An anti matter planet behaving like a star

The scenario you describe - accreting matter being expelled by radiation pressure - will occur if the object exceeds the Eddington luminosity, a limit derived from hydrostatic equilibrium based on the balance between gravity and radiation pressure. The Eddington luminosity $L_{\mathrm{Edd}}$ is proportional to the mass of the accreting object $M$. If we assume that the mass of the planet is, say, roughly that of Jupiter, we find a maximum luminosity of $L_{\mathrm{Edd}}\sim126L_{\odot}$ - pretty significant!

Let's make our model a little more detailed. Let's assume the planet is made of pure antihydrogen, and that the outer layers are completely neutral. (The validity of the latter assumption is a point of contention in my view; low surface temperatures lead to higher accretion rates and a temperature increase, while high surface temperatures lead to lower accretion rates and a temperature decrease). We can also assume that the planet is embedded in a cloud of neutral hydrogen. If the energy generation of the planet at high energies is only due to matter-antimatter annihilation, then the luminosity should be $L=2\epsilon\dot{M}c^2$, where the $\epsilon$ describes the fraction of energy radiated away, and the factor of two arises from the fact that part of the planet is annihilated, too. Let's be conservative and say $\epsilon=1$. We then find that if the planet is radiating at the Eddington luminosity, the mass accretion rate is $\dot{M}=L_{\mathrm{Edd}}/(2c^2)\approx6.5\times10^{13}\text{ g s}^{-1}$.

Is the planet likely to be accreting at this rate? Let's assume the gas cloud is part of a cool, high-density portion of the interstellar medium - say a number density of $n\sim10^4\text{ cm}^{-3}$ and $T\sim10^2\text{ K}$. Assuming the accretion is spherically symmetric and transonic, this leads to an accretion rate of $\dot{M}_t\approx1.8\times10^{13}\text{ g s}^{-1}$, which is sub-Eddington. This would produce $L=3.2L_{\odot}$, which is fairly significant! However, I would guess that the true accretion rate (and therefore the true luminosity) would be lower, as the high-energy emission would quickly ionize and heat up nearby atoms, which in turn would lower the accretion rate until an equilibrium is reached.

Therefore, it seems likely that the planet's luminosity would be decidedly sub-Eddington, and the accretion would proceed roughly in equilibrium. Rather than pulsations, there would likely be steady emission (relative to the dynamics suggested in the original question).

Some notes:

• If accretion proceeded at the rate given by the Eddington limit, the planet would be destroyed on a timescale $\tau\sim M_J/\dot{M}\sim10^9$ years. The true lifetime will be higher given that the accretion rate is inversely dependent on mass.
• I'm curious about the ionization balance surrounding the planet. I believe $T\sim10^4\text{ K}$ would be required for us to assume essentially full ionization, and I would be surprised if the equilibrium surface temperature ends up being that high. Presumably, it would be quite higher than the temperature of a typical giant planet! A luminosity of $L=3.2L_{\odot}$ actually produces surface temperatures of $T\sim10^4\text{ K}$.
• The photons created from matter-antimatter annihilation should have energies of about $938\text{ GeV}$. That's more than enough energy to ionize a hydrogen atom; at the Eddington limit, this object will emit $3.2\times10^{38}$ of them per second. If we naively apply the Strömgren sphere model of HII regions, we find that ionization-recombination equilibrium requires that the planet be surrounded by a region of ionized hydrogen about $\sim1000\text{ AU}$ in radius. As the accretion should be sub-Eddington, the actual radius will be correspondingly lower.

posted over 4 years ago

HDE 226868‭

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