Constructing the Galactic Rotation Curve

Physics Department Earth, Atmospheric, and Planetary Sciences Department Problem Set 8 Reading: Zeilik & Gregory: Chapters 16 & 17. Problem 1 “Constructing the Galactic Rotation Curve” The Table below gives the maximum radial velocity, vrad,max, that is observed for neutral hydrogen in the Galaxy as a function of galactic longitude, �. Use the equation: vrad,max = vrot − v� sin � to construct the rotation curve of our Galaxy. Plot vrot as a function of sin �. (Note that sin � = R/R� , where R and R� are the radial distances from the galactic center of an arbitrary point and the position of the Sun, respectively.) � (degrees): vrad,max (km/s): 15 147 20 25 30 35 40 45 50 145 128 123 106 96 82 81 � (degrees): vrad,max (km/s): 55 67 60 58 65 45 70 34 75 27 80 85 90 22 16 14 For the purpose of constructing this graph, assume that v� = 225 km/s. Problem 2 “Oort Constants” Given the definitions of the Oort A and B constants: � � � � dω R0 A ≡ − , dR 0 2 B ≡ A − ω0 a. Show that A/B would equal −3 if the mass of our Galaxy were almost entirely concen­ trated in the center (i.e., the case where ω ∝ R−3/2 ). 1 b. Show that A/B = −1 for the case where our Galaxy is assumed to have a “flat” rotation curve (i.e., vrot = constant; ω ∝ R−1 ) Problem 3 ­Optional “Kinematic Distances” Once the rotation curve, vrot (R) � ω(R) × R, of our Galaxy is determined, we can use radial velocity (vrad ) measurements to determine “kinematic distances” to objects. Recall the formula: vrad = R0 (ω − ω0 ) sin � , where R0 is the distance from the Sun to the galactic center, and ω0 is the Sun’s angular velocity about the center of the Galaxy. (See the sketch below for the appropriate geometry.) a. Assume that our Galaxy has a “flat” rotation curve (i.e., vrot (R) = constant ≡ v0 ), and that we know the values of R0 and v0 . Derive a formula for R, the distance from an object to the galactic center, in terms of the known and observable quantities (R0 , v0 , �, and vrad ). b. The kinematic distance d (from the Sun to the object) can then be derived from R, R0 , and �. Find this relation. c. A star at a galactic longitude � = 20◦ is observed to have a radial velocity vrad = 100 km/sec (away from the Earth). Use the expressions found in parts (a) and (b) to find the kinematic distance to the star. [Take R0 = 8.5 kpc and v0 = 225 km/sec.] Sun l R0 Object R Galactic Center 2 Problem 4 “Simplified Model of a Star” Consider the following (somewhat unphysical) model of a star that is composed of an in­ compressible fluid – one in which the density ρ is independent of the pressure exerted on it. (Such an approximation is actually much better suited for constructing a model of a planet.) a. Use the equation of hydrostatic equilibrium, dP (r)/dr = −g(r)ρ, and Newton’s theorem that the local acceleration of gravity inside of a spherical distribution is GM (r)/r2 [where M (r) is the mass enclosed within radius r] to derive the pressure as a function of radius within the star. Assume a total stellar mass, M , a radius, R, and a uniform (constant) density ρ = 3M/(4πR3 ). Express your answer for P (r) in terms of M , R, and G. b. Sketch P (r). Note that the radius of the stellar surface, R, is defined by P (R) = 0. c. Evaluate your expression for the pressure at the center of a star [P (r = 0)] for the case where M = 1M� and R = 1R� . Express your answer in units of the atmospheric pressure on Earth (106 dynes cm−2 = 105 Newtons m−2 ). As an interesting application of your expression for P (R), you can use it to compute the pressure at the center of the Earth. Take M⊕ � 6 × 1027 g and R⊕ � 6378 km. The pressure at the center of the Earth is given in the texts as 3.9 × 1012 dynes cm−2 or 3.9 × 1011 Newtons m−2 . Your answer should agree to within a factor of ∼2. d. Now find the temperature at the center of the model star, Tc . Utilize the fact that the “fluid” within a star actually obeys the ideal gas law to a high degree of accuracy: P = nkT , where P is the pressure, T the temperature, n the particle number density, and k is Boltz­ mann’s constant. (Note that this is hardly consistent with the assumption of an incompress­ ible fluid that was made above, but we shall proceed anyway.) Use the ideal gas law and the result of part (c) to find Tc . Note that n = ρ/m, where m is the average weight of a gas particle, and take m � 10−24 g. e. In more accurate models of the Sun, the central temperature turns out to be higher than the value you found in part (d) because the Sun does not have a uniform density and is, in fact, centrally condensed. The actual central density in the Sun is approximately 150 grams cm−3 , again reflecting the high degree of central concentration. Assume that most of the energy generation via the nuclear burning of hydrogen occurs in the central region which contains about 20% of the total mass of the Sun. The total power generated by the Sun is L� = 4 × 1033 ergs/sec. Use the following expression for nuclear energy production per gram of matter to estimate what the temperature near the center of the Sun, Tc , must be in order to produce the observed power output, L� : � � 3381 5 ε = 4.4 × 10 ρ exp − 1/3 erg g−1 s−1 Tc (In evaluating this expression, assume that the central 20% of the mass of the Sun has a uniform temperature Tc and a uniform density ρ = 150 g cm−3 .) 3 The following parts are optional: • Compute the gravitational potential energy, V , of our uniform density stellar model. Ex­ press your answer in terms of M , R, and G. [Hint: the potential energy lost in adding a shell of mass δM to an existing sphere of mass M (r) is δV = GM (r)δM/r.] • Evaluate V for M = 1M� and R = 1R� . • Find the Kelvin­Helmholtz timescale, τKH (the time for a star to radiate away half of its stored energy) for the model star. Estar τKH = L where Estar = |V |/2 (from the Virial theorem) and L is the luminosity of the star (use L = L� = 3.9 × 1033 ergs s−1 ). Express your answer in years. Problem 5 “Fueling the Sun” Optional Zeilik & Gregory; Problem 1, Chapter 16, page 330. Problem 6 “Main­Sequence Lifetimes” Zeilik & Gregory; Problem 6, Chapter 16, page 330. Take the nuclear burning efficiency of the p­p chain to be 0.007. That is, for every mass of hydrogen, m, that is burned to helium, the energy generated is 0.007 mc2 . Problem 7 “Nuclear Binding Energies” Use the following table of Atomic Mass Excesses1 (expressed in energy units of MeV) to compute how much energy is liberated in each of the following reactions from the p­p and CNO chains. a. b. c. d. e. H + D → He3 + γ He3 + He3 → He4 + 2H1 N15 + H1 → O16 + γ He4 + He4 + He4 → C12 + γ He4 + C12 → O16 + γ [Hint: Add up the values of the mass excesses, A − M , for the nuclei on the left­hand­side of the reaction and subtract the sum of the mass excesses for the nuclei on the right­hand­side. The mass of a gamma ray (γ) is zero.] 1 from “Principles of Stellar Evolution and Nucleosynthesis”, by Donald D. Clayton; McGraw­Hill Book Company 4 5