arxiv.org/ftp/arxiv/papers/1209/1209.5323.pdf" rel="nofollow noreferrer">https://arxiv.org/ftp/arxiv/papers/1209/1209.5323.pdf</a> has a paragraph discussing the mass range for habitable worlds on pages 3 to 4:</p>
<blockquote>
<p>A minimum mass of an exomoon is required to drive a magnetic shield on a billion-year timescale (Ms ≳ 0.1M⊕,
Tachinami et al. 2011); to sustain a substantial, long-lived atmosphere (Ms ≳ 0.12M⊕, Williams et al. 1997; Kaltenegger
2000); and to drive tectonic activity (Ms ≳ 0.23M⊕, Williams et al. 1997), which is necessary to maintain plate tectonics and
to support the carbon-silicate cycle. Weak internal dynamos have been detected in Mercury and Ganymede (Kivelson et al.
1996; Gurnett et al. 1996), suggesting that satellite masses &gt; 0.25M⊕ will be adequate for considerations of exomoon
habitability. This lower limit, however, is not a fixed number. Further sources of energy – such as radiogenic and tidal heating, and the effect of a moon’s composition and structure – can alter our limit in either direction. An upper mass limit is
given by the fact that increasing mass leads to high pressures in the moon’s interior, which will increase the mantle viscosity
and depress heat transfer throughout the mantle as well as in the core. Above a critical mass, the dynamo is strongly
suppressed and becomes too weak to generate a magnetic field or sustain plate tectonics. This maximum mass can be placed
around 2M⊕ (Gaidos et al. 2010; Noack &amp; Breuer 2011; Stamenković et al. 2011). Summing up these conditions, we expect
approximately Earth-mass moons to be habitable, and these objects could be detectable with the newly started Hunt for
Exomoons with Kepler (HEK) project (Kipping et al. 2012).</p>
</blockquote>
<p>And the articles cited should give reasons for those mass limits.</p>
<p>I note that Titan, the largest moon of Saturn, has a mass of only 0.022t that of Earth, a surface gravity of 0.138 <em>g</em>, and an escape velocity of 2.641 kilometers per second, 0.236 times that of Earth.  And it has dense atmopshere, a little denser than Earth's, and billions of times more dense than the atmospheres of hte similar satellites Ganymede and Callisto.</p>
<p>Of course Titan gets only about 0.01 as much energy from the Sun as Earth does, and its surface temperature is 93.7 K, much colder than Earth, which means that its exopshere temperatures, and gas velocities, should also be much lower than Earth's.</p>
<p>And if there was a way for a world with the escpae veloicity of Titan to have EArthlike surface temperatures and also exosphere temperatures a low as those of Titan, it would be able to retain an oxygen atmopshere for geological eras of time. But I don not know of any way to make that possible.</p>
<p>In 2019, a new study suggested a new lower mass limit for habitable worlds.   But it applied only to water worlds, with world wide oceans many kilometers deep covering their solid surfaces. Such worlds could be habitable down to a lower mass limit of 0.0268 Earth mass.</p>
<p><a href="https://earthsky.org/space/small-rocky-exoplanets-can-still-be-habitable/" rel="nofollow noreferrer">https://earthsky.org/space/small-rocky-exoplanets-can-still-be-habitable/</a></p>
<p>And that is smaller than your 0.13 Earth mass smaller world.</p>
<p>You might possibly be interested in the Worldbuilding Stack exchange:</p>
<p><a href="https://worldbuilding.stackexchange.com/users/34461/m-a-golding">https://worldbuilding.stackexchange.com/users/34461/m-a-golding</a></p>

CC BY-SA 4.0
2022-12-09T22:38:40.433Z
I'd be a little skeptical about relying on a book from almost 60 years ago; there's been at least some scientific progress since 1964...
Peter Erwin
CC BY-SA 4.0
2022-12-16T17:55:59.093Z
@Peter Erwin  Yes.  But it was the only scientific study specifically about habitability for humans.  Other scientific studies seem to be for habitabiity for liquid water using life in general and so count as habitable worlds with conditions which would be instantly fatal for humans.
M. A. Golding
CC BY-SA 4.0
2022-12-08T21:51:18.077Z
Funny you should mention that, I previously posted another question about the same planetary system to the Worldbuilding Stack Exchange. <https://worldbuilding.stackexchange.com/questions/238998/how-can-a-planet-remain-in-perpetual-algae-bloom>
Thoth
CC BY-SA 4.0
2022-12-08T22:04:58.240Z
I plugged in the escape velocities for both planets, assuming both have the same exosphere temperature as Earth the smaller body's atmosphere should only retain oxygen for a few thousand years (EV = 5.401km/s, ratio = 4.3028), while the larger one has permanent oxygen (EV = 14.15km/s, ratio = 11.32). However, their average surface temperatures are different, ~298°K for the larger one and ~266.7°K for the smaller one. I'm not sure what their exosphere temperatures would be as a result.
Thoth
CC BY-SA 4.0
2022-12-08T20:00:59.010Z
2022-12-08T20:00:59.010Z
M. A. Golding
<p>A spinoff of my previous question, regarding the <a href="https://astronomy.stackexchange.com/questions/51185/orbital-stability-of-a-double-planet-system">Orbital Stability of a Double Planet System</a>. Imagine two closely-orbiting bodies with a semi-major axis of 79,250 km. One of which is 2<em>M</em>, with a large enough iron core to support a substantial magnetic field as strong if not stronger than Earth's. The other body is 10.4<em>M</em>☽, or about .13<em>M</em>, slightly larger than Mars and lacking a large iron core. Both orbit a .64<em>M</em>☉ star at .4 AU.</p>
<p>Does the smaller body orbit close enough for the larger one's magnetosphere to provide significant protection from solar wind? By significant, I mean enough to hold onto as much of an atmosphere as its gravitational pull would allow, if it had the same magnetosphere strength on its own. If not, how close does a sub-earth have to be to receive such protection from its partner?</p>

CC BY-SA 4.0
2022-12-09T16:13:04.090Z
@PeterErwin Ah, that makes more sense. Either way, from what I've gathered from Golding's answer, magnetic protection is only relevant if the planet has no way of replenishing its atmospheric content. If a planet can replenish atmospheric mass faster than solar wind can erode it, it needs no such protection. Therefore, if my smaller body is geologically active enough, it may not need magnetic protection at all.
Thoth
CC BY-SA 4.0
2022-12-13T21:28:31.983Z
@PeterErwin I've recently found out Venus has a weak magnetic field induced by its ionosphere, rather than an internal dynamo. This protects heavier gases like CO2 from being blown away, as well as light trace gases found at lower altitudes. I am not sure whether Mars or a similarly-sized planet could support such a field were it closer to the Sun.
Thoth
CC BY-SA 4.0
2022-12-14T11:39:48.727Z
@Thoth Mars, too, has a weak induced field (so does Saturn's moon Titan). https://www.nature.com/articles/s41550-020-1099-y
Peter Erwin
CC BY-SA 4.0
2022-12-08T06:18:20.267Z
It's not part of the answer, but the whole "magnetic protection" is overhyped.  Venus has no magnetic field at all, and a nice thick atmosphere.
James K
CC BY-SA 4.0
2022-12-08T09:36:42.243Z
@James K Fair, but Venus's atmosphere is extremely thick, meaning its rapid degradation from solar wind is negligible compared to its total mass. If it had the same atmospheric mass as Earth, it would be stripped away in less than a century. The planet in my question likely has a thinner initial atmosphere than Earth's due to lower surface gravity, meaning unless it becomes a Mini-Venus it would need a magnetosphere to retain it.
Thoth
CC BY-SA 4.0
2022-12-08T09:54:43.527Z
Venus has also lower surface gravity than earth, albeit marginally. Yet planetary mass is by this example not necessarily a good proxy for atmospheric thickness or mass
planetmaker
CC BY-SA 4.0
2022-12-08T10:03:15.977Z
@Thoth The mass of Venus’s atmosphere is about 90 times that of Earth’s. If Venus is, as you suggest, currently losing atmosphere at a rate of one Earth atmosphere per century, then it would lose its entire atmosphere in only nine thousand years. Since Venus has been around for four and a half billion years, it’s astonishingly unlikely that we just happen to be here at the moment it’s about to vanish completely.
Peter Erwin
CC BY-SA 4.0
2022-12-08T10:20:51.740Z
Also, spacecraft measurements indicate that Venus, Earth, and Mars are currently losing atmosphere to space at *approximately the same rate*, despite the fact that only one of those planets has an intrinsic magnetic field.
Peter Erwin
CC BY-SA 4.0
2022-12-08T10:36:27.437Z
@PeterErwin I was unaware of this information, thanks for the correction. I am curious about why Mars has such a thin atmosphere if not due to a lack of a magnetosphere, as I always assumed that was the cause. Maybe it just formed that way?
Thoth
CC BY-SA 4.0
2022-12-08T10:46:17.100Z
@planetmaker I always thought Venus' atmosphere only reached a significant mass once its surface temperature reached a tipping point, turning a former hydrosphere into an early cloud layer that would gradually become acidic due to volcanic outgassing. Beforehand, I figured it would be similar to early Earth's, if not slightly thinner. Also, note that the secondary planet in my question is far smaller than Venus, at roughly 0.13 Earth masses.
Thoth
CC BY-SA 4.0
2022-12-08T12:10:01.947Z
@Thoth The key thing about Mars is probably its low surface gravity (about 40% of Earth's or Venus's), which makes it harder for it to hang on to atmospheric molecules. Lacking a magnetic field *might* have played a role, too (though it probably had a magnetic field early in its history) -- my understanding is that modern studies suggest intrinsic magnetic fields can both retard *or* promote atmosphere loss, so it's currently hard to say.
Peter Erwin
CC BY-SA 4.0
2022-12-08T18:16:30.853Z
@Peter Irwin  The key thing about Mars is not its low surface gravity but its low escape velocity.  Tehe escape velocity, and not the surface gravity, is important for retaining an atmosphere.
M. A. Golding
CC BY-SA 4.0
2022-12-08T21:26:54.983Z
@PeterErwin I'd imagine it depends on the intensity of the field, as while one can block solar wind it may also ionize the upper atmosphere. On a related note, I recently found an article claiming flares from Red Dwarves might encourage a thick ozone layer on their orbiting planets. In those cases, stellar wind actually *encourages* atmosphere building rather than deterring it.
Thoth
CC BY-SA 4.0
2022-12-08T21:39:34.327Z
@PeterErwin Also, as M.A. Golding mentioned below, a planet may have methods of replacing gas to counteract the loss, such as volcanic outgassing here on Earth. I'm starting to suspect that, while the terrestrial planets in our system lose gas at the same rate, the two with substantial atmospheres have them *because they can replenish their atmospheric mass,* whereas Mars cannot due to its stagnant tectonic activity. Perhaps the magnetosphere isn't completely irrelevant to atmosphere retention, it's just one of multiple factors.
Thoth
CC BY-SA 4.0
2022-12-09T12:18:20.060Z
@M.A.Golding They're directly related: lower surface gravity means lower escape velocity.
Peter Erwin
CC BY-SA 4.0
2022-12-09T12:23:44.820Z
@Thoth I expect that "flares from red dwarfs might encourage a thick ozone layer" is about UV and X-ray radiation from the flares producing ozone *from other molecules already in the atmosphere*, not magically adding mass to the atmosphere.
Peter Erwin
<natural-satellites><planetary-atmosphere><magnetic-field><planetary-systems>
How close does a double-planet have to be to provide magnetic protection to a smaller body?
Thoth
# How close does a double-planet have to be to provide magnetic protection to a smaller body?

(Asked by: username_0 on 2022-12-07)

A spinoff of my previous question, regarding the [Orbital Stability of a Double Planet System](https://astronomy.stackexchange.com/questions/51185/orbital-stability-of-a-double-planet-system). Imagine two closely-orbiting bodies with a semi-major axis of 79,250 km. One of which is 2*M*, with a large enough iron core to support a substantial magnetic field as strong if not stronger than Earth's. The other body is 10.4*M*☽, or about .13*M*, slightly larger than Mars and lacking a large iron core. Both orbit a .64*M*☉ star at .4 AU.

Does the smaller body orbit close enough for the larger one's magnetosphere to provide significant protection from solar wind? By significant, I mean enough to hold onto as much of an atmosphere as its gravitational pull would allow, if it had the same magnetosphere strength on its own. If not, how close does a sub-earth have to be to receive such protection from its partner?

**Tags:** natural-satellites, planetary-atmosphere, magnetic-field, planetary-systems
> **Comments:**
> **username_2**: It's not part of the answer, but the whole "magnetic protection" is overhyped.  Venus has no magnetic field at all, and a nice thick atmosphere.  
> **username_0**: @James K Fair, but Venus's atmosphere is extremely thick, meaning its rapid degradation from solar wind is negligible compared to its total mass. If it had the same atmospheric mass as Earth, it would be stripped away in less than a century. The planet in my question likely has a thinner initial atmosphere than Earth's due to lower surface gravity, meaning unless it becomes a Mini-Venus it would need a magnetosphere to retain it.  
> **username_3**: Venus has also lower surface gravity than earth, albeit marginally. Yet planetary mass is by this example not necessarily a good proxy for atmospheric thickness or mass  
> **username_1**: @Thoth The mass of Venus’s atmosphere is about 90 times that of Earth’s. If Venus is, as you suggest, currently losing atmosphere at a rate of one Earth atmosphere per century, then it would lose its entire atmosphere in only nine thousand years. Since Venus has been around for four and a half billion years, it’s astonishingly unlikely that we just happen to be here at the moment it’s about to vanish completely.  
> **username_1**: Also, spacecraft measurements indicate that Venus, Earth, and Mars are currently losing atmosphere to space at *approximately the same rate*, despite the fact that only one of those planets has an intrinsic magnetic field.  
> **username_0**: @PeterErwin I was unaware of this information, thanks for the correction. I am curious about why Mars has such a thin atmosphere if not due to a lack of a magnetosphere, as I always assumed that was the cause. Maybe it just formed that way?  
> **username_0**: @planetmaker I always thought Venus' atmosphere only reached a significant mass once its surface temperature reached a tipping point, turning a former hydrosphere into an early cloud layer that would gradually become acidic due to volcanic outgassing. Beforehand, I figured it would be similar to early Earth's, if not slightly thinner. Also, note that the secondary planet in my question is far smaller than Venus, at roughly 0.13 Earth masses.  
> **username_1**: @Thoth The key thing about Mars is probably its low surface gravity (about 40% of Earth's or Venus's), which makes it harder for it to hang on to atmospheric molecules. Lacking a magnetic field *might* have played a role, too (though it probably had a magnetic field early in its history) -- my understanding is that modern studies suggest intrinsic magnetic fields can both retard *or* promote atmosphere loss, so it's currently hard to say.  
> **username_4**: @Peter Irwin  The key thing about Mars is not its low surface gravity but its low escape velocity.  Tehe escape velocity, and not the surface gravity, is important for retaining an atmosphere.  
> **username_0**: @PeterErwin I'd imagine it depends on the intensity of the field, as while one can block solar wind it may also ionize the upper atmosphere. On a related note, I recently found an article claiming flares from Red Dwarves might encourage a thick ozone layer on their orbiting planets. In those cases, stellar wind actually *encourages* atmosphere building rather than deterring it.  
> **username_0**: @PeterErwin Also, as M.A. Golding mentioned below, a planet may have methods of replacing gas to counteract the loss, such as volcanic outgassing here on Earth. I'm starting to suspect that, while the terrestrial planets in our system lose gas at the same rate, the two with substantial atmospheres have them *because they can replenish their atmospheric mass,* whereas Mars cannot due to its stagnant tectonic activity. Perhaps the magnetosphere isn't completely irrelevant to atmosphere retention, it's just one of multiple factors.  
> **username_1**: @M.A.Golding They're directly related: lower surface gravity means lower escape velocity.  
> **username_1**: @Thoth I expect that "flares from red dwarfs might encourage a thick ozone layer" is about UV and X-ray radiation from the flares producing ozone *from other molecules already in the atmosphere*, not magically adding mass to the atmosphere.  
> **username_0**: @PeterErwin Ah, that makes more sense. Either way, from what I've gathered from Golding's answer, magnetic protection is only relevant if the planet has no way of replenishing its atmospheric content. If a planet can replenish atmospheric mass faster than solar wind can erode it, it needs no such protection. Therefore, if my smaller body is geologically active enough, it may not need magnetic protection at all.  
> **username_0**: @PeterErwin I've recently found out Venus has a weak magnetic field induced by its ionosphere, rather than an internal dynamo. This protects heavier gases like CO2 from being blown away, as well as light trace gases found at lower altitudes. I am not sure whether Mars or a similarly-sized planet could support such a field were it closer to the Sun.  
> **username_1**: @Thoth Mars, too, has a weak induced field (so does Saturn's moon Titan). https://www.nature.com/articles/s41550-020-1099-y  

---

## 1 answers

### Answer by username_4 on 2022-12-08. Score: 1

You ask about the larger body's magnetosphere providing protection from the solar rwind for the smaller body. Presumably you want the magnetosphere to prevent the solar wind from stripping away the atmosphere of the smaller body.

You describe the smaller body:

> The other body is 10.4M☽, or about .13M, slightly larger than Mars and lacking a large iron core.

And possibly that world would be so small that it would lose atmopshere so fast that worrying about the solar wind would be pointless.

Wikipedia has an article about atmospheric escape which should be the first place to look to find out about different possible methods of atmospheric loss.

<https://en.wikipedia.org/wiki/Atmospheric_escape>

If you have any desire for either of your hypothetical planets to be habitable for oxygen brathing animals like humans in particular, or for liquid water using life in general, you should do research on planetary habitability.

There is one scientific study about the requirements for planets to be habitable for human beings from Earth (and thus for any lifeforms with the same requirements) *Habitable Planets for Man*, Stephen H. Dole, 1964.

<https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf>

On pages 34 to 35 Dole mentions complex equations for how fast a world's atmopshere will escape into space. And also mentions a simpler rule of thumb based on the ratio of the planet's escape velocity divided by the of the particles of a gas in the exosphere of the atmosphere, the layer that gas particles escape into from. The root-mean-square velocity of gas particles in the exosphere depends on their temperature in the exosphere, which seems to be higher than the surface temperature of a world.

According to table 5 on page 35, a comparatively minor change the ratio can make a comparatively big change in the length of time it takes for the amount of gas to be reduced to 1/*e*, or 0.368 of its original amount.

According to table 5. if the ratio is 1 or 2 the time to reduce the gas to 0.368 of hte original amount is zero, if the ratio is three a few weeks, if the ratio is four a few thousand years, if the ratio is five about 100 million years, if the ratio is six infinite time.

If an atmospheric gas is lost slowly enough, it might be replaced from various sources as fast or faster than it is lost. And there are various pocesses which can speed up atmospheric loss.
So for each kilometer per second of escape velocity a world has, the mximum root-mean-square velocity of gas particles in its exosphere should be no more than 0.1666 to 0.2 kilometers per second, if the world is going to retain that gas for signifiantly long time periodss.

On pages 53 to 58 Dole discusses the mass range for a human habitable planet. He decides that humans wouldn't settle a planet with a surface gravity higher than 1.5 *g*, 1.5 times the surface gravit of Earth. Dole calculated a planet with a surface gravity of 1.5 *g* would have 2.35 the Mass of Earth, 1.25 the radius of Earth, and an escape velocity of 15.03 kilometers per second (which is 1.367 times Earth escape veloity of 11.186 kilometers per second).

The average surfce temperaure of Earth is about 288 degrees K (Kelvin). on page 54 Dole says the expshere temperatures of Earth are bout 1000 K to 2000K. Dole says that if a planet can have an Earth like surface temperature of about 288 K and a maximum exopshere stemperature of 1000k, the root-mean-square velocity of oxygen atoms in the exosphere would be about 1.25 kilometers per second. A world would need to have an escape velocity five times that, or 6.25 kilometers per second, to retain 0.368 of its original oxygen atmosphere for about 100 million years.

So Dole selects an escape velocity of 6.25 kilometers per second (0.558 tha tof Earth) that of Earth) as the minimum for a human habitable world. According to Dole, that corresponds to a world with 0.195 the mass of Earth, 0.63 the radius of Earth, and a surface gravity of 0.49 *g*.

Dole didn't believe that a world that small could produce a dense oxygen atmosphere. Two different calculations on page 56 gave lower mass limits of 0.25 and 0.57 Earth mass to produce a dense oxygen rich atmosphere, and Dole rather arbitaraily selected 0.4 the mass of Earth as the minimum mass to produce an oxygen atmosphere, correponding to 0.78 Earth radius and a surfce gravity of 0.68 *g*.

More recent discussions of planetary habitiablity discuss habitability of liquid water using life in general. Any sort of atmosphere will do for such a planet, as long as it is dense enough to support liquid water on the surface and has the right influence on planetary temperature for water to be liquid on the surface. So an atmosphere with a lot of oxygen is not a requiement for a world to be habitable in such discussions.

This article from 2013 <https://arxiv.org/ftp/arxiv/papers/1209/1209.5323.pdf> has a paragraph discussing the mass range for habitable worlds on pages 3 to 4:

> A minimum mass of an exomoon is required to drive a magnetic shield on a billion-year timescale (Ms ≳ 0.1M⊕,
> Tachinami et al. 2011); to sustain a substantial, long-lived atmosphere (Ms ≳ 0.12M⊕, Williams et al. 1997; Kaltenegger
> 2000); and to drive tectonic activity (Ms ≳ 0.23M⊕, Williams et al. 1997), which is necessary to maintain plate tectonics and
> to support the carbon-silicate cycle. Weak internal dynamos have been detected in Mercury and Ganymede (Kivelson et al.
> 1996; Gurnett et al. 1996), suggesting that satellite masses > 0.25M⊕ will be adequate for considerations of exomoon
> habitability. This lower limit, however, is not a fixed number. Further sources of energy – such as radiogenic and tidal heating, and the effect of a moon’s composition and structure – can alter our limit in either direction. An upper mass limit is
> given by the fact that increasing mass leads to high pressures in the moon’s interior, which will increase the mantle viscosity
> and depress heat transfer throughout the mantle as well as in the core. Above a critical mass, the dynamo is strongly
> suppressed and becomes too weak to generate a magnetic field or sustain plate tectonics. This maximum mass can be placed
> around 2M⊕ (Gaidos et al. 2010; Noack & Breuer 2011; Stamenković et al. 2011). Summing up these conditions, we expect
> approximately Earth-mass moons to be habitable, and these objects could be detectable with the newly started Hunt for
> Exomoons with Kepler (HEK) project (Kipping et al. 2012).

And the articles cited should give reasons for those mass limits.

I note that Titan, the largest moon of Saturn, has a mass of only 0.022t that of Earth, a surface gravity of 0.138 *g*, and an escape velocity of 2.641 kilometers per second, 0.236 times that of Earth. And it has dense atmopshere, a little denser than Earth's, and billions of times more dense than the atmospheres of hte similar satellites Ganymede and Callisto.

Of course Titan gets only about 0.01 as much energy from the Sun as Earth does, and its surface temperature is 93.7 K, much colder than Earth, which means that its exopshere temperatures, and gas velocities, should also be much lower than Earth's.

And if there was a way for a world with the escpae veloicity of Titan to have EArthlike surface temperatures and also exosphere temperatures a low as those of Titan, it would be able to retain an oxygen atmopshere for geological eras of time. But I don not know of any way to make that possible.

In 2019, a new study suggested a new lower mass limit for habitable worlds. But it applied only to water worlds, with world wide oceans many kilometers deep covering their solid surfaces. Such worlds could be habitable down to a lower mass limit of 0.0268 Earth mass.

<https://earthsky.org/space/small-rocky-exoplanets-can-still-be-habitable/>

And that is smaller than your 0.13 Earth mass smaller world.

You might possibly be interested in the Worldbuilding Stack exchange:

<https://worldbuilding.stackexchange.com/users/34461/m-a-golding>

> **Comments:**
> **username_0**: Funny you should mention that, I previously posted another question about the same planetary system to the Worldbuilding Stack Exchange. <https://worldbuilding.stackexchange.com/questions/238998/how-can-a-planet-remain-in-perpetual-algae-bloom>  
> **username_0**: I plugged in the escape velocities for both planets, assuming both have the same exosphere temperature as Earth the smaller body's atmosphere should only retain oxygen for a few thousand years (EV = 5.401km/s, ratio = 4.3028), while the larger one has permanent oxygen (EV = 14.15km/s, ratio = 11.32). However, their average surface temperatures are different, ~298°K for the larger one and ~266.7°K for the smaller one. I'm not sure what their exosphere temperatures would be as a result.  
> **username_1**: I'd be a little skeptical about relying on a book from almost 60 years ago; there's been at least some scientific progress since 1964...  
> **username_4**: @Peter Erwin  Yes.  But it was the only scientific study specifically about habitability for humans.  Other scientific studies seem to be for habitabiity for liquid water using life in general and so count as habitable worlds with conditions which would be instantly fatal for humans.  

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