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Name and student number: ASTC22 (GALACTIC AND EXTRAGALACTIC ASTROPHYSICS) FINAL EXAM - PROBLEMS. 18 APR 2009. Points in the square brackets give the relative weight with which the problems count toward the final score. This part of the exam is worth 30% of the total course score. Various physical constants are given at the end of this part. Please write legibly and explain what you are doing. Describe any obstacles you might encounter - if you get stuck toward the end or with numerical evaluation of your result, but were going in the right direction, you'll get a partial credit. 1 [3 points] NGC 7331 The galaxy in the picture is NGC 7331. The galaxy size is 10.5 arcminutes by 3.7 arcminutes on the sky. It is often thought of as a twin to our Milky Way. Its apparent magnitude is +10.4 mag, and radial velocity from redshift is +816 km/s. [A] Using Hubble's law with the constant H0 = 65 km/s/Mpc, estimate the distance to NGC 7331 in Mpc, and the physical distance in kpc that the galaxy spans (using the cited angular distance). [B] Calculate the absolute magnitude M of NGC 7331 and knowing that the absolute magnitude of the sun equals +4.83 mag, the luminosity of that galaxy in solar luminosities. [C] If mass-to-light ratio of NGC 7331 is ϒ = 4, what is the mass of the luminous matter in the observed disk? SOLUTION: [A] 816 km/s / H0 = 12.55 Mpc. The angular extent in radians is (10.5/60) 180/p rad, and (3.7/60) 180/p rad. This gives physical (projected) scale of (10.5/60) 180/p 12.55 Mpc = 38.5 kpc, and 13.6 kpc, correspond- ingly. [B] M m = 5logd=(10 pc). Hence, M = 20m:1 . Also, M M = 2:5log L=L , therefore L=L = − − − − − 0:93 1010. · [C] The units of mas-to-light ratio are M =L (that wasn't stated, but is clear from lectures and textbook.) Galaxy's luminous mass is therefore about 4e10 L . 1 2 [5 points] Dark Matter in a galaxy A sperically symmetric galaxy's rotation curve, V (R), levels off at large distance R, in apparent contradiction to a limited extent of luminous matter, which would predict a falling rotation curve. As usual, astronomers assigned the difference to the presence o dark matter. Analysis of the rotation curve allowed it to be split into two components, V (due to stars) and Vh (due to dark halo), where V = V x=(1 + x3)1=2 ∗ and 2 1=2 Vh = V0y=(1 + y ) where x = r=R , y = r=Rh. Numerical values of constants are V = 180 km/s, V0 = 230 km/s, R = 4 kpc, and ∗ ∗ ∗ Rh = 8 kpc. Analyze the ratio of the spatial densities of luminous and dark matter as a function of radius, r (r)=rh(r). ∗ Derive the formula and do appropriate sketches. What is the numerical value of the ratio near the center of the galaxy and at r ∞? ! 2 2 HINT: Show that 4pGr = r− d[rV ]=dr: SOLUTION For unknown reasons, many students ignored the hint and have gotten wrong results. There are many ways to actually solve this problem, for instance via V 2 = GM(r)=r relations for two separate components, where dM=dr = 4pGr. The most direct proof of the hint is this. Looking at Poisson equation and recognizing that 2 2 2 2 2 2 2 V = rdΦ =dr is the part of the divergence, we obtain 4pGr = ∇ Φ = r− d(r dΦ=dr)=dr = r− d=dr rV , which proves the hint. Application via the necessary differentiation of both formulae for rV 2, the taking their ratio, gives 2 2 2 2 r V Rh 3(1 + y ) ∗ = ∗ : r V 2 R2 (1 + x3)2(3 + y2) h h∗ 0 This ratio decreases steadily from a value of 2.45 to 0, as r varies from zero to infinity. 3 [4 points] Tidal destruction Prove that SMBH (supermassive black hole) below a certain threshold mass M cannot swallow a sun-like star whole, that is without first tearing it apart by tidal forces, by comparing the Roche lobe radius rL = (M =3M)1=3a, where a is the distance between the star of mass M and the black hole of mass M, with the ∗ ∗ solar radius. Consider the smallest a possible around a black hole (its Schwarzchild radius), to allow for a max- imum tidal force. Find the threshold M in units of solar masses. Does the SMBH in the center of our galaxy (which has a mass of 3 million M ) chews or swallow stars whole? SOLUTION M 1=3 2GM = R 3M · c2 from which, after a short algebra, M R 3=2 = p3 M Rg 2 2 where Rg = 2GM =c (gravitational radius of the sun). Numerically, Rg 3 km, and the critical mass of a ' SMBH able to tidally destroy a sun is M = 2 108 M . · 4 The Possibly Useful Constants c = 2:99792 108 m/s, = 2:99792 1010 cm/s 3 108 m/s, (speed of light) · · ' · G = 6:67259 10 11 m3 kg 1 s 2 = 6.67259 10 8 cm3 g 1 s 2 (gravity) · − − − · − − − M = 1:9891 1030 kg = 1:9891 1033 g (solar mass) · · R = 6:9598 108 m = 6:9598 1010 cm (solar radius) · · L = 3:8515 1026 J/s = 3:8515 1033 erg/s (solar luminosity) · · 1 AU = 1.496 1011 m = 1:496 1013 cm · · 1 pc = 206256 AU = 3:09 1018 cm = 3:09 1016 m · · 3 Name and student #: ————————————————————————- Final exam in ASTC22, 18 Apr 2009. QUIZ. ————————————————————————- This part of the final exam is worth up to 15% of the total course score. It contains approx. 50 questions, and should take about 40 min. Be very careful, some questions can be rather tricky: a single wrong word is enough to invalidate a long, perfectly-looking sentence. However, ignore any typos: a 'wrong word' means scientifically incorrect word, not a misspelled one. Please put Y or N in the brackets for Yes or No answers. Please avoid symbols T/F. Thanks and good luck. ——————————————————————————- [ ] There is only 1 elliptical galaxy (as opposed to dwarf elliptical) in the Local Group. [ ] Quasars are an extreme example of AGNs (Active Galactic Nuclei), fed by a large amount of gas from the merging protogalactic clouds. [ ] Unseen galaxies act as gravitational lenses, distorting an image of a distant quasar into the form of an arc or several separate images. [ ] AGN unification scheme assumes that all types of AGNs including Seyfert galaxies, blazars and quasars, are really the same type of objects seen from a different viewing angle [ ] Seyfert galaxies correspond to a top view of the accretion disk and a small angle between the jet and the line of sight. [ ] Mergers of galaxies in a cluster of galaxies are much more frequent than physical collisions of stars in a galaxy: a typical galaxy has a fairly large chance to encounter another galaxy but a typical star has nearly zero chance of striking another star, at least in time interval equal to 10 Gyr. [ ] Metal enrichment of stars has been observed in our Galaxy as a metallicity-age relationship for F-stars in the solar neighborhood. [ ] Assumptions of the closed-box model of metal enrichment of the interstellar medium ISM) are: (1) no inflow or outflow of gas from a volume element, (2) initial mass is all in the form of gas, (3) yield of metals per one evolved star is constant in time. [ ] G-dwarf problem is the name for a disagreement between the theory of closed-box enrich- ment of ISM, and observations. Theory predicts 30-50of G-type stars to have Z ¡ 0.25 Z , while observations give a much smaller fraction. [ ] A possible solution to the G-dwarf problem is that Z was not zero at the time of formation of the first Milky Way stars, but of order Z = 0.15 Z . [ ] William Herschel built the largest telescope of the 18th century (126 cm diameter mirror) and discoverd the spiral structure of 14 nebulae, including NGC 7331 that is dicussed in the textbook. [ ] The thickness of the so-called thick disk of the Milky Way is ¿ 1300 pc. [ ] The thickness of the so-called thin disk of the Milky Way is ¡ 800 pc. [ ] One of the main reasons why we think supermassive black holes occupy the very centers of many galaxies, is that velocity of objects within a small radius (inside the central parsec) often rise to 1000 km/s. [ ] Most of the energy is emitted by galaxies in the ultraviolet spectral range. [ ] Most of the energy is emitted by galaxies in the infrared and visible parts of spectrum. [ ] Practical issues with CCDs include cosmic ray strikes, overexposure, and flat-fielding of a CCD (removing the effect of uneven sensitivity of the pixels in an array) 4 [ ] Read-out noise is independent of the counts produced by an observed object. [ ] Signal-to-noise ratio involves a combination of readout noise and the statistical photon noise. [ ] Seeing in the ground-based observations is typically about 1” [ ] Surface brightness of disk galaxies is typically 15 mag/arcsec2 at their center, falling expo- nentially outside. The disk radial scale over which the surface brightness decreases by a factor of e=exp(1)=2.718 is normally several pc. [ ] In general, galaxies become bluer and fainter along the sequence from S0 to Sd and Sm (Magellanic type). [ ] Type S0 galaxies are typically bluer than Sc type, because the Sd galaxies are usually at larger distances (hence, redshifted).
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