Astrophysical Fluid Dynamics Solution Set 5 By Eric Bellm, Jeff Silverman, and Eugene Chiang

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1 Astophysical Fluid Dynamics Solution Set 5 By Eic Bellm, Jeff Silveman, and Eugene Chiang Readings: Shu pages 73 8; pages 4 3 of Couse Reade, photocopied fom Fank, King, and Raine Poblem. Thee Can Be Only One (Not Counting Sequels) Conside spheically symmetic in/out-flow nea an object of mass M. Assume the flow is isothemal at sound speed c s. Define the Mach numbe M = u/c s fo adial velocity u. (a) Show that ( ) M M M = GM c s () whee measues adial distance fom the object and G is the gavitational constant. In class, we deived fom continuity, momentum consevation, and P ρ γ that a steady spheical flow satisfies ( M ) u = GM ( ) (M c s) M = GM Fo an isothemal flow, c s is independent of. ( M ) c sm M = c s ( c s GM ) ( c s GM ) () (3) GM (4) ( ) M M M = GM c s (5) (b) Show that M ln M = s + ln s + C (6) whee s GM/c s gives the adius of the sonic point and C is a constant of integation. Equation (5) is sepaable, so we integate and obtain

2 Let C = C + ln s, whee s GM. Then c s ln M M = GM c s ln + C (7) M ln M = s + ln s + C (8) Plot seveal solutions of M vesus, dawing fom all mathematically possible families. Label the solutions by thei espective C values. Identify those subset of solutions that ae physically ealizable. Multiply though by and exponentiate. M em = exp{4 s + 4 ln + C} s (9) ( s ) 4 s exp{ 4 C} = M e M (0) Now, this has the fom x = W (x)e W (x). W (x) is a special function known as the Lambet W function; discussion of its application to this poblem can be found moe extensively in astoph/ The Lambet function has two solutions when x < 0 as it is hee: W 0 (x) and W (x). These ae implemented in Mathematica as PoductLog[k,x]. Theefoe, W 0 ( ( ) s 4 M = exp{ 4 s C}) W ( ( ) () s 4 exp{ 4 s C}) We obtain solutions by consideing combinations of these banches. Fo tans-sonic banches, M = at = s, which equies C = 3. Since the foce at s is continuous, the deivative of M should be continuous also. Plotting W 0 and W fo C = 3 gives a cusp at = s. Instead, we join the two functions to give two physical solutions: { W0, < M W s (), > s fo a wind, and M { W, < s W0, > s (3) fo Bondi accetion. Fo W 0, C > 3, we obtain subsonic beezes. Fo W, C > 3, we obtain unphysical flows which ae supesonic eveywhee. Fo C < 3, by joining W 0 and W we obtain unphysical, multivalued solutions which stat and end at = 0 o = without passing though s. See plots. Poblem. The Sola Wind

3 Figue : Plot of the solution families fo spheically symmetic flows. We have labelled the values of the constants, the banch of the Lambet W function, and the physical significance of the solutions. 3

4 At the base of the sola coona at R, outwad flow speeds ae still subsonic. Spectal line obsevations of the coonal base eveal tempeatues of about T 0 6 K, a fully ionized plasma, and a numbe density of potons of about 0 8 cm 3. Hint: You may use the esults of Poblem. (a) Estimate Ṁ fo the sola wind, assuming the wind is isothemal. Expess in M y. Fo a steady flow, Ṁ is constant. Continuity implies Ṁ = 4π ρu (4) Hee, Ṁ = 4π ρmc s (5) Ṁ = 4πR µ pn p M µ 0.6m p fo sola abundances. Assuming the wind is isothemal, we can find Mfom ou solution to b. s = GM ; c c s = cm/s, so s =.5 0 cm, and R / s = Solving s using the C = 3/ tans-sonic wind fom b, we obtain M (R = R ) = kt µ (6) Plugging it all in, Ṁ = 4. 0 g/s (7) Ṁ = M /yea (8) (b) Estimate the numbe density of sola wind potons flying past the Eath. Expess in cm 3. Ṁ is conseved fo a steady flow. Ṁ = 4π µn p ( )M(R = R )c s (9) Using the methods of b again, M(R = AU) = 5.98, so n p ( ) = 5 cm 3 (0) Poblem 3. Jet Collimation by Pessue Confinement fom an Extenal Medium Adapted fom Pingle and King, poblem 3.4. The poblem stats by consideing a de Laval nozzle, and ends by consideing how astophysical jets might behave like such nozzles with the tube of given coss-sectional aea eplaced by an extenal medium of a given pessue pofile. 4

5 Conside a γ-law gas moving inviscidly though a de Laval nozzle. The gas initially is vey subsonic, but eventually moves supesonically. Take the sound speed of the gas initially to be c s0. (a) Solve fo the gas velocity u and the sound speed c s at the pinched thoat of the nozzle, in tems of c s0 and γ. Fo a γ-law gas, dp/ρ = c s/(γ ). Benoulli gives: c s0 γ = u + c s γ Set u = c s at the pinched thoat and solve fo u = c s = c s0 /(γ + ). (b) Solve fo the gas velocity u at infinity, in tems of c s0 and γ. At infinity, the gas has zeo density, and since this is a γ-law gas, it must have zeo sound speed thee as well. So Benoulli gives: c s0 γ = u and we solve fo u = c s0 /(γ ). (c) Solve fo the pessue P at the pinched thoat, in tems of the initial P 0 and γ. Does the pessue decease monotonically eveywhee along the tube? Incease monotonically? Neithe? Justify you answe. P ρc s and P ρ γ which means ρ c /(γ ) s which means P c γ/(γ ) s. We saw in pat (a) that the atio of sound speeds at minus-infinity and at the pinched thoat is [/(γ + )] /. So the atio of pessues is P/P 0 = [(/(γ + )] γ/(γ ). Since γ >, this means the pessue at the thoat is less than the pessue at minus-infinity: the pessue deceases fom minus-infinity to the thoat. Past the thoat, the pessue continues to decease, because we know the density continues to decease (and P ρ γ ). So the pessue deceases monotonically down the tube. (d) Now conside an astophysical jet. How ae jets collimated? One poposed mechanism is pessue confinement by an extenal, static medium though which the jet tavels. Fo example, the jet could be a stella jet (poweed by accetion fom a disk), and the extenal medium could be the emnant gas cloud fom which the sta fomed. O the jet could be an AGN (supemassive black hole) jet (poweed by accetion fom a disk), and the extenal medium could be gas in the coe of the galaxy. Thee ae two ideas hee. The fist is that the extenal, static medium has a themal pessue that deceases monotonically in the jet diection. The second is that the jet and the extenal medium each pessue equilibium. That is, the themal pessue at any point within the jet equals the local themal pessue of the extenal suounding medium. Once in pessue equilibium, the jet stops expanding in the tansvese diection (some papes efe to the jet as being cocooned. ) These two ingedients, coupled with the usual assumption of steady inviscid flow, imply a vey close analogy 5

6 between de Laval nozzles and jets. Instead of being given the tube s coss-sectional aea A(x) and solving fo (say) P (x), we ae instead given P (x) and solve instead fo the jet coss-sectional aea A(x). Conside a steady jet composed of a gas having γ = 7/5. Fa along the jet, we ae so fa fom the cental engine (i.e., whateve is diving the jet) that we can neglect gavity. Moeove, the jet at this stage has achieved a constant supesonic velocity (analogous to pat b of this poblem). The jet buows into an extenal medium whose pessue scales with distance along the jet as x (like that of a singula isothemal sphee; o an isothemal wind). Unde these conditions, how does the opening angle of the jet (tansvese length divided by longitudinal length x) scale with x? Take the jet to have a cicula coss-section, and justify whateve assumptions you make. (The idea of jet as de Laval nozzle was intoduced by Blandfod & Rees 974 in the context of elativistic AGN jets. Thee also exists a Benoulli constant fo elativistic flows, but it looks diffeent fom ou non-elativistic fom u / + dp/ρ + Φ. Afte thinking about this poblem, you should find at least the fist few pages of thei pape to be faily eadable; you will also appeciate thei Figue.) Mass flux is conseved, so ρau = constant, whee A j is the coss-sectional aea of the jet. Now fa along the jet, u is constant (as can be seen fom Benoulli), so that means ρa = constant. Since inside the jet P inside ρ γ, outside the jet P outside x, and P inside = P outside by pessue equilibium, we must have ρ x /γ, which implies by mass flux consevation ρa x /γ j = constant, o j x /γ. The opening angle is j /x x ( γ)/γ x /7. It deceases with x, ego collimation. This poblem is eminiscent of wate unning out of a faucet. The steam is obseved to collimate! The explanation is given in Shu, pages In the faucet case, P is actually constant along the flow because the extenal atmospheic pessue hadly changes along the 0 cm of the steam. But because u(x) inceases along the steam because of gavity we find that A(x) shinks. 6