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velchange.mma
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(* velocity transformation from equatorial to galactic
based on http://arxiv.org/pdf/1306.2945v2.pdf *)
alphag=192.85948*Pi/180
deltag=27.12825*Pi/180
lomega=32.93192*Pi/180
alpha=129.27351*Pi/180
delta=6.17071*Pi/180
C1=Sin[deltag]*Cos[delta]-Cos[deltag]*Sin[delta]*Cos[alpha-alphag]
C2=Cos[deltag]*Sin[alpha-alphag]
cosb=Sqrt[C1^2+C2^2]
rotation={{C1,C2},{-C2,C1}}/cosb
muequatorial={2.14,51.64}
mugalactic=rotation.muequatorial
GL=219.72*Pi/180
GB=26.27*Pi/180
(* from http://en.wikipedia.org/wiki/Stellar_kinematics km/s *)
U=10. (* moving away from pulsar *)
V=5.23 (* towards l=90, results in positive apparent \mu_l at anticenter *)
W=7.17 (* towards NGP, results in negative apparent \mu_b *)
(* 1 km/sec at 625 pc is 3.1e10m/yr/(625*3.086e16) = 0.32mas/yr *)
(* 1 km/sec at 640 pc is 3.1e10m/yr/(625*3.086e16) = 0.33mas/yr *)
(* 1 km/sec at 394 pc is 3.1e10m/yr/(625*3.086e16) = 0.53mas/yr *)
earth motion:
observation on Nov 12, 39 days from winter solstice.
Sun is at RA=15:26, dec=-18.4, earth is moving towards
ra=09:26, dec=15 degrees, this is 15 degrees from pulsar direction.
Projected velocity is 10km/s, about 3mas/yr for pulsar.
about 2mas/yr for difference.
earth motion direction = 2.47, 0.26 (radians)
results in reflex motion of negative \mu_\delta, negative \mu_alpha for
object to the other side of sun (i.e. b0834).
define
z=sin(delta)
x=cos(delta) sin(alpha)
y=cos(delta) cos(alpha)
earth motion:
velalpha=2.47
veldelta=0.26
pulsarhat={Cos[delta] Sin[alpha],Cos[delta]Cos[alpha],Sin[delta]}
velhat={Cos[veldelta] Sin[velalpha], Cos[veldelta] Cos[velalpha],Sin[veldelta]}
velperp=velhat - pulsarhat*(velhat.pulsarhat)
velperphat=velperp/Sqrt[velperp.velperp]
dechat0={0,0,1}
dec0=dechat0-(pulsarhat.dechat0)*pulsarhat(*orthogonal to pulsar, pointing up*)
dechat=dec0/Sqrt[dec0.dec0]
veldec=velperp.dechat
velalpha=Sqrt[velperp.velperp-veldec^2]
apparent motion of pulsar: \mu_dec=-30*0.33*0.154 = -1.52 mas/yr
\mu_alpha=-30*0.33*0.205 = -2.03mas/yr
of lens: \mu_dec=-30*0.53*0.154 = -2.45mas/yr
\mu_alpha=-30*0.53*0.205 = -3.26mas/yr
relative velocity of pulsar to lens:
\mu_dec = 0.92 mas/yr
\mu_alpha = 1.23 mas/yr
mumodel={-6.07,62.48 }
mumodelgalactic=rotation.mumodel