Habitable moon of a gas giant: working out the sizes and distances

Question

I am attempting to create fictional, stable P-Type binary system, featuring a gas giant in a stable orbit, with a habitable Earth-like moon. “Is a Jupiter-sized planet plausible in a habitable zone?” has some interesting and useful information about gas giants in the circumstellar habitable zone of a system, and “Can a gas giant have its own habitable zone?” has some good information about the potential of a gas giant having its own habitable zone, separate from the circumstellar habitable zone.

Within the constraints of this fictional system, I have a habitable zone spanning between 1.976 AU and 2.808 AU, and the following considerations.

• Gas giant would need a stable magnetosphere. Jupiter and Saturn may be useful examples.

• The moon's mass must be great enough to sustain an atmosphere. In this instance, a nitrogen/oxygen atmosphere. It is estimated that a moon with a Mars-like density, would need at least 7% of Earth's mass in order to support such an atmosphere for several billion years.

• Both gas giant and the habitable moon must maintain a stable orbit. Simulations would suggest that to maintain a stable orbit to a gas giant, or a brown dwarf that orbits 1 AU from a sun-like star, would require a moon orbital period of less than 45–60 days.

• The moon itself must be capable of generating its own magnetosphere in order to deflect stellar wind and the gas giants' naturally generated radiation belts.

• There is a high likelihood that the moon would be tidally locked with its parent world. Monoj Joshi, Robert Haberle, and their colleagues suggest that the effect of tidal heating could support conditions amenable to habitability. Additionally, tidal effects may allow for plate tectonics, causing volcanic activity and a regulation of the moon's surface temperature. The potential, resulting geodynamo effect would allow for a strong magnetic field.

• Balance: The moon should be large enough to support tectonic activity, dense enough to support a strong protective magnetosphere, close enough to the gas giant to maintain a stable orbit, and be far enough away that its own magnetosphere may better protect from sputtering caused by its parent worlds' radiation belts.

• It is suggested that the larger and denser a terrestrial, water-rich world, the further out its habitable zone extends.

• The moon does not necessarily need to be an earth analog, and may simply be demonstrated as habitable to human life.

• The gas giant does not necessarily need to be within the habitable zone and may cradle the outer limits of the circumstellar habitable zone, or be further out provided it can be demonstrated that the orbiting moon could feasibly support human life unassisted by technology. i.e. Robin Crusoe could become stranded on the moon, and survive.

Ready for the fun part?

If the terrestrial moon must be of a certain size to display tectonic activity throughout its life, as demonstrated in the difference between Earth and Venus (Venus being about 85% the size of Earth), then an Earth-sized moon (or larger) would be preferable.

To the best of my understanding this paper suggests that a world of this size, wouldn't be formed in the accretion disk of a gas giant (but I may have misunderstood), however, after the migration of a gas giant, the mixing to inner system and outer system debris has shown in simulations, to allow for the formation of water-rich terrestrial worlds. The paper does at least suggest that larger bodies may be captured, and pulled into orbit by a gas giant.

So, let's say our gas giant migrated from the frostline of the system, to somewhere near the circumstellar habitable zone, and afterword, as the orbit began to stabilize, a new terrestrial world began to take shape. Its orbit took it near enough to the gas giant to be pulled into orbit of the planet, and over time, their mutual orbits stabilized.

• How can I figure out how large the gas giant must be in order to capture this moon, and establish a stable orbit?

• Tidal locking of the moon may be an issue, but also may be compensated by its orbit around the gas giant. How can I determine how far the moon would need to orbit the gas giant, to not be tidally locked? This topic has some interesting points.

Here is my hypothetical.

The gas giant cradles the outer limits of the circumstellar habitable zone in such a way that the captured moon passes through the circumstellar habitable zone during each rotation. The size of the moon is large enough for tectonic activity, which may in turn, be aided by gravitational forces from its primary. The moon is also dense enough, with an iron/nickel core, to produce a strong magnetosphere, which if further aided by tectonic activity. Tidal forces are affected by the gravitational pull of the primary, throughout the moons orbit. Keeping the world warm enough to sustain liquid water, I don't think will be an issue, and would instead be a matter of striking a balance between orbital distance between the primary and the system's stars.

I feel like I am missing a few things. What are your thoughts on how I can work out a viable, habitable moon in this scenario?

_{Originally I asked this on Astronomy, and it was suggested that I ask here in Worldbuilding instead.}

Answer 1

Let's work out some factors.

• Luminosity

  You gave the radius of the inner edge of the habitable zone as 1.976 AU and the outer edge as 2.808 AU. From this, we can calculate the luminosity of the star. There's an explanation of how to do this on Planetary Biology. The formulae are $$r_i=\sqrt{\frac{L_{\text{star}}}{1.1}}$$ $$r_o=\sqrt{\frac{L_{\text{star}}}{0.53}}$$ Plugging in your numbers, I get a luminosity of $$4.295 L_{\odot}\text{ (inner radius)}$$ $$4.179 L_{\odot}\text{ (outer radius)}$$ I'll average those, giving us a luminosity of $4.237$ times the luminosity of the Sun. But a P-type orbit is around two stars, as you said, so we divide by two to get an average luminosity of $2.112$ solar luminosities. We can assume that the two stars are similar because they most likely formed together, and have similar properties.

• Mass

  The mass-luminosity relation can tell us the masses of the stars. It is $$\left(\frac{L}{L_{\odot}} \right)=\left(\frac{M}{M_{\odot}} \right)^a$$ The stars likely have masses similar to the Sun, so we can assume $a \approx 4$. The left side is $4.179$. We write $$4.179^{\frac{1}{4}} \times M_{\odot}=M\approx 1.430M_{\odot}$$ So each star is about $1.430$ solar masses, leaving a combined mass of $2.860$ solar masses.

• Orbital period of the gas giant

  Kepler's Third Law tells us that $$T=\sqrt{\frac{4 \pi ^2}{GM_{\text{star}}}r^3}$$ Here, $M_{\text{Star}}$ is actually the mass of both the stars. If the radius is in the middle of the zone (at about $r=2.392$ AU) $$T=\sqrt{\frac{4 \pi}{6.673 \times 10^{-11} \times 5.689 \times 10^{30}}(3.578 \times 10^{11})^3}=6.902 \times 10^7 \text{ seconds}= 800 \text{ days}$$

• Orbital radius of the moon

  Here we just go in reverse. We do need the mass of the gas giant, though - so going from the graph on TimB's answer here, I'll pick about 5 Jupiter masses, or $9.49 \times 10^{27}$ kilograms. The period will be in between the values you said, so about 52.5 days, which is $4.536 \times 10^6$ seconds. We put this all in and get $$r=\left( \frac{6.673 \times 10^{-11} \times 9.49 \times 10^{27}}{4 \pi ^2}(4.536 \times 10^6)^2 \right)^{\frac{1}{3}}=6.911 \times 10^{6} \text{ kilometers}$$ Obviously, it's still in the habitable zone. But it's far away - although that's because the gas giant is so massive. You may want to opt for a shorter period.

  This system setup appears to be viable if you move the moon closer to the gas giant, giving it a smaller orbital period.

• Tidal Locking

  The formula for the time it takes for a satellite to be tidally locked is $$t \approx \frac{wa^5IQ}{3Gm_{\text{planet}}^2k_2R^5}$$ The factors are described on the Wikipedia page. Here, we can say that $I \approx 0.4m_sR^2$, so $$t \approx \frac{0.4 wQR^2a^6}{3Gm_{\text{planet}}^2k_2r^5}$$ Since $$k_2 \approx \frac{1.5}{1+\frac{19 \mu}{2 \rho gR}}$$ and $g \approx \frac{Gm_s}{R^2}$, $$k_2 \approx \frac{1.5}{1+\frac{19 \mu R}{2 \rho Gm_s}}$$ $$k_2 \approx \frac{3 \rho Gm_s}{2 \rho GM_s+19 \mu R}$$ $$t \approx \frac{0.4 wQR^2a^6(2 \rho GM_s+19 \mu R)}{9G^2m_{\text{planet}}^2 \rho R^5}$$ With $Q \approx 100$, $\mu = 3 \times 10^{10}$, $R \approx R_{\text{Earth}}$ and $\rho = \rho_{\text{Mars}}$, you can figure out the tidal locking time. I'm in a rush, so I don't have time to do the calculation, but I may include it later.

How can I determine how far the moon would need to orbit the gas giant, to not be tidally locked?

Tidal locking will occur at some point in time. You can't get around it.

Tidal forces will also be problematic because moons orbiting gas giants will likely experience tidal forces so strong that tidal heating can render the moon uninhabitable (see Heller & Barnes (2013)).

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Capture - Corrections

In my original post, I naively said that there are a bunch of scenarios where capture would be possible. This, as HopDavid pointed out, is blatantly false, because the planet would be traveling in a hyperbolic orbit relative to the gas giant, and so would escape its pull rather easily.

So it has to have its orbit modified somehow.

My suggestion would be an interaction with another body, preferably another gas giant. This could change its orbit such that gravitational capture by the original gas giant is possible. Without this sort of interaction, the planet will just scoot away.

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Section on the moon's properties

This may be list-like, but it's the best I can do.

• Mass: You required an atmosphere and a magnetosphere. Both of those require a planet with the right mass and size, as well as composition (which I'll get to). Not many moons have atmospheres complex enough and dense enough to support life. In fact, Mercury can't support an atmosphere. But mass isn't the only thing that plays into this. Titan, one of Saturn's moons, has a mass less than twice that of Mercury, yet it has a rich atmosphere. As Jim2B pointed out, though, such a planet wouldn't be able to hold onto water vapor, as this chart shows, because its escape velocity would be too low:

^{Image courtesy of Wikipedia user Cmglee under the Creative Commons Attribution-Share Alike 3.0 Unported license.}

Also, the maximum mass of the moon is related to the mass of the parent planet, meaning that for a more massive moon, you'll need a much more massive gas giant for it to orbit.

You can attribute this to a few factors:

• The presence of Saturn's magnetosphere
• Low temperatures

• A weak solar wind at that distance from the Sun.

  You've got the distance, the weak stellar wind, and the presence of a gas giant and its magnetosphere. So you really want to aim for a mass similar to that of Titan, at $1.3452 \times 10^{23}$ kilograms.

  • Size: You don't want anything too tiny, because the density of such a body is far greater than expected. Conversely, you don't want anything too big, because the surface gravity would be weaker than you'd like. So go for a surface acceleration of perhaps $0.5g$ - half that of Earth. You can figure out your average radius using $$g=\frac{MG}{r}$$ So you can see why we needed the mass.

  • Composition: You don't want an environment that's hostile to life, so perhaps it would be best to mimic Earth as much as possible. Choose silicate materials for the outer layers, but remember to have nickel and iron for the core. These can help produce that moon's magnetosphere - a crucial component to retaining an atmosphere. Mars' lack of a magnetosphere has contributed to it slowly losing its atmosphere.

Answer 2

An object's speed about a star is given by the Vis Viva equation:

$V=\sqrt{GM(2/r - 1/a)}$ Where a = semi major axis of ellipse GM = Gravitational constant times mass of star r = distance from star's center.

Presumably a earth sized body and a gas giant in their own star centered orbits would have different values for a, semi-major axis. When they cross orbits they'd both be the same distance from the star so we could use the same value of r for both moon and gas giant when they cross orbits. So they would have different speeds with regard to the central star. Their orbits would also likely intersect at an angle. Here is a pic:

The velocity difference in their star centered orbits is shown in red. I will call the velocity indicated in red $V_{infinity}$. Here is an explanation of Vinf

When the incoming body enters the gas giant's sphere of influence, the big planet's gravity is the dominant influence so it no longer makes sense to model the path as an ellipse about the star. Now the moon's path is better modeled as a hyperbola about the gas giant.

Speed of a hyperbola: $V_{hyperbola}=\sqrt{V_{escape}^2+V_{infinity}^2}$

If $|V_{infinity}|>0$ then the incoming body's speed will exceed the gas giant's escape velocity. If $V_{infinity}=0$ then the incoming body's speed will equal escape velocity and the orbit will be parabolic.

Unless there is another influence beside the star's and gas giant's gravity, an incoming body from another orbit won't be captured by the gas giant.

Other influences are possible. If the gas giant already has moons, collisions or swing bys could shed the incoming body's speed wrt gas giant. Or if the moon passes through the gas giant's upper atmosphere and sheds velocity via aerobraking.

Large stable moons probably would have formed in the gas giant's neighorhood during accretion of the protoplanetary disk. See Wikipedia's article on the formation and origin of Jupiter's moons.

Answer 3

This scenario appears to cover most of the factors, though my reading of the Water Ice Lines and the Formation of Giant Moons Around Super-Jovian Planets paper you quoted seems to indicate that formation of moons such as this is not all that unlikely, just that it would be necessary for the super-jovian planet to migrate to a closer solar orbit with its moons, which would most likely be retained.

Tidal locking is a factor of distance and mass. The larger the mass and the distance, the less effect gravitic tides would have on the rotation of the moon, and hence the longer the period required for it to spin down to a synchronous rotational period. An earth-sized moon may still have a non-synchronous rotational period. You need to determine the period over which life evolved or arrived due to some sort of panspermia in order to determine the likelihood of tidal locking by the time your world becomes the focus of your story.

Certainly, you would get plate tectonics in such a system. Also, the super-jovian planet eclipsing the sun(s) would provide a major seasonal factor, probably greater than any circumstance of axial tilt or binary star position. However, since a super-jovian planet is likely to be a brown dwarf, this factor may not be as severe as the IR output of the brown dwarf may provide some minimal heating, preventing rapid or total freezing.

It is likely that such a world could be human-habitable, albeit with severe seasonal variations in climate related to the moon's position relative to the super-jovian planet and the primary stars.