EdX.org/Greatest Unsolved Mysteries of the Universe/Simulation/Random Galaxies Influenced by Mutual Gravity

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from visual import * from time import clock from random import random,uniform import numpy import math

Sets up a bunch of masses and lets them move under the influence of their

mutual gravity.

Masses are put in random positions within a sphere originally. An additional

force is applied which simulates an equal density distributed over the

rest of the universe outside this initial sphere.

def randomdirection(): # Generates random direction on sky. # RA (in radians) # Dec (in radians) ra = 2.0math.pirandom() dec = math.acos(2.0random()-1.0)-0.5math.pi return ra,dec

def ranvec(): # Generates a randomly orientated unit vector. theta, phi = randomdirection() z = math.sin(phi) x = math.cos(phi)*math.sin(theta) y = math.cos(phi)*math.cos(theta) vec = numpy.array([x,y,z]) return vec

Stars interacting gravitationally

Program uses numpy arrays for high speed computations

Nstars = 200 # change this to have more or fewer stars

G = 6.7e-11 # Universal gravitational constant

Typical values

Msun = 1.5E30 # 2E30 is good Rsun = 3E8 Rtrail = 2e8 L = 4e10 vsun = 0.9sqrt(GMsun/Rsun) h0 = 1.0e-5 # Hubble's constant - expansion rate 8.0e-6 is good.

scene = display(title="Stars", width=1320, height=830, range=L, forward=(-1,-1,-1))

Stars = [] poslist = [] plist = [] mlist = [] rlist = [] p0 = 0.0Msun100000.0

for i in range(Nstars): vec = Lranvec()(random()**0.3333) x = vec[0] y = vec[1] z = vec[2] r = Rsun col0 = (uniform(0.7,1.0),uniform(0.7,1.0), uniform(0.7,1.0)) Stars = Stars+[sphere(pos=(x,y,z), radius=r, color=col0)] mass = Msun px = p0uniform(-1,1) py = p0uniform(-1,1) pz = p0*uniform(-1,1) poslist.append((x,y,z)) plist.append((px,py,pz)) mlist.append(mass) rlist.append(r)

pos = array(poslist) p = array(plist) m = array(mlist) m.shape = (Nstars,1) # Numeric Python: (1 by Nstars) vs. (Nstars by 1) radius = array(rlist)

vcm = sum(p)/sum(m) # velocity of center of mass p = p-m*vcm # make total initial momentum equal zero

dt = 50.0 Nsteps = 0 pos = pos+(p/m)*(dt/2.) # initial half-step time = clock() Nhits = 0

while 1: rate(50)

L *= 1.0+h0*dt
con = 1.0*G*Nstars*Msun/(L*L*L)# strength of force to allow for external mass

# Compute all forces on all stars
try:  # numpy
    r = pos-pos[:,newaxis] # all pairs of star-to-star vectors
    for n in range(Nstars):
        r[n,n] = 1e6  # otherwise the self-forces are infinite
    rmag = sqrt(add.reduce(r*r,-1)) # star-to-star scalar distances
    hit = less_equal(rmag,radius+radius[:,newaxis])-identity(Nstars)
    hitlist = sort(nonzero(hit.flat)[0]).tolist() # 1,2 encoded as 1*Nstars+2
    F = G*m*m[:,newaxis]*r/rmag[:,:,newaxis]**3 # all force pairs
except: # old Numeric
    r = pos-pos[:,NewAxis] # all pairs of star-to-star vectors
    for n in range(Nstars):
        r[n,n] = 1e6  # otherwise the self-forces are infinite
    rmag = sqrt(add.reduce(r*r,-1)) # star-to-star scalar distances
    hit = less_equal(rmag,radius+radius[:,NewAxis])-identity(Nstars)
    hitlist = sort(nonzero(hit.flat)) # 1,2 encoded as 1*Nstars+2
    F = G*m*m[:,NewAxis]*r/rmag[:,:,NewAxis]**3 # all force pairs
    
for n in range(Nstars):
    F[n,n] = 0  # no self-forces
p = p+sum(F,1)*dt+pos*con*dt*m

# Having updated all momenta, now update all positions         
pos = pos+(p/m)*dt

# Expand universe
pos *= 1.0+h0*dt

# Update positions of display objects; add trail
for i in range(Nstars):
    Stars[i].pos = pos[i]

# If any collisions took place, merge those stars
for ij in hitlist:
    i, j = divmod(ij,Nstars) # decode star pair
    if not Stars[i].visible: continue
    if not Stars[j].visible: continue
    # m[i] is a one-element list, e.g. [6e30]
    # m[i,0] is an ordinary number, e.g. 6e30
    newpos = (pos[i]*m[i,0]+pos[j]*m[j,0])/(m[i,0]+m[j,0])
    newmass = m[i,0]+m[j,0]
    newp = p[i]+p[j]
    newradius = Rsun*((newmass/Msun)**(1./3.))
    iset, jset = i, j
    if radius[j] > radius[i]:
        iset, jset = j, i
    Stars[iset].radius = newradius
    m[iset,0] = newmass
    pos[iset] = newpos
    p[iset] = newp
    Stars[jset].visible = 0
    p[jset] = vector(0,0,0)
    m[jset,0] = Msun*1E-30  # give it a tiny mass
    Nhits = Nhits+1
    pos[jset] = (10.*L*Nhits, 0, 0) # put it far away

Nsteps += 1
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