tokens: external stream, they nevertheless dominate the broad-band opacity at FUV, optical, and IR wavelengths if they are present. They also dominate at EUV wavelengths (), if the hydrogen neutral fraction is less than ≈0.001. The strong coupling assumption means that all the radiative forces applied to the dust grains are directly felt by the gas also. 2.1 Bow shells supported by radiation and wind The internal pressure is the sum of wind ram pressure and the effective radiation pressure that acts on the bow shell. The radiative momentum loss rate of the star is L / c and the wind momentum loss rate can be expressed as where η w is the momentum efficiency of the wind, which is 1 in all cases except WR stars (Lamers Cassinelli 1999 ). If the optical depth at UV/optical wavelengths is sufficiently large, then all of the stellar radiative momentum, emitted with rate L / c , is trapped by the bow shell. At the infrared wavelengths where the absorbed radiation is re-emitted, the grain opacity is much lower, so the bow shell is optically thin to this radiation. Combined with the fact that the UV single-scattering albedo is only ≈0.5, this means that the single-scattering limit is approximately valid, and we adopt it here. We define a fiducial bow shock radius R * in the optically thick, radiation-only limit by balancing stellar radiation pressure against stream ram pressure at the bow apex along the symmetry axis from the star: which yields We now consider the opposite, optically thin limit. If the total opacity (gas plus dust) per total mass (gas plus dust) is κ (with units of ), then the radiative acceleration is Therefore, an incoming stream with initial velocity, v ∞ , can be brought to rest by radiation alone at a distance R ** where yielding On the other hand, we can also argue as in the optically thick case above by approximating the bow shell as a surface, and balancing stellar radiation pressure against the ram pressure of the incoming stream. The important difference when the shell is not optically thick is that only a fraction 1 − e −τ of the radiative momentum is absorbed by the bow, so that equation ( 2 ) is replaced with In the optically thin limit, 1 − e −τ ≈ τ, so these two descriptions can be seen to agree ( R 0 → R ** ) so long as which we will assume to hold generally. We define a fiducial optical depth, With these fiducial values established, we return to the combined radiation plus wind scenario. Adding the stellar wind ram pressure term from equation ( 1 ) allows us to write the general bow shell radius in terms of the fiducial radius as where x is the solution of Since this is a transcendental equation, x must be found numerically, but we can write explicit expressions for three limiting cases: The first case, x ≈ (1 + η w ) 1/2 , corresponds to a radiation bow shock (RBS); the second case, x ≈ 2τ * , corresponds to a radiation bow wave (RBW); and the third case, , corresponds to a wind bow shock (WBS). The two bow shock cases are
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