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```					Propagation of Light About
Rapidly Rotating Neutron
Stars
Sheldon Campbell
University of Alberta
Motivation
• Telescopes are now precise enough to
detect thermal spectra from compact
stars.
• What flux is measured by an observer
looking at a rapidly rotating relativistic
star?
• How accurate are measurements of
neutron star radii, estimated from
measurements of flux and distance?
General Relativistic Effects
• gravitational redshift
• bending of light
– the rear of the star becomes partially
visible.
• doppler effects due to the star’s rotation
– one side of the star is blue-shifted while the
other side is red-shifted.
• frame-dragging
– zero angular momentum orbits have
nonzero angular deflection.
To discuss flux received by an observer,
we must know:

• from where on the star each photon originated
• the red/blue shift of each photon

Ray-tracing must be implemented to
accomplish this.
Rotating Neutron Star Metric
• Spherical coordinates (r, θ, φ)
• r corresponds to the isotropic radial
coordinate.

2ν          2α
ds = − e dt + e ( dr + r d θ )
2          2             2      2    2

+ e 2 β r 2 sin 2 θ ( d φ − ω dt ) 2

• Metric potentials ν, α, β, and ω are
functions of r and θ only.
Rigid Rotator of Perfect Fluid
• 4-velocity of rotator with angular velocity Ω
−ν
µ                     e                       µ     µ
u =                                            (t + Ωφ )
1 − e β −ν (Ω − ω ) 2 r 2 sin 2 θ

• Perfect fluid Stress-Energy Tensor
µν                 µ ν          µν
T         = (ε + p )u u + pg
• Relationship ε(p) is given by EOS
• Many candidate equations of state exist
that conform with current observed data.
Solution for Metric
• Einstein’s equations are solved
numerically to determine the metric
potentials.

• Sufficient conditions to state a model
are
– the equation of state,
– gravitational mass M, and
– frequency of rotation f.
Geodesic Equations
dt
= e − 2ν (1 − ω b)
dλ
dφ          − 2ν       −2 β      b
= ω e (1 − ω b) + e
dλ                          r 2 sin 2 θ
2
2
d r          1                  1  dr      dr dθ
= α , r +  P − R −  2α , r +   − 2α ,θ
dλ 
2
r                  r  dλ      dλ dλ
2
dθ 1
(α ,θ P − Θ ) − 2α ,θ  dθ  − 2α ,r + 1  dr dθ
2
= 2                                        
dλ2
r                         dλ            r  dλ dλ

b2
P = e −2α Q                Q = e − 2ν (1 − ω b) 2 − e − 2 β
r 2 sin 2 θ
1 − 2α ∂Q                  1 − 2α ∂Q
R=− e                      Θ=− e
2      ∂R                  2      ∂θ
Initial Conditions
• Initial position:
to , ro ,θ o , φo
• Initial velocity:
&
ro ,θ o , and the sign of φ o
& &
Numerical Geodesic Solutions
• Use 5th order Runge-Kutta Solver with
variable step size.
• Accuracy of solutions tested by:
– 4-momentum conservation requires
2         2
 dr       dθ 
     + r2     −P=0
 dλ       dλ 
which I use as an independent error
estimator,
– comparison with 3rd and 4th order solvers,
– verify Schwarzschild geodesics when Ω=0.
Geodesics from the Star’s
Surface

φ =0
π
0 ≤θ ≤
2

rp
•Surface parametrized rs(θ)
1 drs               re
•Let K (θ ) ≡ −
rs dθ
Light Emitted from Surface
Altitude-Azimuth System
&                   cos h sin A
φ o = e − ( β +ν )
r sin θ
D          1 − cos 2 h sin 2 A
ro = ε e − (α +ν )
&
1+ D2       1 − e 2 ( β −ν )ω 2 r 2 sin 2 θ
ε                      1          1 − cos 2 h sin 2 A
θ&o =       e − (α +ν )
r                     1+ D2    1 − e 2 ( β −ν )ω 2 r 2 sin 2 θ

2
K − ε N (1 + K )ξ 1 − ξ    2
ε N = sign (cos A)
D=
ξ 2 (1 + K 2 ) − 1
 sign ( D + K ), D + K ≠ 0                                      sin h
ε =                                                ξ=
ε N            , D+K =0                                  1 − cos 2 h sin 2 A
The Star’s Sky

Observer
θS
∆φ S

When a photon’s destination is the same
latitude as the observer, a photon
originating at the same latitude, altitude
and azimuth, at a longitude equal to ∆φ S
will reach the observer.
Specific Intensity I. o
ν

The energy of radiation emission
• with frequency between νo and νo+dνo
• passing through an area dA with normal n  ˆ
• in time dt
• emitted within a solid angle dΩ about the
ˆ
direction k
is

ˆ          ˆ ˆ
dE = Iν o (k , x, t ) k ⋅ n dA dΩ dν o dt
Photon Number
ˆ, x , t ) = h ν n ( x , p , t )
4   3
r             o   r r
I vo ( k               2
c
Iν o
Since n is an invariant, 3 is an
invariant.              νo

We know the photon number for
blackbody emission, giving the specific
intensity at the surface of the star:
2h        νo
3
Iν o = 2 hν o
c e kT − 1
Flux from a Star at the
Observer
4
2h  ν ∞  sin h ν 0  3
dF = 2                    dν o dA
c  ν o  d e kT − 1
2  hν o

 

The shift in frequency is given by
2ν    2β
ν∞   e − e (Ω − ω ) r sin θ 2   2   2
=                        .
νo         1 − Ωb
The area element is
α +β
dA = e         1 + K r sin θ dθ dφ .
2   2       2
The Distance Seen Behind Some Stars

140                                           1.400 MSUN, eos A,
rotating away
Extremal Longitudes (degrees)

135
130                                           1.400 MSUN, eos A,
rotating toward
125                                           1.400 MSUN, eos L,
120                                           rotating away
115                                           1.400 MSUN, eos L,
rotating toward
110
1.657 MSUN, eos L,
105                                           rotating away
100                                           1.657 MSUN, eos L,
95                                            rotating toward
2.000 MSUN, eos L,
90
rotating away
0   200    400     600     800   1000
2.000 MSUN, eos L,
Frequency of Rotation (Hz)          rotating toward
Percent Increase in Flux as Rotation Increases

50
Percent Increase in Flux from

40
Spherical Star

1.400 MSUN,   eosA
1.400 MSUN,   eosL
30
1.657 MSUN,   eosL
20                                                2.000 MSUN,   eosL

10

0
0       200     400     600     800   1000
Frequency of Rotation (Hz)
Percent Increase of Flux per Area as Rotation
Increases
Percent Increase of Flux per

10
Area from Spherical Star

8
1.400 MSUN,   eosA
6                                              1.400 MSUN,   eosL
1.657 MSUN,   eosL
4                                              2.000 MSUN,   eosL

2

0
0     200    400     600     800   1000
Frequency of Rotation (Hz)
Determining the Radius of a
Star from its Flux
• A star’s luminosity radius is defined as
2
Fd
RL =      .
σT 4

• In the Schwarzschild metric, the flux is
redshifted. In this case the luminosity
radius, when the flux is viewed from
infinity, is          R
R∞ =      L
.
2M
1−
R∞
• Solving for R∞ gives the estimated
radius.
       RL 
2
R∞ = M 1 + 1 + 2 
       M 
          

• How does this value compare with the
actual equatorial and polar radii of
rapidly rotating neutron stars?
Comparison of Observed Radii with Actual Radii

21
R_e, 1.4 MSUN, eos A
R_p, 1.4 MSUN, eos A
19
Stellar Radius (km)

R_inf, 1.4 MSUN, eos A
R_e, 1.4 MSUN, eos L
17                                              R_p, 1.4 MSUN, eos L
R_inf, 1.4 MSUN, eos L
15                                              R_e, 1.657 MSUN, eos L
R_p, 1.657 MSUN, eos L
13                                              R_inf, 1.657 MSUN, eos L
R_e, 2.000 MSUN, eos L
R_p, 2.000 MSUN, eos L
11
R_inf, 2.000 MSUN, eos L

9
0      200    400     600     800   1000
Rotation Frequency (Hz)
Percent Difference Between R_e and R_inf

18
16
14
Percent Difference

12                                               1.400 MSUN,   eos A
10                                               1.400 MSUN,   eos L
1.657 MSUN,   eos L
8
2.000 MSUN,   eos L
6
4
2
0
0   200     400      600       800   1000
Frequency of Rotation (Hz)
Future Direction for this
Research

• Implement observers at any angle in the
star’s sky.
• Calculate the stellar spectrum received by
the observer.
• Consider the rotational broadening of
spectral lines.
I would like to thank:
• Sharon Morsink for introducing me to
this problem and guiding me through it.

• NSERC for its support of this research.

```
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 views: 3 posted: 11/30/2011 language: English pages: 28