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Since cs is the speed at which pressure disturbances travel through the gas, it limits the rapidity with which the gas can respond to pressure changes. For example, if the pressure in one part of a region of the gas of characteristic size L is suddenly changed, the other parts of the region cannot respond to this change until a time of order L/cs , the sound crossing time, has elapsed. Conversely, if the pressure in one part of the region is changed on a timescale much longer than L/cs the gas has ample time to respond by sending sound signals throughout the region, so the pressure gradient will remain small.
35), using c2s = γP/ρ ∝ ργ−1 : (−v) = M˙ M˙ cs (∞) = 2 2 4πr ρ(r) 4πr ρ(∞) cs (r) 2/(γ−1) . 33) gives an algebraic relation for cs (r); the solution of this then gives ρ(r) and v(r). In practice, the algebraic equation for cs (r) has fractional exponents and must be solved numerically. 33). At large r the gravitational pull of the star is weak and all quantities have their ‘ambient’ values (ρ(∞), cs (∞), v ∼ = 0). As one moves to smaller r, the inﬂow velocity increases until (−v) reaches cs (∞), the sound speed at inﬁnity.
46), with σ assumed constant, we obtain ∇ ∧ (∇ ∧ B) = cµ0 4π σ(∇ ∧ E + [c]∇ ∧ (v ∧ B)/c). 4π c On the right hand side Faraday’s law, ∇ ∧ E = −[c] 1 ∂B , c ∂t can be used to eliminate the electric ﬁeld E. The left hand side is ∇ ∧ (∇ ∧ B) = −∇2 B − ∇(∇ · B) = −∇2 B; so we obtain 4π c2 2 ∂B = ∇ ∧ (v ∧ B) + 2 ∇ B. 47) Thus, the time rate of change of the magnetic ﬁeld is governed by diﬀusion of the ﬁeld (∝ σ −1 ∇2 B) and convection (∇∧(v∧B)) of the ﬁeld by the ﬂuid. e. for suﬃciently large σ. Rm is known as the magnetic Reynolds number.