The TROJAN Test

0:01 Most moons in the solar system are thousands of  times lighter than the planets they orbit – but  

0:05 The moon is only 80 times lighter than earth.  So how do we know the moon is actually a moon,  

0:10 and not say – a binary planet together with Earth?  What about Pluto and its moon, Charon? For that  

0:14 matter, how do we know that Jupiter actually  orbits the sun, rather than Jupiter and the sun  

0:14 together being a binary system? And what about the  planet TOI-2379b, which is 5 times heavier than  

0:14 Jupiter and whose star is smaller than our sun? Questions like these are best answered with the  

0:15 Trojan test, which I suspect you haven't heard of  before, in part because I only just now gave it  

0:15 that name, but also because it’s not usually  mentioned when defining moons – normally, we  

0:15 say “the earth and moon orbit around their  common center of mass, called their barycenter,  

0:19 and the earth is sufficiently heavier than  the moon so the center of mass is actually  

0:22 inside the earth, which means we can say it’s  the moon that orbits the earth (rather than  

0:26 the two orbiting each other). Done.” But this  barycenter criteria has two major problems. 

0:31 The first problem is that (perhaps surprisingly)  the barycenter criteria doesn’t actually tell us  

0:35 anything about the movement of the objects. On  one extreme, take two objects with the same mass  

0:39 on opposite sides of an identical orbit around  their center of mass – this is the quintessential  

0:43 definition of a binary system. But, if one  of the objects had a low enough density,  

0:47 its radius could be big enough that the  center of mass would be inside that object,  

0:51 and suddenly our quintessential binary system  becomes – according to the barycenter test – a  

0:55 satellite/planet system. And at the other  extreme… A star could be a million times  

0:59 more massive than an orbiting planet, and  so the planet’s orbit around the barycenter  

1:02 would be a million times further out than  the star’s – the polar opposite of a binary  

1:06 system. But in spite of all that, if the orbit  was ten million times the radius of the star,  

1:10 then the barycenter would be located at ten times  the radius of the star… which is outside the star,  

1:14 and the system is then – according to the  barycenter test – a binary system. The problem is,  

1:21 the barycenter criteria isn’t testing the right  thing: binary-looking orbits and satellite-looking  

1:24 orbits can both get labeled as either binaries  or satellites. What's more, objects with  

1:28 elliptical orbits move closer or farther from  the barycenter throughout their orbit, and so  

1:32 can have the barycenter move from outside them to  inside to outside to inside over and over again. 

1:36 The second problem with the “barycenter being  inside one object” criteria is that it’s an  

1:40 intellectual threshold, not a physical one  with physical consequences. There’s nothing  

1:44 different that happens when the center of mass  of a system moves outside of an object’s radius.  

1:48 The barycenter of the solar system regularly  moves in and out of the sun without any effect  

1:52 whatsoever. This contrasts with other physically  meaningful thresholds, such as the definition of  

1:55 a star (where below a certain mass you can’t  do fusion and above it you can), or how an  

1:59 elliptical orbit (where the object returns again  and again) becomes – once the eccentricity goes  

2:03 above 1 – a hyperbolic trajectory where the  object will escape, passing by only once. So,  

2:07 when it comes to binaries, a much more  natural and physically meaningful cutoff  

2:11 for being able to say one object orbits another  is – you guessed it – the Trojan Test – which  

2:15 is a name I’ve just now given to something  normally called the “L4/L5 instability.”  

2:19 If you have two planets, or stars, or whatever,  orbiting each other, there are always – no matter  

2:23 the relative masses or distances of the objects –  there are always five points where the combination  

2:27 of their gravitational attractions together  with the centrifugal completely cancel out,  

2:30 and so an asteroid or spacecraft at one of the  points can in principle orbit “along with” the  

2:34 smaller planet or star. These points are called  “Lagrange Points” and are labeled L1 through L5.  

2:39 But it turns out that only L4 and L5 are what’s  called “stable” – if you park your spacecraft  

2:43 near points 1 through 3, it will slowly but  eventually drift off and stop orbiting along with  

2:47 the smaller planet or star, or get ejected from  the system entirely. L4 and L5, on the other hand,  

2:52 are stable [Due to coriolis force being dominant  over the gravitational forces], so your spacecraft  

2:53 (or space rock) can be parked there indefinitely,  orbiting in tandem ahead of or behind you. 

2:58 For example, Jupiter has tons of asteroids  orbiting in the vicinity of its L4 and L5  

3:02 points relative to the sun, called the “Trojan  asteroids” because the first ones discovered  

3:06 were named after figures from the Trojan war.  And Earth has a few small asteroids orbiting  

3:10 along with us at our L4 and L5 points relative to  the Sun, which we also call Trojan asteroids in  

3:14 imitation of the ones in Jupiter’s orbit. A couple  of Saturn’s moons even have smaller moons orbiting  

3:19 at their Trojan points relative to Saturn! But there’s a less well-known property of  

3:23 the Trojan points L4 and L5: they’re not always  stable –  their stability requires the bigger  

3:28 object to be more than 25 times the mass of the  smaller one. If the two objects are too close in  

3:32 size – too close to being a binary system – then  the L4 and L5 points become unstable like L1,  

3:38 L2 & L3. Though unstable doesn’t mean un-useful  – the “unstable” L1 and L2 lagrange points  

3:43 around Earth are regularly used for positioning  spacecraft. The spacecraft do gradually drift away  

3:48 from the L1 & L2 lagrange points, but only enough  to need a small correction from a rocket thruster  

3:51 every few months. Asteroids don’t have thrusters,  so they can only collect at the L4 and L5 points,  

3:56 and only, as we’ve mentioned, if the bigger  object is more than 25 times the mass of the  

3:59 smaller one. In short, you can only have Trojan  asteroids or moons (or park your spacecraft  

4:04 with the engine off) if you’re at least 25  times less massive than the thing you’re  

4:08 orbiting. This is the trojan test: if you can in  principle have Trojan asteroids (regardless of  

4:12 whether or not you actually do) then you’re  a little thing orbiting a big thing. If the  

4:16 big object is less than 25 times more massive  than you, then you can’t have Trojan asteroids  

4:19 and you should be considered a binary system. Case in point: both Jupiter and the Earth can  

4:24 in principle have Trojan asteroids (and both do)  – so by the Trojan test, they're orbiting the Sun  

4:28 (unlike the faulty barycenter test that thinks  Jupiter and the sun might be binary companions  

4:32 because their shared barycenter is outside of  the sun). In contrast, Pluto is only around eight  

4:36 times more massive than Charon, so Charon can’t  have Trojan asteroids, and therefore (according  

4:40 to the Trojan test) the two are a binary planet. The great thing about the Trojan test is that  

4:44 it’s purely about the relative masses of the two  objects. It doesn’t matter how dense they are,  

4:48 or how far apart you put them: Earth, for  example, would still be able to have trojan  

4:51 asteroids regardless of how far you moved  it away from the sun (in contrast with  

4:54 the barycenter center test where if you move the  Earth far enough away from the Sun, like really,  

4:57 really far away, eventually the barycenter of the  Earth-Sun system will move outside of the Sun’s  

5:00 radius). Another great thing is that the Trojan  test is a cutoff with an actual physical effect:  

5:04 on one side of the cutoff, an object can have  Trojan asteroids orbiting along with it. On the  

5:08 other side of the cutoff, Trojans are impossible.  The Trojan test is so good it has been proposed to  

5:19 be used as part of a criteria to determine if  an exoplanet is really orbiting a star instead  

5:22 of being a binary companion. But I think that we  should use it even more broadly! In my mind, if  

5:26 you have ANY two objects gravitationally orbiting  each other, anywhere in the universe, regardless  

5:30 of what they are, we should use the Trojan Test to  distinguish between the two possible situations:  

5:34 either one of the objects is less than a 25th  the mass and is actually orbiting the other  

5:38 (like a moon around a planet, or primary planet  around a star), or they’re closer in mass and  

5:43 therefore orbiting each other as a binary pair. And what does the Trojan Test say about our moon?  

5:48 Well, the earth is about 80 times heavier than  the moon, well above the Trojan test cutoff,  

5:52 which means the moon can in principle have  trojan asteroids… (even though we haven’t  

5:55 discovered any yet). Therefore, according to  the Trojan test the moon is indeed orbiting the  

5:59 earth – that is, the moon is a moon. Hello, I’m Josh from MinutePhysics,  

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