# Binary World Mechanics

I’m trying to develop a realistic system with a limited understanding of astrodynamics.

Is it possible to have two binary worlds capable of sustaining atmosphere and complex life orbiting one another at such close proximity that they dominate each other’s skyline with even city lights of the other world in the sky being clearly visible at night? Does the scale of the worlds or the distances involved mean that that can simply never exist?

• Define "binary" in this context. By most definitions, our Earth and Moon are a binary system; both planet-sized bodies that are co-orbiting the same star and have a barycenter. Since you want to contrast your system to our world, it seems you're using "binary" in a specific way, and narrowing that down will help people to answer this question. Nov 19, 2019 at 22:19
• @Ghedipunk everything that orbits has a barycentre and peturbs the other body to some degree. The Earth-Moon barycentre is well within Earth's radius. The Pluto-Charon system is much more interesting in that regard, but strangely my idea of calling the former setup a "boring binary" doesn't seem to have caught on. Nov 19, 2019 at 22:21
• It is generally considered to be OK to make sweeping changes to a question before anyone posts an answer. So if that's what you want to ask, make the change quickly ;-) Nov 19, 2019 at 22:36
• Sorry about that guys. I realised that what I was asking appeared simple on the surface but probably relied on lots of little factors that require an astrophysicist to describe. My attempt to shorthand it all while being excited about finally finding a site where I can ask questions did not go well. Nov 19, 2019 at 22:55
• If the Earth and Moon were any closer to each other, the Earth would be tidally locked, too. The Earth originally had 9 hour days, and it's due to the effect of tides slowing our planet down that we have 24 hour days now... and we are still slowing down, by about a second every few years. (Tidal forces may not be the most immediate effects, but it is the most persistent effect: en.wikipedia.org/wiki/Leap_second ) Nov 20, 2019 at 0:27

# Hmm, it might be possible...but unlikely.

To respond to the clarified question (can be found in the comments), let's assume that by "take up the sky" you mean that each planet has almost a 90 degree angular diameter when viewed from the other planet. This will take up 50% of the night's sky. (Getting 100% is impossible due to the curvature of a spherical object)

I have calculated that the distance necessary between two Earth-sized objects for the above scenario is about 15380 km. This leaves only 2638 km of distance between the outer surfaces of each planet! Uh-oh! That might be too close for each planet to be habitable!

With each planet taking up 25% of the sky, this distance increases to 23019 km, leaving around 10277 km of space between the surfaces of the planets. This is still extremely close.

However, you'd be surprised at how big objects will look, even if they do not take up most of the visable sky. For example, with an angular diameter of 5 (about 2.7% of the sky), the planets would appear to be about 10 times bigger in apparent size than the real-life full moon. The distance required for this sight is around 152429 km (about half the Earth-moon distance), which is a significantly greater distance and would almost certainly be habitable.

If you want to calculate the distance necessary for two Earth-sized objects, use the following formula, which I derived from the formula for angular diameter: $$D =\frac{d}{2\;sin\left( \frac{\pi a}{360} \right)}$$ where $$D$$ is the distance between the two objects, $$d$$ is the actual diameter of the two objects (assuming they are identical), and $$a$$ is the desired angular diameter, measured in degrees.

The result (distance) of this formula will always be in the same units that you provided for the actual diameter.

Note: this part of my answer was written before the original poster clarified his question in the comments. It has been kept here because it offers Roche limit calculations:

### What you are looking for is the Roche limit of both planets in your system.

Let us assume you have 2 exact replicas of the Earth: For a rigid-satellite calculation, the Roche limit is determined by the ratio of the densities of the two objects, and is given by the following formula: $$d=R_M \sqrt[3]{2\frac{p_M}{p_m}}$$ where $$R_M$$ is the radius of the primary (larger/more massive) body, $$p_m$$ is the density of the primary body, and $$p_M$$ is the density of the satellite.

Since $$p_M$$ and $$p_m$$ are the same, they cancel out and become $$1$$. The simplified equation then becomes:
$$d = R_M \sqrt[3]{2}$$

According to this simplified equation (it does not consider inertia force and rigid structure), there are not really limitations on the distance. This doesn't seem correct to me, but it might be due to the fact they are of comparable sizes so the force of one body does not dominate the other.

• Geostationary orbit of the pair each having exactly 1 Earth Mass each puts them at an apparent diameter of about 10 degrees, and since they're tidally locked (how can't they be?) their orbital period will be 1 sidereal day, which makes a solar day 24 hours. (Radius of orbit: 42,164 km. Distance between each planet's surface and its twin's center: 35,786 km) Nov 20, 2019 at 0:20
• @Ghedipunk Great calculations! I love number crunching. An angular diameter of around 10 degrees would be about 20x larger in apparent size than the full moon! You'd definitely be able to see the city lights! Nov 20, 2019 at 1:12

The answer to your question is yes it is possible. The roche limit for 2 equally sized planets is very close. Close enough to allow a shared atmosphere even. Although that is another question and there are other issues, suffice to say that two planets could orbit each other in very close proximity.

However such a situation would not be without consequences. For starters it would be unstable as slight orbital perturbations would in all likelihood lead to catastrophe within a few hundred or a few thousand years depending on circumstances.

In addition planets orbiting in such close proximity would generate monstrous tides that would rise and fall by hundreds if not thousands of metres twice a day. Such a situation would also destabilise the orbits.

So in summary the worlds you imagine can exist but would be precarious and liable to destruction within a relatively short period. Life on both planets would likely share a common origin due to the close proximity and the ease of transferring material between the two via large meteorite strikes on either planet.

If you push the planets slightly further appart their life span would increase dranatically.

• I WANT TO THANK ALL OF YOU FOR TAKING THE TIME TO ANSWER MY QUESTION AND PROVIDING ME WITH MATHMATICAL FORMULAS THAT DEMONSTRATE THE LIMITS OF REALISM IN THIS SCENARIO. Nov 20, 2019 at 11:29