Changes between Version 12 and Version 13 of u/erica/d


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Timestamp:
12/14/15 14:11:57 (9 years ago)
Author:
Erica Kaminski
Comment:

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  • u/erica/d

    v12 v13  
    44444. there is no field within the cylinder of the colliding flows themselves (which is a good approximation given how dynamically weak the field is there)
    45455. to track the ring we define 2 radii, the outer radius of the ring [[latex($r_o$)]], and the inner radius of the ring, [[latex($r_i$)]] (both relative to the center of the collision region)
    46 6. this ring contains the same amount of flux as it spreads out, by flux freezing. this means that as the ring grows the field strength is geometrically diluted
     466. this ring contains the same amount of flux as it spreads out, by flux freezing. this means that as the ring grows the field strength is geometrically diluted.
    47477. as far as the ram pressure pushing this ring out, we assume the velocity of the ejecta is a constant with radius, and only the density is decreasing by geometrical dilution.
    48488. we take the ram pressure of the ejecta to be the incoming ram pressure. this is a reasonable approximation as the shocked sound speed should be of order the incoming velocity.
    49 9. we consider the ring to be formed from a spherical distribution of field lines centered on the collision region. thus, the radius of curvature is just r, and the thermal pressure gradient is along r, but perpendicular to the field directly above the collision region.
     499. we consider the ring to be formed from a spherical distribution of field lines centered on the collision region. thus, the radius of curvature is just r, and the magnetic pressure gradient is along r, but perpendicular to the field directly above the collision region.
     5010.  we ignore gravity out in the ambient.
    5051
    5152This gives the following momentum equation for the ring (i.e. for [[latex($r_i < r < r_o$)]]):
     
    6162[[latex($B^2 = B_0^2(\frac{r_0}{r})^2$)]]
    6263
    63 where the field is the initial value in the un-perturbed medium beyond the ring's outer radius !r0.
     64where the field is the initial value in the unperturbed medium beyond the ring's outer radius !r0.
     65
     66More fully then, the total solution for B^2^(r) in this model is:
     67
     68[[latex($B^2 = B_0^2(\frac{r_0}{r})^2 ~for ~r_i<r<r_0$)]][[br]]
     69[[latex($B^2 = B_0^2 ~for ~r>r_0$)]][[br]]
     70[[latex($B^2 = 0 ~for ~r<r_i$)]]
     71
     72Thus, we envision a ring of flux surrounding the colliding flows cylinder R that is carried outward away from the cylinder boundary. It comes to some steady state in the ambient by developing a magnetic pressure gradient that counterbalances magnetic tension. At the rings outer boundary, the magnetic pressure is balanced with the initial unperturbed magnetic pressure. At the rings inner boundary, the magnetic pressure is balanced with ram pressure from the colliding flows.
     73