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
24-2-2014see http://www.strw.leidenuniv.nl/˜ franx/college/ 3. Galactic Dynamics mf-sts-2014-c3-1 24-2-2014see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2014-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 24-2-2014see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2014-c3-3 24-2-2014see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2014-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. 24-2-2014see http://www.strw.leidenuniv.nl/˜ franx/college/ 3.3 Virial Theorem: mf-sts-2014-c3-5 (not in BT) 24-2-2014see 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-2014-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 24-2-2014see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2014-c3-7 Hence XX i F~ij ~xi = XX i j6=i i (F~ij ~xi + F~ji ~xj ) j>i j>i F~i ~xi = − i X X Gmi mj (~xi − ~xj )(~xi − ~xj ) 3 |~ x − ~ x | i j i j>i which equals =− 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 W =− 1 X X Gmi mj 2 i |~xi − ~xj | j6=i Gmi mj ~ xi −~ xj For gravitational force F~ij = − |~xi −~ xj |2 |~ xi −~ xj | X mf-sts-2014-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 24-2-2014see http://www.strw.leidenuniv.nl/˜ franx/college/ Proceed by assuming that 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. Explain each step in this calculation. 2) Derive how empty galaxies are. First calculate the distance between neighbor stars, by assuming they are located on a 3-dimensional grid. Next get the radius of the sun from literature. Take the ratio of the two. Compare this to the same ratio of galaxies: calculate the average distance between neighboring galaxies in the same way, and get the radius of galaxies. 24-2-2014see http://www.strw.leidenuniv.nl/˜ franx/college/ 3.4 Applications mf-sts-2014-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! 24-2-2014see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2014-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) The kinetic energy is given by 1/2M hv 2 i if the velocities of the stars are NOT correlated with ther mass. Give the correct expression for the total kinetic energy if the velocities of stars ARE correlated with their mass. This is a simple, single expression without additions of various terms. 24-2-2014see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2014-c3-11 24-2-2014see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2014-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 steady state galaxies? • 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 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) 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, the scaled system is in equilibrium if the scaling parameters satisfy: am ax a2t or a2m = 2, ax am = a3x a2t . 24-2-2014see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-2014-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 up in size by a factor of 2, and reduce the temparature by a factor of 2 ? Which equilibrium is maintained, which equilibrium is lost ?