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I'm a worldbuilder and my setting has a lot of brown dwarfs.

While I'm trying to keep my setting as science-compliant as possible, I can't seem to find any resources for brown dwarfs like there are for main sequence stars, just basic encyclopedia pages describing what brown dwarfs are.

I do know that brown dwarfs undergo deuterium fusion for a short period of time after they form, cf. Wikipedia's Brown dwarf; deuterium fusion.

Does that process lead to a mass-radius-luminosity relation like hydrogen fusion does in main sequence stars?

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Yes, there are monotonic relationships between mass and luminosity and radius on the "deuterium burning main sequence".

Deuterium "burning" begins when the core temperature exceeds just over $10^6$ K. This happens very early in the life of a contracting protostar and, because it is fully convective and thoroughly mixed, all the D is "burned" in less than a couple of million years. The process takes much longer (5-50 million years) in lower mass brown dwarfs. Thus only systems younger than 50 Myr could have a D-burning brown dwarf. Below about 13 Jupiter masses, the brown dwarf core never gets hot enough to ignite D.

D burning mimics hydrogen burning in that it is possible to stabilise the contraction of a protostar if the D burning rate can match that which would be released by gravitational contraction. The burning process then acts as a thermostat, keeping the core at roughly constant temperature, and the protostar/brown dwarf at constant luminosity, until all the D has been depleted. The big difference between D and H burning though, is that the initial D/H ratio is something like $2\times 10^{-4}$, so D burning does not last long.

The rate of gravitational contraction of a protostar/brown dwarf increases with mass. Hence the rate of D burning, and hence the luminosity, must increase with mass. The virial theorem used with an ideal gas approximation then tells us that since the central temperature is roughly proportional to mass/radius, and since the D burning reaction rate is very temperature sensitive ($\propto T^{12}$), the D burns at almost the same temperature, regardless of mass, and so the radius during D burning will be roughly proportional to the mass. This is only rough because the cores of such objects are close to degenerate and ideal gas approximations stop working.

The plot below (from [Tout et al. 1999]) 1) shows the H-burning (lower) and D-burning (upper) "main sequences" for stars as dotted lines in the Hertzsprung- Russell diagram. You can use this plot to estimate the luminosity and radius for a star of a given mass on the D burning sequence. Note the isochrones (line of constant age, running across the diagram, labelled in Myr). There is no convenient formulae that I am aware of.

H and D burning main sequences from Tout et al. 1999

For brown dwarfs I can't find anything similar. Below are cooling curves from Burrows et al. (1997), showing how luminosity changes with time. D-burning manifests itself as a plateau at a luminosity that increases with increasing mass, and with a duration that increases with decreasing mass. The brown dwarfs are the green curves.

Burrows et al. 1997 cooling curves

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  • $\begingroup$ And I suppose metallicity has an impact on the rate too. $\endgroup$ – PM 2Ring Jun 12 at 7:02
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    $\begingroup$ @Pm2Ring minor for fully convective objects. $\endgroup$ – Rob Jeffries Jun 12 at 7:16

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