1. Define the following: molecular cloud, molecular core, protostar. Include typical properties when necessary.

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1 1 Soluions o PH6820 Miderm 1. Define he following: molecular cloud, molecular core, proosar. Include ypical properies when necessary. A molecular cloud is a disinc, self-graviaing cloud comprised primarily of H 2 and He. The sizes range from 10 M (for a dark globule) o 10 6 M for he mos massive gian molecular clouds. Typical kineic emperaures range from K. The average densiies are around cm 3. They ofen show filamenary and clumpy morphologies. A molecular core is a self graviaing globule wihin a molecular cloud. The ypical sizes (0.1 pc), masses (a few M ), emperaures (10 K) and subsonic non-hermal velociies sugges ha hese are hermally suppored, self graviaing eniies, ha will collapse o form a single sar or sellar sysem. The presence of infrared sources in half of all cores indicaes ha hey are ofen in a sae of collapse. A proosar is a sar undergoing rapid mass accreion. They are found in he middle of collapsing envelopes ha absorb heir luminosiy and reprocess i ino he infrared. 2. Imagine a proosar wih a consan infall rae given by he soluion for an infalling Singular Isohermal Sphere. a. Assuming he infall rae is equal o he accreion rae, how does he mass of he cenral sar change wih ime? Wrie he answer in erms of he sound speed. Soluion: The infall rae of a collapsing, hermally suppored, core is given by (approximaely!): Ṁ = c3 s G Thus, he oal mass accumulaed is simply he infall rae ime he ime over which infall has occurred: (1) M = c3 s G (2) where is he age of he proosars. I m assuming ha he iniial mass is negligible, bu here probably had o be some iniial mass. b. Assuming he cenral sar has a consan densiy, how does accreion luminosiy vary wih ime? Wrie he answer in erms of he sound speed, and he densiy of he sar. Soluion: The accreion luminosiy is given by:

2 2 L acc = GṀM R (3) Wha is he radius? If we assume a consan densiy, R = (3M/4πρ) 1/3. Thus, he luminosiy is L acc = G ( ) ( c 3 s c 3 s G G ) ( 4πGρ 3c 3 s ) 1/3 = (4π/3) 1/3 G 2/3 c 5 sρ 1/3 2/3 (4) c. Also assume ha he inrinsic luminosiy of he cenral proosar is given by he relaionship L = L (M/M ) 3.5? A wha mass does he inrinsic luminosiy of he source equal he accreion luminosiy? Adop a densiy of 3.5 gm cm 3. Also assume he gas emperaure is 20 K and µ = 2.7. Soluion: The sound speed is given by c s = kt/µm H = 0.25 km s 1. Wih a densiy of 3.5 gm cm 3, his gives us: while he inrinsic luminosiy is: hen L in = L acc means ( ) 2/3 L acc = 77 L 10 5 (5) yrs ( ) c ( ) 3.5 L in = s L = 0.02 L GM 10 5 (6) yrs ( yrs ) 3.5 = 77 ( ) 2/3 (7) 10 5 yrs which has he soluion of = 1.8 Myr. A his poin, he Mass is hen given by equaion 2. ( ) c 3 s ( G = cm s 1 ) Myr 10 6 yr π 10 7 s = 2 M cm 3 gm 1 s 2 (8)

3 3 3. In he very early universe, he primary coolan of molecular clouds is molecular hydrogen (here were few if any meals). Thus, molecular clouds would be much warmer since H 2 does no have any low-lying energy levels. Les adop a cloud emperaure of 100 K. a. Assuming ha he gas is molecular hydrogen and helium wih he curren abundance of He (µ = 2.7), wha is he ypical sound speed? Soluion: he sound speed is given by: c s = kt µm H = erg K K gm (9) b. Wha would be he Jeans mass for a 10,000 cm 3 molecular cloud? Soluion: M J = π 3/2 c 3 s (G3 ρ) 1/2 = 200 M (10) c. How would i compare o ha in a ypical molecular cloud oday? Soluion: I would be significanly greaer, a leas by a facor of 10 (for a cloud wih gas emperaures of 20 K) d. How would infall raes differ beween proosars now and hen? Soluion: The infall raes would be significanly higher. Since he sound speed is a facor of 2.2 greaer, he infall rae (c 3 s /G) would be 11 imes greaer. e. How migh sars and he iniial mass funcion be differen? Soluion: The masses of he firs sar are expeced o be sysemaically higher due o he larger Jeans mass and he higher infall raes. 4. Below are millimeer-wave specra oward proosars. They are observed in hree lines: op is a specrum of he molecule CS (parially opically hick), he middle is a specrum of he molecule H 2 CO (parially opically hick) and he boom is a line of he molecular ion H 2 H + (opically hin). Which of he following cores show infall, which show ouflow, and which show neiher? Soluion: Red-shifed self-absorpion implies infall, blueshifed self-absorpion implies ouflows. These sources have redshifed self-absorpion (i.e. exhibi infall): L1172, L1251a, L1251b, L1152. The following show blue-shifed self-absorpion (i.e. exhibi ouflow): L43, L146 and L1262. The following exhibi neiher: IRS32.

4 4 5. Define he following: Pre-main sequence sar, T Tauri sar, Herbig Ae/Be sar. Pre-main sequence sars: he pre-main sequence phase occurs afer he proosellar phase and before he main sequence. During he pre-main sequence, he sar has accreed mos of is mass, bu does no ye have cenral emperaures needed for Hydrogen fusion o occur. T Tauri sar: a low mass pre-main sequence sar. Classical T Tauri sars have disks, are accreing small amouns of gas, and ofen show srong line emission from accreion and winds. Weak line T Tauri sars do no have disks and do no show srong line emission. Herbig Ae/Be sar: an inermediae mass pre-main sequence sar. These sars show emission line (by definiion) and have disks. The original definiion by George Herbig was an A or B specral ype, srong line emission, and proximiy o nebulosiy (o show ha hey are near heir naal molecular gas). 6. a. Calculae he energy released in an accreion disk as gas moves from a radius of r +dr o r. Assume ha he gas is in Keplerian roaion. Soluion: In Keplerian roaion, he velociy of a roaing parcel of gas is v θ = GM /R. Thus, an annulus of gas wih mass M in Keplerian roaion would have he oal energy and he change in energy is hen: E = 1 2 dmv2 θ GM sar r = 1 2 GM dm r (11) de = 1 GM dm dr (12) 2 r 2 b. Now, convering dr/d o Ṁ, show ha he luminosiy of a given annulus in an accreion disk is given by L = GMM/2R 2. Give he deails of your derivaion. A wha radius does mos of he luminosiy come from? Soluion: Le s consider a disk wih a surface densiy Σ. Then he mass in a given ring is dm = Σ2πrdr. We can hen wrie he mass flow a a given radius as dm = Σ2πrdr and Ṁ = Σ2πrdr d = Σ2πrv r (13) where v r is he velociy of he gas in he radial direcion. The rae of energy loss is hen Ė = 1 GM Σπrdr v 2 r 2 r = 1 GM Σπrv r dr = 1 GM Ṁ (14) 2 r 2 2 r 2

5 5 Noe ha v r would be negaive in his case. c. Derive he rae a which he same annulus loses angular momenum in erms of G, M, Ṁ and R. Soluion: he angular momenum of he ring is given by: L ang = dmv θ r = Σr 2 GM dr = Σ GM r r 3 dr (15) we can hen wrie he change in angular momenum as ( ) 1/2 ( ) 1/2 ( ) 1/2 3GM r 3GM L ang = Σ drv r = Σr drv r = 2 2r Ṁ 3GM dr (16) 2r 7. Consider a 1 solar luminosiy sar wih a narrow bel of planeesimals a 1 A.U. from he sar. a. Calculae he emperaure of he planeesimals. Assume ha he planeesimals are spherical and grey: hey reflec 10% of he incoming radiaion and absorb he res a all wavelenghs. Soluion: The emperaure is given by he balance of absorpion and emission. For an absorpion of a λ of 0.9 a visible ligh and ǫ λ = 1 a infrared wavelenghs, and assuming a grain radius of r and a luminosiy and disance o he sar of L and R resuling in: a λ πr 2 L 4πR = ǫ λ4πr 2 σt 4 (17) 2 T = ( ) 1/4 aλ L = 273 K (18) ǫ λ 16πσR 2 b. Nex calculae he wavelengh of peak flux densiy, F λ, for he sar and for he planeesimals. Adop a emperaure of 6000 K for he sellar phoosphere. Soluion: use Wien s law ha λt = 2900 µm K giving λ = 10 µm. c. If he planeesimals where each 1 km in diameer, wrie an expression for he raio of he luminosiy densiy, L λ (he luminosiy per wavelengh, his is jus F λ 4πD 2 where D is he disance of he sar o he Earh) of he bel relaive o he sar a he wavelengh of peak

6 6 flux densiy for he bel. Wrie his in erms of he oal combined mass of he aseroids, he densiy of he aseroids (5 gm cm 3 ), he diameer of he aseroids, he luminosiy, emperaure and radius of he sar, and he disance of he bel o he sar. Soluion: For a blackbody sar, L λ is given by: L λ (sar) = 4πR 2 1 π2hc2 λ 5 e hc/λkt 1 = 4π2 R 2 2c λ4kt (19) for he R-J limi. I m making he approximaion ha we are in he R-J limi, alhough ha isn really rue for he planeesimals. Neverheless, he approximaion will work well enough for his exercise. For he planeesimals he luminosiy is L λ (bel) = N p 4πRp 2 1 π2hc2 λ 5 e hc/λkt 1 = N p4π 2 Rp 2 2c λ4kt (20) where R p is he radius of he planeesimals and N P is he number of planeesimals given by N P = 3M P /4πρ P R 3 P where ρ P is he planeesimal densiy and M P is he oal mass of planeesimals. We can hen wrie he luminosiy of he bel as: L λ (bel) = 3M P 4πρ P R 3 P 4π 2 Rp 2 2c λ 4kT = 6cπM P ρ P R P λ4kt (21) and he raio is L λ (bel) L λ (sar) = 3M P 4πρ P R P R 2 T P T (22) d. Assuming ha he oal combined mass of all he planeesimals was equal o he mass of he Earh, calculae he luminosiy densiy of he bel as a fracion of he sellar luminosiy densiy a he peak wavelengh of he bel. Skech he SED of he sar plus he bel. The mass of he Earh is gm. Assume he disk is very opically hin. L λ (bel) L λ (sar) = gm ( ) 273 K 4π 5 gm cm cm ( cm 2 ) 6000 K (23) for he M P and R P and ρ P given above, he raio is e. Now assume ha he planeesimals were ground up ino 1 cm diameer dus paricles (perhaps by collisions), calculae he luminosiy densiy of he bel as a fracion of he sellar

7 7 luminosiy densiy a he peak wavelengh of he bel. Skech he SED of he sar plus he bel. Assume he disk is sill very opically hin. Soluion: in his case he raio is 10 5 imes larger, i.e. he bel is 2660 more luminous han he sar a ha wavelengh. The reason is ha you dramaically increased he surface area of he planeesimals by grinding hem up ino smaller pieces. 8. Using he SEDs below, idenify one proosar (class I), one sar wih disk (class II), and one pre-main sequence sar wihou disk (pure phoosphere or class III). Idenify sars by heir phone number (73-???) Soluion: Class I is The Class III are , , and maybe The remainder are Class II.