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Home , Galaxy rotation curve, NGC 7331

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 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 . Its is +10.4 mag, and radial velocity from 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 of that galaxy in solar . [C] If mass-to-light ratio of NGC 7331 is ϒ = 4, what is the mass of the luminous in the observed disk? SOLUTION: [A] 816 km/s / H0 = 12.55 Mpc. The angular extent in radians is (10.5/60) 180/π rad, and (3.7/60) 180/π rad. This gives physical (projected) scale of (10.5/60) 180/π 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] 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 ) 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, ρ (r)/ρh(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 4πGρ = 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 = 4πGρ. 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 4πGρ = ∇ 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 ρ V Rh 3(1 + y ) ∗ = ∗ . ρ 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 () below a certain threshold mass M cannot swallow a sun-like 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 = √3 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 () · − − − · − − − 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 (as opposed to dwarf elliptical) in the . [ ] are an extreme example of AGNs (Active Galactic Nuclei), fed by a large amount of gas from the merging protogalactic clouds. [ ] Unseen act as gravitational lenses, distorting an image of a distant into the form of an arc or several separate images. [ ] AGN unification scheme assumes that all types of AGNs including Seyfert galaxies, 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 -age relationship for F-stars in the solar neighborhood. [ ] Assumptions of the closed-box model of metal enrichment of the 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 . [ ] 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 of the Milky Way is ¿ 1300 pc. [ ] The thickness of the so-called 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 ) 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). [ ] Type Sc galaxies are typically bluer than S0 type, because they formed later and contain younger (bluer) stars [ ] Surface brightness of a galaxy is independent of its distance, as the example with lamps in a park demonstrates (they appear smaller but the same surface brightness, when distant.) [ ] Lenticular galaxies have so little gas that it’s hardly observable. [ ] In a few exceptional lenticular galaxies, there are two stellar disks one embedded in the other, rotating in the opposite directions. [ ] Spider diagram is a name for the iso-velocity contours map of a disk galaxy. [ ] The relationship V2 = G M(R) /R (where M(R) is the mass inside radius R) for the circular velocity in disk galaxy is only approximate, because they are not spherical, in which case the formula would be exact. In practice, using this formula results typically in a lare error (more than a factor of 1.5). [ ] The rotation curves of galaxies show that the dark halos lack central concentration (core radii are many to dozens of kpc). [ ] Little dark matter is needed to understand the rotation in the inner parts of galaxies. Within the visible/radio radii of galaxies, the fraction of dark to light matter grows to 90 [ ] Both leading and trailing arms are frequently observed in spiral galaxies. [ ] The observed spiral patterns cannot be made of objects fixed to these arms, as the spirals would quickly wrap up and disappear due to the differential rotation of the disk. [ ] If, by magic, all the stars were to start living shorter lives (say, by a factor of 2), then the spiral arms we see would become two times narrower, as they are outlined by the short-lived massive stars. [ ] Stability criterion (Safronov-Toomre criterion) can be derived from the WKB dispersion rela- tion for waves in a galactic disk. [ ] Self-gravity of the disk participates in establishing the spiral wave pattern in a disk galaxy, hence the waves we see (short leading waves) are never exact analogs of sound waves. However, they are indeed supported mostly by pressure (velocity dispersion) effects, and in that sense represent a large-scale analog of compression sound waves in a rotating medium. [ ] Galaxies are observed to stay close to the gravitational instability situation, but always a safe factor of 1.5 or 2 away from the Toomre parameter Q = 1. This is a result of ther ability to curb the approaching instability (mainly because nonlinear spiral waves heat the disk kinematically and stabilize it.) [ ] Lindblad resonance occurs wherever the frequency with which stars in the disk encounter spiral waves equals the natural frequency of radial epicyclic oscillations (called kappa). [ ] Galaxies which do have inner Lindblad resonances may have an amplification mechanism called WASER, in which waves are only present outside the ILR. [ ] About 1/10 of disk galaxies have bars. The Milky Way is one example.

5 [ ] If a group of galaxies (like Stefan’s quintet) is emmersed in a hot intergalactic gas cloud, we can use the measurements of its temperature to compute the total mass of the system. The resultant mass is 10 or more times larger than the mass of visible matter, the rest must be in the form of dark matter. [ ] Dark matter may consist of fast-moving elemenatry particles which have not been discovered yet. [ ] Dark matter may consist of molecular hydrogen which is difficult to observe. [ ] Mergers of disk systems as well as elliptical systems invariably lead to triaxial, roundish remnants resembling elliptical galaxies. [ ] Dynamical friction is most efficient when the encounter velocity of a small stellar system is much smaller than the velocity disparsion of the target system. [ ] Dynamical friction may have cause many globular clusters to merge with the bulge/nucleus of our and Andromeda galaxies, contributing to their growth. [ ] Dynamical froction is too weak to cause the Large and Small Magellanic Clouds to approach our Galaxy and merge with it within 10-20 Gyr. [ ] Elliptical galaxies, although looking very smooth out, are not dynamically relaxed to Maxwellian velocity state. They retain the memory of initial conditions during formation episode, triaxiality etc. [ ] de Vaucouleurs law of brightness of elliptical systems can be derived from Jeans equations of stellar dynamics. [ ] M87 is the central, giant elliptical of the Virgo cluster of galaxies. [ ] VLBI (Very Large Baseline Interferometer) using radio telescopes on different continents of Earth allows angular resolutions corresponding to the Schwarzschild radius of the M87’s black hole. That’s how this black hole was spotted. [ ] Faber-Jackson relationship relates the luminosity of an elliptical galaxy to the (approximately) fourth power of its observed velocity dispersion. [ ] Tully-Fisher relationship relates the luminosity of the galaxy to the approximately third power of its observed velocity dispersion. [ ] In an isothermal system, circular velocity is 20.5 times the velocity dispersion sigma. It is independent of radius (flat rotation curve). [ ] Microwave backgroud radiation is slightly anisotropic (like a dipole) because our Galaxy moves with respect to it at the speed of 1024 km/s. [ ] After removing the Doppler effect from motion of the Galaxy and the microwave radiation of its dust, as well as of dust in the solar system, what remains is a tiny anisotropy of order 1e-5, which is the record of how nonuniform the was at the epoch of recombination. [ ] COBE and WMAP are space observatories that proved that the curvature of the spacetime is very close to zero. Closed spacetime universe would make the peaks of CMB radiation appear at angular scale of about 2 degrees of arc, open universe at 0.5 degrees, whereas the flat spacetime should produce typical scale of 0.75 degree, which was observed. [ ] Supernovae type Ia can be used as very bright and accurate standard candles for the purpose of distance maserurements. [ ] Ia research combined with the COBE/WMAP resulted in the determination that 20-25with normal matter (barionic matter) and 75-80 [ ] Supernova Ia research combined with the COBE/WMAP resulted in the determination that about 20-25associated with dark (non-barionic) matter, and 75-80energy represented by constant Lambda in Einstein’s equations.

6 [ ] The cosmological constant problem consists in the fact that the value of Lambda (cosmological constant) measured in is very much smaller than the prediction of the quantum theory. [ ] The standard model of universal expansion (k=0, Omega=1 universe model) has a problem with the observation-derived ages of stars in globular clusters reaching 14 Gyr, while the model predicts the age of the universe of only 11 Gyr (the inverse of Hubble constant of roughly 70 km/s/Mpc) [ ] There were two inflationary periods in our universe, one at extremely short time (1e-33 to 1e- 30 seconds after big bang), and the other in which we are now (much slower inflation, or accelerated expansion). [ ] Up to about a quarter of the rest mass (m c2) is released before matter is swallawed by a black hole. [ ] Scharzschild radius is a radius from below nothing, includung light, can escape. [ ] Toomre criterion predicts gravitational stability of disks against collapse and fragmentation of galactic disks if the dispersion of stellar velocities in the disk is sufficiently small for a fixed surface density of the disk, or conversly if the surface density is small enough for a given dispersion of velocities. [ ] About 1/2 of galaxies have central bars. [ ] de Vaucouleurs law of brightness fits many galaxy profiles well but has no theoretical deriva- tion/justification. [ ] Isothermal sphere is a special case of the so-called singular isothermal sphere, in which density falls with the inverse square of radius, at large distances. [ ] Typical density law of stellar density in stellar clusters and galaxies has a nearly constant central part (known as flat density core). [ ] A central cusp in a density distribution of a stellar system may be connected with a supermas- sive black hole in the center. [ ] Sagittarius A* is a satellite of the Milky Way and a member of the Local Group. [ ] The discovery of accelerating expansion of the universe was made in the middle of the 20th century by the use of Cepheid variable stars to establish the distance scale. [ ] The speed with which we move relative to the microwave cosmic backgroud radiation is 36.8 km/s. [ ] dwarf galaxies tend to contain a larger percentage of dark matter than regular, bright ones. [ ] the radius of the molecular ring around the central engine of an is on the order 1 kpc. [ ] whenever we see only one jet, we may suspect a special viewing geometry of a galactic nucleus, hiding the oppsite jet, which would appear heavily redshifted and less luminous to us to the extent that it is actually unobservable. [ ] Metal-poor globular clusters are predominantly found in the spherical subsystem, whereas the more metallic globular clusters tend to gather in a disk subsystem. [ ] is a lenticular S0-type galaxy. [ ] Van Maanen was an observer who, in the beginning of the 20th century first measured the rotation speed of the spiral pattern in galaxies via proper-motion studies. [ ] Leading spiral arms are those which turn clockwise in an image as we trace them from the center outwards. [ ] There is a way to tell which side of an edge-on disk galaxy is appraching us and which is receding. One side of a such galaxy is normally brighter (we see the bright convex edge of the arms, where stars form and shine just after passing through an arm) – that is the approaching side.

7 [ ] Quasars are an extreme example of AGNs (active galactic nuclei), in which we look directly down on an outflowing jet.

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