-
Notifications
You must be signed in to change notification settings - Fork 3
/
kinematics.py
375 lines (314 loc) · 15.6 KB
/
kinematics.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
import numpy as np
#import galpy
#from scipy import weave
# Functions:
# gal_uvwxyz: converts equatorial to galactic coordinates.
# gal_rdpmuv: converts galactic coordinates into equatorial.
# gal_tester: checks to make sure gal_uvwxyz and gal_rdpmuv are inverses
# ballistic: computes a straight-line linear traceback of equatorial coordinates to a specified time in the past, for a specified number of monte carlo iterations
# epicyclic: the same as ballistic, but uses an epicyclic approximation of galactic orbits from Makarov et al. (2004).
# potential: the same as ballistic, but uses a galactic gravitational potential from galpy
# random: returns uniform random variables from -1 to 1 (rather than 0 to 1)
# ballistic_uniform: the same as ballistic, but for uniformly distributed uncertainties rather than normally distributed uncertainties.
# epicyclic_uniform: uniform monte carlo version of epicyclic
# potential_uniform: uniform monte carlo version of potential
# from astrolibpy
# AR 2013.0910: Fixed this script to run with vectors. Mostly, I put back pieces of
# http://code.google.com/p/astrolibpy/source/browse/astrolib/gal_uvw.py
# from whence the original translation comes.
def gal_uvwxyz(distance=None, lsr=None, ra=None, dec=None, pmra=None, pmdec=None, vrad=None, plx=None):
"""
NAME:
GAL_UVWXYZ
PURPOSE:
Calculate the Galactic space velocity (U,V,W) and position (X,Y,Z) of star
EXPLANATION:
Calculates the Galactic space velocity U, V, W of star given its
(1) coordinates, (2) proper motion, (3) distance (or parallax), and
(4) radial velocity.
CALLING SEQUENCE:
GAL_UVW [/LSR, RA=, DEC=, PMRA= ,PMDEC=, VRAD= , DISTANCE=
PLX= ]
OUTPUT PARAMETERS:
U - Velocity (km/s) positive toward the Galactic *anti*center
V - Velocity (km/s) positive in the direction of Galactic rotation
W - Velocity (km/s) positive toward the North Galactic Pole
REQUIRED INPUT KEYWORDS:
User must supply a position, proper motion,radial velocity and distance
(or parallax). Either scalars or vectors can be supplied.
(1) Position:
RA - Right Ascension in *Degrees*
Dec - Declination in *Degrees*
(2) Proper Motion
PMRA = Proper motion in RA in arc units (typically milli-arcseconds/yr)
PMDEC = Proper motion in Declination (typically mas/yr)
(3) Radial Velocity
VRAD = radial velocity in km/s
(4) Distance or Parallax
DISTANCE - distance in parsecs
or
PLX - parallax with same distance units as proper motion measurements
typically milliarcseconds (mas)
OPTIONAL INPUT KEYWORD:
/LSR - If this keyword is set, then the output velocities will be
corrected for the solar motion (U,V,W)_Sun = (-8.5, 13.38, 6.49)
(Coskunoglu et al. 2011 MNRAS) to the local standard of rest.
Note that the value of the solar motion through the LSR remains
poorly determined.
EXAMPLE:
(1) Compute the U,V,W coordinates for the halo star HD 6755.
Use values from Hipparcos catalog, and correct to the LSR
ra = ten(1,9,42.3)*15. & dec = ten(61,32,49.5)
pmra = 627.89 & pmdec = 77.84 ;mas/yr
dis = 144 & vrad = -321.4
gal_uvw,u,v,w,ra=ra,dec=dec,pmra=pmra,pmdec=pmdec,vrad=vrad,dis=dis,/lsr
===> u=154 v = -493 w = 97 ;km/s
(2) Use the Hipparcos Input and Output Catalog IDL databases (see
http://idlastro.gsfc.nasa.gov/ftp/zdbase/) to obtain space velocities
for all stars within 10 pc with radial velocities > 10 km/s
dbopen,'hipparcos,hic' ;Need Hipparcos output and input catalogs
list = dbfind('plx>100,vrad>10') ;Plx > 100 mas, Vrad > 10 km/s
dbext,list,'pmra,pmdec,vrad,ra,dec,plx',pmra,pmdec,vrad,ra,dec,plx
ra = ra*15. ;Need right ascension in degrees
GAL_UVW,u,v,w,ra=ra,dec=dec,pmra=pmra,pmdec=pmdec,vrad=vrad,plx = plx
forprint,u,v,w ;Display results
METHOD:
Follows the general outline of Johnson & Soderblom (1987, AJ, 93,864)
except that U is positive outward toward the Galactic *anti*center, and
the J2000 transformation matrix to Galactic coordinates is taken from
the introduction to the Hipparcos catalog.
REVISION HISTORY:
Written, W. Landsman December 2000
fix the bug occuring if the input arrays are longer than 32767
and update the Sun velocity Sergey Koposov June 2008
vectorization of the loop -- performance on large arrays
is now 10 times higher Sergey Koposov December 2008
"""
n_params = 3
if n_params == 0:
print 'Syntax - GAL_UVW, U, V, W, [/LSR, RA=, DEC=, PMRA= ,PMDEC=, VRAD='
print ' Distance=, PLX='
print ' U, V, W, X, Y, Z - output Galactic space velocities (km/s) and positions'
return None
if ra is None or dec is None:
raise Exception('ERROR - The RA, Dec (J2000) position keywords must be supplied (degrees)')
if plx is None and distance is None:
raise Exception('ERROR - Either a parallax or distance must be specified')
if distance is not None:
if np.any(distance==0):
raise Exception('ERROR - All distances must be > 0')
plx = 1 / distance #Parallax in arcseconds
if plx is not None and np.any(plx==0):
raise Exception('ERROR - Parallaxes must be > 0')
cosd = np.cos(dec*np.pi/180.)
sind = np.sin(dec*np.pi/180.)
cosa = np.cos(ra*np.pi/180.)
sina = np.sin(ra*np.pi/180.)
k = 4.74047 #Equivalent of 1 A.U/yr in km/s
a_g = np.array([[-0.0548755604, +0.4941094279, -0.8676661490],
[-0.8734370902, -0.4448296300, -0.1980763734],
[-0.4838350155, 0.7469822445, +0.4559837762]]) # rotation matrix for J2000 -->Galactic
pos1 = cosd*cosa
pos2 = cosd*sina
pos3 = sind
#AR 2013.0910: In order to use this with vectors, we need more control over the matrix multiplication
x = 1/plx * (a_g[0,0] * pos1 + a_g[1,0] * pos2 + a_g[2,0] * pos3)
y = 1/plx * (a_g[0,1] * pos1 + a_g[1,1] * pos2 + a_g[2,1] * pos3)
z = 1/plx * (a_g[0,2] * pos1 + a_g[1,2] * pos2 + a_g[2,2] * pos3)
vec1 = vrad
vec2 = k * pmra / plx
vec3 = k * pmdec / plx
#AR 2013.0910: In order to use this with vectors, we need more control over the matrix multiplication
u = (a_g[0,0] * cosa * cosd + a_g[1,0] * sina * cosd + a_g[2,0] * sind) * vec1 + (-a_g[0,0] * sina + a_g[1,0] * cosa) * vec2 + (-a_g[0,0] * cosa * sind - a_g[1,0] * sina * sind + a_g[2,0] * cosd) * vec3
v = (a_g[0,1] * cosa * cosd + a_g[1,1] * sina * cosd + a_g[2,1] * sind) * vec1 + (-a_g[0,1] * sina + a_g[1,1] * cosa) * vec2 + (-a_g[0,1] * cosa * sind - a_g[1,1] * sina * sind + a_g[2,1] * cosd) * vec3
w = (a_g[0,2] * cosa * cosd + a_g[1,2] * sina * cosd + a_g[2,2] * sind) * vec1 + (-a_g[0,2] * sina + a_g[1,2] * cosa) * vec2 + (-a_g[0,2] * cosa * sind - a_g[1,2] * sina * sind + a_g[2,2] * cosd) * vec3
lsr_vel = np.array([8.5, 13.38, 6.49])
if (lsr is not None):
u = u + lsr_vel[0]
v = v + lsr_vel[1]
w = w + lsr_vel[2]
return (u,v,w,x,y,z)
def gal_rdp(u,v,w,x,y,z):
# conversion between galactic and equatorial coordinates
k = 4.74047 #Equivalent of 1 A.U/yr in km/s
a_g = np.array([[-0.0548755604, -0.8734370902,-0.4838350155],
[+0.4941094279, -0.4448296300, 0.7469822445],
[-0.8676661490, -0.1980763734, +0.4559837762]]) # rotation matrix for Galactic-->J2000
radeg = 180/np.pi
#AR 2014.0123: First, rotate galactic xyz back to equatorial xyz
pos1 = (a_g[0,0] * x + a_g[1,0] * y + a_g[2,0] * z)
pos2 = (a_g[0,1] * x + a_g[1,1] * y + a_g[2,1] * z)
pos3 = (a_g[0,2] * x + a_g[1,2] * y + a_g[2,2] * z)
dist = np.sqrt(pos1**2 + pos2**2 + pos3**2)
ra = np.arctan2(pos2,pos1)*radeg
try:
flip = np.where(ra < 0)
ra[flip] = 360 + ra[flip]
except IndexError:
if ra < 0:
ra = 360 + ra
dec = 90.0 - np.arccos(pos3/dist)*radeg
cosd = np.cos(dec/radeg)
sind = np.sin(dec/radeg)
cosa = np.cos(ra/radeg)
sina = np.sin(ra/radeg)
a_c = np.array([ [cosa*cosd,-sina,-cosa*sind],
[sina*cosd,cosa,-sina*sind],
[sind,0,cosd] ]) # rotation matrix for cartesian to spherical
b = np.dot(a_g,a_c)
#vec = np.dot(vec,b)
vec1 = (b[0,0] * u + b[1,0] * v + b[2,0] * w)
vec2 = (b[0,1] * u + b[1,1] * v + b[2,1] * w)
vec3 = (b[0,2] * u + b[1,2] * v + b[2,2] * w)
vrad = vec1
pmra = vec2 / (k * dist)
pmdec = vec3 / (k * dist)
return ra,dec,dist,pmra,pmdec,vrad
def gal_tester():
for i in xrange(1000):
ra1 = np.random.rand() * 360
dec1 = (np.random.rand()-0.5) * 180
pi1 = np.random.rand() * 100
pmra1 = (np.random.rand()-0.5) * 20
pmdec1 = (np.random.rand()-0.5) * 20
vrad1 = (np.random.rand()-0.5) * 200
u,v,w,x,y,z = gal_uvwxyz(distance=pi1,ra=ra1,dec=dec1,pmra=pmra1,pmdec=pmdec1,vrad=vrad1)
ra2,dec2,pi2,pmra2,pmdec2,vrad2 = gal_rdp(u,v,w,x,y,z)
print ra1,dec1,1/pi1,pmra1,pmdec1,vrad1
print ra2,dec2,pi2,pmra2,pmdec2,vrad2
print ra1-ra2,dec1-dec2,1/pi1-pi2,pmra1-pmra2,pmdec1-pmdec2,vrad1-vrad2
print
def ballistic(ra,era,dec,edec,dist,edist,pmra,epmra,pmdec,epmdec,rv,erv,timespan,timestep,n_int):
n_time = np.int(np.ceil(timespan/timestep))
times = np.arange(0,timespan,timestep)
px = np.zeros((n_int,n_time))
py = np.zeros((n_int,n_time))
pz = np.zeros((n_int,n_time))
pc = []
for j in xrange(n_int):
bra = ra + np.random.randn()*era
bdec = dec + np.random.randn()*edec
bdist = dist + np.random.randn()*edist
bpmra = pmra + np.random.randn()*epmra
bpmdec = pmdec + np.random.randn()*epmdec
brv = rv + np.random.randn()*erv
tu,tv,tw,tx,ty,tz = gal_uvwxyz(ra=bra,dec=bdec,distance=bdist,pmra=bpmra,pmdec=bpmdec,vrad=brv)
# backtrack this one simulation through all of time
px[j] = tx + tu * times * 1.0226 # conversion between km/s and pc/Myr
py[j] = ty + tv * times * 1.0226
pz[j] = tz + tw * times * 1.0226
return px,py,pz
def epicyclic(ra,era,dec,edec,dist,edist,pmra,epmra,pmdec,epmdec,rv,erv,timespan,timestep,n_int):
n_time = np.int(np.ceil(timespan/timestep))
times = np.arange(0,timespan,timestep)
px = np.zeros((n_int,n_time))
py = np.zeros((n_int,n_time))
pz = np.zeros((n_int,n_time))
pc = []
# Oort constants from Bobylev et al. 2010 2010MNRAS.408.1788B
A = 0.0178
B = -0.0132
vos = 2*np.pi/85 # Vertical Oscillations
kap = np.sqrt(-4*B*(A-B)) # planar epicyclic velocity
for j in xrange(n_int):
bra = ra + (np.random.randn()*era)*np.cos(dec*np.pi/180.)
bdec = dec + np.random.randn()*edec
bdist = dist + np.random.randn()*edist
bpmra = pmra + np.random.randn()*epmra
bpmdec = pmdec + np.random.randn()*epmdec
brv = rv + np.random.randn()*erv
tu,tv,tw,tx,ty,tz = gal_uvwxyz(ra=bra,dec=bdec,distance=bdist,pmra=bpmra,pmdec=bpmdec,vrad=brv)
# backtrack this one simulation through all of time
# Calculations taken from Section 4 of Makarov, Olling, and Teuben (2004) 2004MNRAS.352.1199M
# AR 2013.1110 I'm multiplying tx and ty by ones because Python is having trouble broadcasting.
px[j] = tx + tu*kap**(-1)*np.sin(kap*times) + (tv - 2*A*tx)*(1-np.cos(kap*times))*(2*B)**(-1)
py[j] = ty - tu*(1-np.cos(kap*times))*(2*B)**(-1) + tv*(A*kap*times - (A - B)*np.sin(kap*times))*(kap*B)**(-1) - tx*2*A*(A-B)*(kap*times - np.sin(kap*times))*(kap*B)**(-1)
pz[j] = tz*np.cos(vos*times) + tw*(vos)**(-1)*np.sin(vos*times)
return px,py,pz
#def potential(ra,era,dec,edec,dist,edist,pmra,epmra,pmdec,epmdec,rv,erv,timespan,timestep,n_int):
# galacticpotential = galpy.potential.MWPotential2014()
# times = np.arange(0,timespan,timestep)
#
# for j in xrange(n_int):
# bra = ra + np.random.randn()*era
# bdec = dec + np.random.randn()*edec
# bdist = dist + np.random.randn()*edist
# bpmra = pmra + np.random.randn()*epmra
# bpmdec = pmdec + np.random.randn()*epmdec
# brv = rv + np.random.randn()*erv
#
# orbit = galpy.orbit.Orbit(vxvv=[bra,bdec,bdist/1000,bpmra,bpmdec,brv],radec=True)
# # integrate the orbit
# orbit.integrate(times,galacticpotential)
# # get the orbit
# orbitresult = orbit.getorbit()
#
# return px,py,pz
def random():
return (np.random.rand()*2)-1
def ballistic_uniform(ra,era,dec,edec,dist,edist,pmra,epmra,pmdec,epmdec,rv,erv,timespan,timestep,n_int):
n_time = np.int(np.ceil(timespan/timestep))
times = np.arange(0,timespan,timestep)
px = np.zeros((n_int,n_time))
py = np.zeros((n_int,n_time))
pz = np.zeros((n_int,n_time))
pc = []
for j in xrange(n_int):
bra = ra + random()*era
bdec = dec + random()*edec
bdist = dist + random()*edist
bpmra = pmra + random()*epmra
bpmdec = pmdec + random()*epmdec
brv = rv + random()*erv
tu,tv,tw,tx,ty,tz = gal_uvwxyz(ra=bra,dec=bdec,distance=bdist,pmra=bpmra,pmdec=bpmdec,vrad=brv)
# backtrack this one simulation through all of time
px[j] = tx + tu * times * 1.0226
py[j] = ty + tv * times * 1.0226
pz[j] = tz + tw * times * 1.0226
return px,py,pz
def epicyclic_uniform(ra,era,dec,edec,dist,edist,pmra,epmra,pmdec,epmdec,rv,erv,timespan,timestep,n_int):
n_time = np.int(np.ceil(timespan/timestep))
times = np.arange(0,timespan,timestep)
px = np.zeros((n_int,n_time))
py = np.zeros((n_int,n_time))
pz = np.zeros((n_int,n_time))
pc = []
# Oort constants from Bobylev et al. 2010 2010MNRAS.408.1788B
A = 0.0178
B = -0.0132
vos = 2*np.pi/85 # Vertical Oscillations
kap = np.sqrt(-4*B*(A-B)) # planar epicyclic velocity
for j in xrange(n_int):
bra = ra + random()*era
bdec = dec + random()*edec
bdist = dist + random()*edist
bpmra = pmra + random()*epmra
bpmdec = pmdec + random()*epmdec
brv = rv + random()*erv
tu,tv,tw,tx,ty,tz = gal_uvwxyz(ra=bra,dec=bdec,distance=bdist,pmra=bpmra,pmdec=bpmdec,vrad=brv)
# backtrack this one simulation through all of time
# Calculations taken from Section 4 of Makarov, Olling, and Teuben (2004) 2004MNRAS.352.1199M
# AR 2013.1110 I'm multiplying tx and ty by ones because Python is having trouble broadcasting.
px[j] = tx + tu*kap**(-1)*np.sin(kap*times) + (tv - 2*A*tx)*(1-np.cos(kap*times))*(2*B)**(-1)
py[j] = ty - tu*(1-np.cos(kap*times))*(2*B)**(-1) + tv*(A*kap*times - (A - B)*np.sin(kap*times))*(kap*B)**(-1) - tx*2*A*(A-B)*(kap*times - np.sin(kap*times))*(kap*B)**(-1)
pz[j] = tz*np.cos(vos*times) + tw*(vos)**(-1)*np.sin(vos*times)
return px,py,pz
#def potential_uniform(ra,era,dec,edec,dist,edist,pmra,epmra,pmdec,epmdec,rv,erv,timespan,timestep,n_int):
# galacticpotential = galpy.potential.MWPotential2014()
# times = np.arange(0,timespan,timestep)
#
# for j in xrange(n_int):
# bra = ra + random()*era
# bdec = dec + random()*edec
# bdist = dist + random()*edist
# bpmra = pmra + random()*epmra
# bpmdec = pmdec + random()*epmdec
# brv = rv + random()*erv
#
# orbit = galpy.orbit.Orbit(vxvv=[bra,bdec,bdist/1000,bpmra,bpmdec,brv],radec=True)
# # integrate the orbit
# orbit.integrate(times,galacticpotential)
# # get the orbit
# orbitresult = orbit.getorbit()
#
# return px,py,pz