Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ 3. Galactic Dynamics mf-sts-2013-c3-1 22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2013-c3-2 handout 3 Solution: assume ensemble of point masses, moving under influence of their own gravity Aim: understand equilibrium of galaxies Point masses can be: stars, mini BHs, planets, very light elementary particles ... 1. 2. 3. 4. What are the dominant forces ? Can we define some kind of equilibrium? What are the relevant timescales? Do galaxies evolve along a sequence of equilibria ? [like stars] ? equilibrium of stars as comparison: • gas pressure versus gravity isotropic pressure at a given point • hydrostatic equilibrium, thermal equilibrium • timescales corresponding to the above, plus nuclear burning • structure completely determined by mass plus age (and metallicity) very little influence from “initial conditions” galaxies: Much harder: what are galaxies are made of ? Questions addressed in this and the next handout: • What are equations of motion ? • Do stars collide ? • Use virial theorem to estimate masses, binding energy, etc. • How many equilibrium solutions for each mass ? • What are the relevant time scales ? dynamical timescale, particle interaction timescale Basic reference: Binney & Tremaine (1987): Galactic Dynamics (BT) chapters 1 to 1.1; 2 selected parts; 3 selected parts, 4 selected parts 22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2013-c3-3 22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2013-c3-4 3.1 Equations of motion assume a collection of masses mi at location xi , and assume gravitational interaction. Hence the force F~i on particle i is given by X ~xj − ~xi d2 ~ Gmi mj Fi = mi 2 ~xi = dt |~xj − ~xi |3 j6=i • Only analytic solution for 2 point masses • “Easy” to solve numerically (brute force) - but slow for 1011 particles [see http://astrogrape.org/] • Analytic approximations necessary for a better understanding of solution In the following we investigate properties of gravitational systems without explicitly solving the equations of motion. 3.2 Do stars collide ? (BT 1 to 1.1) Is it safe to ignore non-gravitational interactions ? Calculate number of collisions between t1 and t1 + dt of star coming in from the left, in galaxy with homogeneous number density n. • incoming star has velocity v • suppose all stars have radius r∗ . • all stars in volume V1 will cause collision −> V1 = π(2r∗ )2 × v × dt • number of stars in V1 : N1 = nV1 • number of collisions per unit time = 4πr∗2 nv Typical values: r⊙ = 7 × 1010 cm n = 1010 /[3kpc]3 = 1.3 × 10−56 cm−3 v = 200 km/sec= 2 × 107 cm/sec which gives a collision rate of 1.6×10−26 sec−1 = 5×10−19 yr−1 −> very rare indeed Hence we can ignore these collisions without too much trouble. 22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ 3.3 Virial Theorem: mf-sts-2013-c3-5 (not in BT) 22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ Since = F~i we have X 1 d2 I F~i ~xi + 2K. = 2 dt2 i relation for global properties: Kinetic Energy and Potential energy. Again consider our system of point masses mi with positions ~xi . P Construct i p~i ~xi and differentiate w.r.t. time: d~ pi dt Now assume the galaxy is ’quasi-static’, i.e. its prop2 erties change only slowly so that ddt2I = 0. Then the equation above implies K=− d X d~xi d X1 d d X p~i ~xi = mi ~xi = (mi x2i ) dt i dt i dt dt i 2 dt where I = P i 1 d2 I = 2 dt2 . X d~ X d~xi d X pi p~i ~xi = ~xi + p~i dt i dt dt i i Then X i X d~xi = mi~vi2 = 2K p~i dt i with K the kinetic energy. 1X~ Fi ~xi 2 i Then assume that the force F~i can be written as a summation over ’pairwise forces’ F~ij F~i = mi x2i , which is the moment of inertia. However, we can also write: mf-sts-2013-c3-6 X F~ij j,j6=i Now realize that X i F~i ~xi = XX i F~ij ~xi j6=i This summation can be rewritten. We sum the terms over the full area 0 < i ≤ N , 0 < j ≤ N , i 6= j. However, we can also limit the summation over just half this area: 0 < i ≤ N , i < j ≤ N , and add the (j, i) term explicitly to the (i, j) term within the summation. Hence instead of summing over F~ij ~xi , we sum over F~ij ~xi + F~ji ~xj 22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2013-c3-7 Hence XX i F~ij ~xi = XX i j6=i i (F~ij ~xi + F~ji ~xj ) j>i j>i Gmi mj ~ xi −~ xj For gravitational force F~ij = − |~xi −~ xj |2 |~ xi −~ xj | X F~i ~xi = − i X X Gmi mj (~xi − ~xj )(~xi − ~xj ) 3 |~ x − ~ x | i j i j>i which equals =− mf-sts-2013-c3-8 which is the virial theorem for quasi-static systems. The more general expression for non-static systems is: This is simply a change in how the summation is done, it does not use any special property of the force field. Because F~ij = −F~ji (forces are equal and opposite for pairwise forces) the last term can be rewritten XX X F~ij (~xi − ~xj ) F~i ~xi = i 22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ X X Gmi mj 1 X X Gmi mj =− =W |~ x − ~ x | 2 |~ x − ~ x | i j i j i j>i i j6=i with W the total potential energy. Therefore for a galaxy in quasi-static equilibrium: 1 K = − W, 2 1 d2 I = W + 2K. 2 dt2 Homework assignments: 1) derive the relation for the potential energy given above. Suppose each particle has mass f mi , 0 ≤ f ≤ 1. Slowly add an extra mass mi df to each particle from infinity and calculate how much energy is released. Integrate f from 0 to 1. 2) How empty are galaxies ? Calculate the distance between neighbor stars, and get the radius of a sun from literature. Take the ratio of the two. Compare this to the same ratio of galaxies: take the average distance between neighbor galaxies, and the radius of galaxies. 22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ 3.4 Applications mf-sts-2013-c3-9 (BT pages 213, 214) Consider a system with total mass M Kinetic energy K = 12 M hv 2 i with hv 2 i = mean square speed of stars (assumption: speed of star not correlated with mass of star) Define gravitational radius rg GM 2 W =− rg Spitzer found for many systems that rg = 2.5rh , where rh is the radius which contains half the mass Virial theorom implies: GM 2 M hv i = rg 2 M = hv 2 irg G−1 Hence, we can estimate the mass of galaxies if we know the typical velocities in the galaxy, and its size! 22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2013-c3-10 Homework Assignments: 3) calculate the mass of our galaxy for two assumed radii: a.) 10 kpc (roughly the distance to the sun) b.) 350 kpc (halfway out to Andromeda) Take as a typical velocity the solar value of 200 km/s. 4) Give the correct expression for the total kinetic energy if the velocities of stars ARE correlated with their mass. 22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2013-c3-11 22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2013-c3-12 3.5 Binding Energy and Formation of Galaxies 3.6. Scaling Relations The total energy E of a galaxy is Consider a steady state galaxy with particles mi at location xi (t). Can the galaxy be rescaled to other physical galaxies? E = K + W = −K = 1/2W • Bound galaxies have negative energy - cannot fall apart and dissolve into a very large homogeneous distribution • A galaxy cannot just form from an unbound, extended smooth distribution −> Etotal = Estart ≈ 0, Egal = −K, so energy must be lost or the structure keeps oscillating: (not in BT) • scaled particle mass m̂i =am mi • scaled particle location x̂i = ax xi (at t) • am , ax , at are scaling parameters In the ’rescaled’ galaxy we have ˆ d2 F~i =m̂i (d2 x̂i /dt2 )=am mi dt x(at t) = am ax a2t F~i,orig 2 ax ~ The gravitational force is equal to P ˆ F~G = j6=i ax ~ xj −ax ~ xi |ax ~ xj −ax ~ xi |3 Gam mi am mj = a2m ~ a2x FG,orig Equilibrium is satisfied if the two terms above are equal Possible energy losses through • Ejection of stars • Radiation (before stars would form) ˆ ˆ F~i = F~G Since we have equilibrium when all scaling parameters are equal to 1, we obtain am ax a2t or a2m = 2, ax am = a3x a2t . 22-2-2013see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2013-c3-13 Now how do velocities scale ? d d v̂= dt x̂ = dt ax x(at t) = ax at vorig Hence, the scaling of the velocities satisfies: av = ax at . We can write the new scaling relation as am = ax a2v Hence ANY galaxy can be scaled up like this ! Notice that it is trivial to derive this from the virial theorem - the expression for the mass has exactly the same form. As a consequence, if we have a model for a galaxy with a certain mass and size, we can make many more models, with arbitrary mass, and arbitrary size. Homework Assignments: 5) Show explicitly that the virial theorem produces the same result for the scaling relations. 6) How does this compare to stars ? What happens to a star when we scale it in the same way ? Which equilibrium is maintained, which equilibrium is lost ?