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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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posted:11/30/2011
language:English
pages:28