Habitable Planets in Unlikely Places

Every now and then, the researcher’s life throws up the odd cherry or two. Today was a case in question.

I’ve been interested in extrasolar planets for a long time. Like a lot of people, I’ve been particularly interested by the idea of potentially-inhabitable exoplanets. I actually do research on brown dwarfs, though, as they’re pretty much as interesting and they don’t involve staring at radial velocity periodograms (eurgh). Anyway, literally several years ago, I wondered in passing if it was possible to have an inhabitable planet around a brown dwarf. And these days, I realised, I have the technical knowledge to attempt an answer. Hurrar!

The short answer is ‘Yes’.

Some definitions and explanations
Before we dig into this any further, let me make clear a few things that underlie all of this maths.

1: These calculations do not indicate likelihood, merely possibility. In fact, I suspect that ‘Earthlike’ planets around brown dwarfs are extremely rare, if they exist at all. (Note though that this too is an assumption – I can’t actually prove that statement. My intuition could be wrong!)

2: The maths I did only describe one particular set of circumstances. Any others would require the calculations to be re-done using appropriate values.

3: I’m only considering life as we know it – i.e. a metabolism based on the Kreb’s Cycle, breathing oxygen, using water as a solvent and needing a more-or-less Earthlike temperature/pressure/gravity regime. Speculative biochemistries are interesting, but hard to factor in.

4: For this analysis, I’m defining an ‘Earthlike’ planet to be possible if it meets two criteria.
a) The planet orbits within the habitable zone, as based on the primary’s luminosity
b) That orbital radius is outside of the primary’s Roche limit.

If condition a) is violated, then the planet’s temperature is wrong for liquid water and thus life. If condition b) is violated, then the planet is so close to the primary that the primary’s tides will break it up. Obviously this isn’t good from a living-world point of view!

5: For simplicity’s sake, I will be assuming a 1 Earth-mass, 1 Earth-radius planet in each case.

6: I’m using the term ‘primary’ to refer to the thing the planet orbits, rather than ‘star’. The primaries here are brown dwarfs and as such, they’re not strictly speaking ‘stellar’. (I don’t want to create any more confusion than is absolutely necessary!)

The Objects
Brown dwarfs don’t fuse hydrogen. Instead, they shine by radiating away residual heat from their formation. In the memorable words of Adam Burrows, they ‘cool like a rock’. This means brown dwarfs evolve quite differently from stars. A star starts its life in a given spectral class and will basically stay there until it expands into a red giant some time later. But spectral class is ultimately dependent on surface temperature. As a brown dwarf ages and cools, it will evolve through different spectral classes.

Now, life on Earth definitely got started by 2.5 billion years ago. I think photosynthesis was up and going by then, too. So I decided to use this date as a limiting case – logic suggests that a planet can’t be meaningfully-Earthlike without photosynthesis. And it was cyanobacteria that started oxygenating the atmosphere about 2.4 bn years ago – so let’s take that as a cut-off date. (Okay, that’s strictly the Earth at +2.1 Gyr, but 2.5 is a tidier number … and remember, all these numbers are only estimates.)

So, we need to see if a 2.5 Gyr old planet can exist in the biozone around various brown dwarfs.

Roughly speaking, more massive brown dwarfs cool more slowly. Let’s consider a range of spectral types, from L0 to T7. (To be precise, L0, L2,L5, L7, T0, T2, T5 and T7, in order of warmest to coldest.) This way we’ve bracketed the full range. An L0 will have a surface temperature of something like 2300 K, a T0 will be ~ 1450 and a T7 will be something like 850 K.

Maths, maths, maths!
So, to the cooling-curve models. According to the Tucson group models, to be a T7 at ages 2.5 Gyr implies a mass of ~0.035 Suns. An L0, by contrast, would have 0.08 solar masses (at this level, depending on metallicity, the object in question may actually be an extremely small star rather than a brown dwarf).

The next thing we need is a luminosity. I just lifted these from Vrba et al (2004). Not the most recent of papers, but it will do for our purposes here! (Also, handily, they have a table with both T_eff and luminosity, so *exactly* what was needed.)

Here’s the full distribution of properties:

Type Mass (M_obj/M_Sun) T_eff/K Luminosity (L_obj/L_Sun)
L0 0.080 2300 3.02e-4
L2 0.078 2100 1.51e-4
L5 0.730 1600 6.03e-5
L7 0.072 1550 3.63e-5
T0 0.070 1450 2.82e-5
T2 0.065 1400 2.88e-5
T5 0.058 1200 1.51e-5
T7 0.035 850 5.50e-6

As you can see, T-dwarfs are rather faint! That T7 average on the end there is less than 6 parts in a million of our Sun’s luminosity. Any planet orbiting such a primary is going to have to be very close in indeed.

The next step is to calculate the radius of the habitable zone. A rigorous treatment would involve looking at the way a standard atmosphere would interact with differing stellar spectra, but that would be a serious modelling job. Instead, I just took the square root of the luminosity and assumed that the atmospheric interactions behave themselves. (This will do for an order-of-magnitude estimate.) Doing this would output the habitable-zone orbital radius in AU.

The next thing I did was calculate the Roche limit. To check compliance with criterion 4b), I divided the habitable-zone orbital radius by the Roche limit.

Also, out of curiosity, I calculated the angular size of the primary as it would appear in the sky of its planet.

Here are the results:

Type HZ radius/AU Roche limit/AU HZ / Roche Primary apparent size in HZ (Sun=1)
L0 0.0174 2.27e-4 76.5 5.9
L2 0.0123 2.29e-4 53.7 8.4
L5 0.0078 2.34e-4 33.1 13.3
L7 0.0060 2.35e-4 25.6 17.1
T0 0.0053 2.38e-4 22.3 19.4
T2 0.0054 2.44e-4 22.1 19.2
T5 0.0039 2.53e-4 15.4 26.5
T7 0.0023 2.99e-4 7.8 44.0 (!)

Discussion
Surprisingly, in every single case the planet is safe from tidal break-up. Even for the T7, the planet orbits at nearly 8 times the critical radius.

The stellar apparent sizes are much weirder, however. In the case of the T7, the view from the surface of its planet would seem bizarre. The ‘sun’ would look bigger than a dinner plate held at arm’s length – or more than 40 times the apparent width of our Sun.

Also, note the apparent reversal in the sizes between T0 and T2. This is an artifact of small-number statistics; Vrba et al’s T2 average luminosity is very slightly higher than their T0 luminosity. In fact, current estimates indicate that T2s are slightly fainter, as we’d expect. (In 2004 there were only around a dozen or so T2s known, and they were all biased toward brighter objects.)

Conclusion
Perhaps somewhat surprisingly, we’ve demonstrated that habitable planets around brown dwarfs are physically-possible. We’ve also shown that the surface environments would likely be very strange, and the sky would look nothing like ours.

Also, hand-coding tables in HTML sucks.

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