Version 4 (modified by 6 years ago) ( diff ) | ,
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Blowoff Threshold
Assumptions for below: photon momentum is directed purely radially. Full extent of torus is optically thick (all photons are absorbed). Torus moves uniformly.
The initial gravitational binding energy of the gas torus is given by
, with the initial average orbital radius (major radius of the torus) and the mass of gas contained in a torus that extends for an angle around the star with minor radius .
The area presented to Lyman-alpha radiation is essentially the rectangle of height
and width , so that.
For some major radius
, the flux incident on the torus is
(and area is
).At this radius, the power provided by the photons is
. (This part is potentially bothersome - the photons are absorbed as momentum, not energy.)
We can calculate the blowoff timescale by determining the amount of time required for the photons to deposit the binding energy of the torus:
We can also calculate the time required for the torus to be fully replenished:
Taking the ratio of these timescales:
, where is the change in major radius we consider sufficient for the material to be "blown off." If , the torus will remain essentially unaffected. If , the torus will be completely blown away. The timescales are equal for a mass loss rate of 3x10^{10} g/s at a flux of ~8.5x10^{12} phot/cm^{2}/s.
Including Ablation
As a first attempt, I've taken the energy required to unbind a thick parabola (seen in the steady state at the end of this movie) compared to the time required to refill it. This needs to be modified still to account for the fact that the material being pushed around the torus pulls new material in behind it, increasing the fill rate compared to the mass loss rate of the planet alone.
The edge of the torus is approximately
from the planet, so the area exposed to Lyman- radiation is,
where the term in square brackets is the arc length of the outside edge of the thick parabola (with a_{1} the half-width (approximately from planet to edge in -z direction) and h_{1} the height (approximately from planet to edge in -x direction). Additionally, we can calculate the volume of the thick portion:
,
where the term in square brackets here is the difference in area between the outer and inner bounding parabolas.
We can calculate the power delivered by the Lyman-
photons as before.
And the ablation timescale is then
s
And the timescale for replenishing (taking into account only the mass loss rate of the planet) is the same:
s
Both of these timescales are linearly related to
, so an increase in density would not explain the factor of 50 difference (though we don't see an increase in density, in any case). However, material appears to accelerate to the outer edge as it reaches steady state. It seems as though the ablated material is pulling new material in to fill the vacuum left by it, which could set up a steady state (although it seems like this would require extra material from somewhere?).Lyman-
Line TransferThis is basically identical to the line transfer for ionizing radiation, except:
and the energy per photon is instead the momentum per photon,
.
To test, I've put neutral hydrogen of uniform density and temperature 10^{4} K in a uniform gravitational field opposed by radiation pressure. To balance, equate volumetric force:
with
. Clearly the value of n_{H} doesn't affect the result, and I've chosen a small enough length scale that absorption doesn't matter and we can take across the grid.With a flux
, we get a result of (tweaking some decimals - flux in physics.data for this is actually ):after ~0.1 seconds, when acceleration under only gravity would give . Radiation pressure and gravity are clearly not perfectly matched, but it also seems clear that radiation pressure is working as it should.
Dynamic tests
Lyman- | flux5.1d13 phot/cm^{2}/s |
lScale | 1 |
Slab
Slab density | 1 CD |
Ambient density | 1d-4 CD |
Slab extent | (0,0.3) |
Domain | {(0,1),(0,1)} |
Clump
Clump density | 1 CD |
Ambient density | 1d-4 CD |
Clump extent | Circle around (0,0) with R = 0.5 |
Domain | {(-1,1),(-1,1)} |
Slab with ionization
Slab density | CD | |
Ionizing flux | 2d8 | ←- Need to test value on smaller length scale |