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I'm trying to figure out the orbital characteristics of a double-planet system, kind of like Pluto and Charon, but much more equal in size.

  • One planet is pretty much a perfect copy of Earth and the other planet is 20% less massive with a radius 10% smaller.

  • Both planets are tidally locked to each other.

  • The planets' orbits around their center-of-mass is perfectly circular and the distance between them (center-to-center of each planet) is a constant 1,000,000 km.

What would be the orbital velocity of each planet around the COM? What is the orbital period? How far even is the COM?

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    $\begingroup$ Reference any college physics textbook. You'll find the relevant equation, then just do the math. $\endgroup$
    – jamesqf
    Oct 4 '20 at 3:31
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That is not stable over geological time. Systems like Pluto and Charon require one to be much more massive than the other so that their orbits don't intersect. If the masses are the same, it may take some millions of years but the planets will collide. That's because any slight perturbation in their orbit will cause one planet to go to a very slightly lower or higher orbit, changing their period, and then it is a matter of time until they catch up. Even their own geography could cause that perturbation.

Also notice that your planets are further apart from each other than Earth's SOI, which is roughly 0.929 million kilometers. With a lower mass than Earth, they might not even be able to orbit a common center of mass like that if they are in the goldilocks zone of a star like our Sun. They would just orbit their parent star.

If you still wish for us to calculate orbital parameters anyway, we need to know the mass of the parent star and these planet's aphelion and perihelion.

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    $\begingroup$ A large difference in mass is not required for stability; ignoring the hill sphere issues for the specified object, two objects close in mass can stably orbit each other for billions of years, orbiting their common center of mass. Examples include Alpha Centauri A & B, and the double asteroid 90 Antiope $\endgroup$
    – notovny
    Oct 4 '20 at 23:13
  • $\begingroup$ @notovny good points, but Alpha Centauri A is 25% more massive than its partner. As for Antiope, it could be a relatively recent breakup of a larger asteroid. I don't know of any stable examples involving planets nor stars with a size and mass similarity like that of Antiope. $\endgroup$ Oct 5 '20 at 1:53
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    $\begingroup$ 25% is a very small difference in mass on binary star scales. But iit's not difficult to find binary stars with similar masse. The two larger stars in EZ Aquarii are both 0.11 Solar Masses. The error bars on the masses of Wolf 424 A &B have a large overlap at 0.14-0.13 solar masses. 36 Ophiuchi A&B are both 0.85 solar masses. And that was just among the 100 nearest star systems. $\endgroup$
    – notovny
    Oct 5 '20 at 2:22
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You write:

One planet is pretty much a perfect copy of Earth and the other planet is 20% less massive with a radius 10% smaller.

So the smaller planet has 0.90 of the radius of Earth and 0.80 of the mass of Earth.

A planet with 0.9 the radius of Earth should have a volume of 0.729 the volume of Earth. If its average density was equal to that of Earth, it would have a mass of 0.729 Earth.

A planet with 0.8 the mass of Earth should have a volume of 0.8 Earth if it has the same denisty as Earth. My rough calculations indicate that a planet with a radius of 0.925 that of Earth would have a volume of 0.791453 Earth, and thus a mass of 0.791453 Earth if it has the same density as Earth.

For two planets with the same ratio of elements in their composition, the more massive planet will have a larger overall density, because its greater gravity will compress materials more.

You want a planet less massive than Earth to be a bit more dense than Earth. That can be done by increasing the proportion of denser, heavier, elements in the composition of the planet. But is the extra proportion of denser elments necessary for that overall density plausible? I don't know, I am not an expert on planetary formation.

So if the two planets formed at the same distance from their star, why would the larger planet, with almost identical mass and radius as Earth, have a lower density than the smaller planet?

Possibly the two planets formed at different distances from their star, and early processes of planetary orbital migration caused their orbits to approach and they eventually captured each other and became a double planet. Of course that seems to be a statisically very improbable event, so maybe you should adjust your figures so that the smaller planet has a similar but lesser density than the larger planet.

As I remember, Habitable Planets for Man, stephen H. Dole, 1964, has a table and possibly a formula for calculated the radius and density of an Earth like planet of a specified mass.

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

And it gives a formula for the average density of a rocky planet calculated from its average surface density and its radius. And of course multiplying the average density of a planet by its volume (calculated from its radius) will give its mass. And there is also a figure on page 30s howing the relationship between the radius of a planet relative to Earth and the planet's average density.

Of course today there are now more precise values known for the masses of Venus and Mercury, and the Dwarf planet Ceres. There are also rather precise values known for the masses, radii, and average densities of the larger moons of Jupiter, saturn, Uranus, and Neptune, but those bodies are likely to be partially made of ice, and so should have much lower densities than the the Earth-like objects in the solar system.

The separation of the two planets at 1,000,000 kilometers.

The Hill sphere or Roche sphere of an astronomical body is the region in which it dominates the attraction of satellites. The outer shell of that region constitutes a zero-velocity surface. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own. Any object within that distance would tend to become a satellite of the moon, rather than of the planet itself. One simple view of the extent of the Solar System is the Hill sphere of the Sun with respect to local stars and the galactic nucleus.1

In the Earth-Sun example, the Earth (5.97×1024 kg) orbits the Sun (1.99×1030 kg) at a distance of 149.6 million km, or one astronomical unit (AU). The Hill sphere for Earth thus extends out to about 1.5 million km (0.01 AU). The Moon's orbit, at a distance of 0.384 million km from Earth, is comfortably within the gravitational sphere of influence of Earth and it is therefore not at risk of being pulled into an independent orbit around the Sun. All stable satellites of the Earth (those within the Earth's Hill sphere) must have an orbital period shorter than seven months.

https://en.wikipedia.org/wiki/Hill_sphere[2]

The Hill sphere is only an approximation, and other forces (such as radiation pressure or the Yarkovsky effect) can eventually perturb an object out of the sphere. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius. The region of stability for retrograde orbits at a large distance from the primary is larger than the region for prograde orbits at a large distance from the primary. This was thought to explain the preponderance of retrograde moons around Jupiter; however, Saturn has a more even mix of retrograde/prograde moons so the reasons are more complicated.3

https://en.wikipedia.org/wiki/Hill_sphere#True_region_of_stability[3]

So the Earth's true region of stability for satellite orbits only extends to about 500,000 or 750,000 kilometers. Thus it seems unlikely that two Earth like planets with a total mass of less than two times that of the Earth could have stable orbits around each other at a distance of 1,000,000 kilometers.

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