The secrets of how planets form

Astronomers who were measuring the length of one day on an exoplanet for the first time were in for a surprise: it was shorter than any planet’s in the Solar System. Beta Pictoris b, orbiting the star Beta Pictoris, has a sunrise every eight hours. On Jupiter, there’s one once every 10 hours; on Earth, every 24 hours.

This exoplanet is located 63.4 light-years from the Solar System. It is a gas giant, a planet made mostly of gases of light elements like hydrogen and helium, and more than 10 times heavier than Earth. In fact, Beta Pictoris b is about eight times as heavy as Jupiter. It was first discovered by the Very Large Telescope and the European Southern Observatory in 2003. Six years and more observations later, it was confirmed that it was orbiting the star Beta Pictoris instead of the star just happening to be there.

On April 30, a team of scientists from The Netherlands published a paper in Nature saying Beta Pictoris b was rotating at a rate faster than any planet in the Solar System does. At the equator, its equatorial rotation velocity is 25 km/s. Jupiter’s equatorial rotation velocity is almost only half of that, 13.3 km/s.

The scientists used the Doppler effect to measure this value. “When a planet rotates, part of the planet surface is coming towards us, and a part is moving away from us. This means that due to the Doppler effect, part of the spectrum is a little bit blueshifted, and part of it a little redshifted,” said Ignas Snellen, the lead author on the Nature paper and an astronomy professor at the University of Leiden.

So a very high-precision color spectrum of the planet will reveal the blue- and redshifting as a broadening of the spectral lines: instead of seeing thin lines, the scientists will have seen something like a smear. The extent of smearing will correspond to the rate at which the planet is rotating.

Bigger is faster

So much is news. What is more interesting is what the Leiden team’s detailed analysis tells us, or doesn’t, about planet formation. For starters, check out the chart below.

This chart shows us the relationship between a planet’s mass (X-axis) and its spin angular momentum (Y-axis), the momentum with which it spins on an axis. Clearly, the heavier a planet is, the faster it spins. Pluto and Charon, its moon, are the lightest of the lot and their spin rate is therefore the lowest. Jupiter, the heaviest planet in the Solar System, is the heaviest and its spin rate is also the highest. (Why are Mercury and Venus not on the line, and why have Pluto and Earth been clubbed with their moons? I’ll come to that later.)

“Apparently the more massive the planet, the more angular momentum it acquires,” Prof. Snellen said. This would put Beta Pictoris b farther along the line, possibly slightly beyond the boundaries of this chart – as this screenshot from the Leiden team’s pre-print paper shows.

Unfortunately, science doesn’t yet know why heavier planets spin faster, although there are some possible explanations. A planet forms from grains of dust floating around a star into a large, discernible mass (with many steps in between). This mass is rotating in order to conserve angular momentum. As it accrues more matter over time, it has to conserve the kinetic and potential energy of that matter as well, so its angular momentum increases.

There have been a few exceptions to this definition. Mercury and Venus, the planets closest to the Sun, will have been affected by the star’s gravitational pull and experienced a kind of dragging force on their rotation. This is why their spin-mass correlations don’t sit on the line plotted in the chart above.

However, this hypothesis hasn’t been verified yet. There is no fixed formula which, when plotted, would result in that line. This is why the plots shown above are considered empirical – experimental in nature. As astronomers measure the spin rates of more planets, heavy and light, they will be able to add more points on either side of the line and see how its shape changes.

At the same time, Beta Pictoris b is a young planet – about 20 million years old. Prof. Snellen used this bit of information to explain why it doesn’t sit so precisely on the line:

Sitting precisely on the line would be an equatorial velocity of around 50 km/s. But because of its youth, Prof. Snellen explained, this exoplanet is still giving off a lot of heat (“this is why we can observe it”) and cooling down. In the next hundreds of millions of years, it will become the size of Jupiter. If it conserves its angular momentum during this process, it will go about its life pirouetting at 50 km/s. This would mean a sunrise every 3 hours.

I think we can stop complaining about our days being too long.

Spin velocity v. Escape velocity

Should the empirical relationship hold true, it will mean that the heaviest planets – or the lightest stars – will be spinning at incredible rates. In fact, the correlation isn’t even linear: even the line in the first chart is straight, the axes are both logarithmic. It is a log-log plot where, like shown in the chart below, even though the lines are straight, equal lengths of the axis demarcate exponentially increasing values.

If the axes were not logarithmic, the line f(x) = x³ (red line) between 0.1 and 1 would look like this:

The equation of a line in a log-log plot is called a monomial, and goes like this: y = ax^k. In other words, y varies non-linearly with x, i.e. a planet’s spin-rate varies non-linearly with its mass. Say, if k = 5 and a (a scaling constant) = 1, then if x increases from 2 to 4, y will increase from 32 to 1,024!

Of course, a common, and often joked-about, assumption among physicists has been made: that the planet is a spherical object. In reality, the planet may not be perfectly spherical (have you known a perfectly spherical ball of gas?), but that’s okay. What’s important is that the monomial equation can be applied to a rotating planet.

Would this mean there might be planets out there rotating at hundreds of kilometres per second? Yes, if all that we’ve discussed until now holds.

… but no, if you discuss some more. Watch this video, then read the two points below it.

1. The motorcyclists are driving their bikes around an apparent centre. What keeps them from falling down to the bottom of the sphere is the centrifugal force, a rotating force that, the faster they go, pushes them harder against the sphere’s surface. In general, any rotating body experiences this force: something in the body’s inside will be fleeing its centre of rotation and toward the surface. And such a rotating body can be a planet, too.
2. Any planet – big or small – exerts some gravitational pull. If you jumped on Earth’s surface, you don’t shoot off into orbit. You return to land because Earth’s gravitational pull doesn’t let you go that easy. To escape once and for all, like rockets sometimes do, you need to jump up on the surface at a speed equal to the planet’s escape velocity. On Earth, that speed is 11.2 km/s. Anything moving up from Earth’s surface at this speed is destined for orbit.

Points 1 and 2 together, you realize that if a planet’s equatorial velocity is greater than its escape velocity, it’s going to break apart. This inequality puts a ceiling on how fast a planet can spin. But then, does it also place a ceiling on how big a planet can be? Prof. Snellen to the rescue:

Yes, and this is probably bringing us to the origin of this spin-mass relation. Planets cannot spin much faster than this relation predicts, otherwise they would spin faster than the escape velocity, and that would indeed break the planet apart. Apparently a proto-planet accretes near the maximum amount of gas such that it obtains a near-maximum spin-rate. If it accretes more, the growth in mass becomes very inefficient.

(Emphasis mine.)

Acting forces

The answer will also depend on the forces acting on the planet’s interior. To illustrate, consider the neutron star. These are the collapsed cores of stars that were once massive but are now dead. They are almost completely composed of neutrons (yes, the subatomic particles), are usually 10 km wide, and weigh 1.5-4 times the mass of our Sun. That implies an extremely high density – 1,000 litres of water will weigh 1 million trillion kg, while on Earth it weighs 1,000 kg.

Neutron stars spin extremely fast, more than 600 times per second. If we assume the diameter is 10 km, the circumference would be 10π = ~31 km. To get the equatorial velocity,

V_spin = circumference × frequency = 31 × 600/1 km/s = 18,600 km/s.

Is its escape velocity higher? Let’s find out.

V_e = (2GM/r)^0.5

G = 6.67×10-11 m³ kg^-1 s^-2

M = density × volume = 10¹⁸ × (4/3 × π × 125) = 5.2×10²⁰ kg

r = 5 km

∴ V_e = (2 × 6.67×10^-11 × 5.2×10²⁰/5)^0.5 =  ~37,400 km/s

So, if you wanted to launch a rocket from the surface of a neutron star and wanted it to escape the body’s gravitational pull, it has to take off at more than 30 times the speed of sound. However, you wouldn’t get this far. Water’s density should have given it away: any object would be crushed and ground up under the influence of the neutron star’s phenomenal gravity. Moreover, at the surface of a neutron star, the strong nuclear force is also at play, the force that keeps neutrons from disintegrating into smaller particles. This force is 10³² times stronger than gravity, and the equation for escape velocity does not account for it.

However, neutron stars are a unique class of objects – somewhere between a white dwarf and a black hole. Even their formation has nothing in common with a planet’s. On a ‘conventional’ planet, the dominant force will be the gravitational force. As a result, there could be a limit on how big planets can get before we’re talking about some other kinds of bodies.

This is actually the case in the screenshot from the Leiden team’s pre-print paper, which I’ll paste here once again.

See those circles toward the top-right corner? They represent brown dwarfs, which are gas giants that weigh 13-75 times as much as Jupiter. They are considered too light to sustain the fusion of hydrogen into helium, casting them into a limbo between stars and planets. As Prof. Snellen calls them, they are “failed stars”. In the chart, they occupy a smattering of space beyond Beta Pictoris b. Because of their size, the connection between them and other planets will be interesting, since they may have formed in a different way.

Disruption during formation is actually why Pluto-Charon and Earth-Moon were clubbed in the first chart as well. Some theories of the Moon’s formation suggest that a large body crashed into Earth while it was forming, knocking off chunks of rock that condensed into our satellite. For Pluto and Charon, the Kuiper Belt might’ve been involved. So these influences would have altered the planets’ spin dynamics, but for as long as we don’t know how these moons formed, we can’t be sure how or how much.

The answers to all these questions, then, is to keep extending the line. At the moment, the only planets for which the spin-rate can be measured are very massive gas giants. If this mass-spin relation is really universal, than one would expect them all to have high spin-rates. “That is something to investigate now, to see whether Beta Pictoris b is the first of a general trend or whether it is an outlier.”

Reference:

Fast spin of the young extrasolar planet β Pictoris b. Nature. doi:10.1038/nature13253