Propagation of Light About Rapidly Rotating Neutron Stars. Sheldon Campbell University of Alberta

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Archibald Collins
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1 Ppagatin f Light Abut Rapily Rtating Nutn Stas Shln Campbll Univsity f Albta

2 Mtivatin Tlscps a nw pcis nugh t tct thmal spcta fm cmpact stas. What flux is masu by an bsv lking at a apily tating lativistic sta? Hw accuat a masumnts f nutn sta aii, stimat fm masumnts f flux an istanc?

3 Gnal Rlativistic Effcts gavitatinal shift bning f light th a f th sta bcms patially visibl. ppl ffcts u t th sta s tatin n si f th sta is blu-shift whil th th si is -shift. fam-agging z angula mmntum bits hav nnz angula flctin.

4 T iscuss flux civ by an bsv, w must knw: fm wh n th sta ach phtn iginat th /blu shift f ach phtn Ray-tacing must b implmnt t accmplish this.

5 Rtating Nutn Sta Mtic Sphical cinats (,, φ cspns t th istpic aial cinat. s t α ( β sin ( φ ωt Mtic ptntials, α, β, an ω a functins f an nly.

6 Rigi Rtat f Pfct Flui 4-vlcity f tat with angula vlcity Ω µ µ µ u ( t Ωφ β ( Ω ω sin Pfct flui Stss-Engy Tns µ µ T ( ε p u u pg Rlatinship ε(p is givn by EOS Many caniat quatins f stat xist that cnfm with cunt bsv ata. µ

7 Slutin f Mtic Einstin s quatins a slv numically t tmin th mtic ptntials. Sufficint cnitins t stat a ml a th quatin f stat, gavitatinal mass M, an fquncy f tatin f.

8 Gsic Equatins ( λ λ α λ α α λ λ λ α λ α α λ ω ω λ φ ω λ β P R P b b b t Θ,,,,,, sin ( ( Q P α ω β sin ( b b Q R Q R α α Θ Q

9 Initial Cnitins Initial psitin: t,,, φ Initial vlcity: &, &, an th sign f φ &

10 Numical Gsic Slutins Us 5 th Rung-Kutta Slv with vaiabl stp siz. Accuacy f slutins tst by: 4-mmntum cnsvatin quis λ λ which I us as an inpnnt stimat, P cmpaisn with 3 an 4 th slvs, vify Schwazschil gsics whn Ω0. 0

11 Gsics fm th Sta s Sufac φ 0 0 π Sufac paamtiz s ( s Lt K( s p

12 Light Emitt fm Sufac

13 Altitu-Azimuth Systm ω ε ω ε φ β α β α β ( ( ( ( ( sin sin cs sin sin cs sin sin cs A h D A h D D A h & & & (cs ε N sign A 0, 0, sign ( K D K D K D ε N ε ( ( K K K D N ξ ξ ξ ε A h h sin cs sin ξ

14 Th Sta s Sky Obsv S φ S Whn a phtn s stinatin is th sam latitu as th bsv, a phtn iginating at th sam latitu, altitu an azimuth, at a lngitu qual t φ S will ach th bsv.

16 Spcific Intnsity I. Th ngy f aiatin missin with fquncy btwn an passing thugh an aa A with nmal in tim t mitt within a sli angl Ω abut th ictin kˆ is nˆ E I ( kˆ, x, t kˆ nˆ A Ω t

17 Phtn Numb,, (, ˆ, ( 3 4 t p x n c h t x k I v Sinc n is an invaiant, is an invaiant. 3 I W knw th phtn numb f blackby missin, giving th spcific intnsity at th sufac f th sta: 3 kt h c h I

18 Flux fm a Sta at th Obsv A h c h F kt h sin Th shift in fquncy is givn by. sin ( b Ω Ω ω β Th aa lmnt is. sin φ β α K A

20 Th Distanc Sn Bhin Sm Stas Extmal Lngitus (gs Fquncy f Rtatin (Hz.400 MSUN, s A, tating away.400 MSUN, s A, tating twa.400 MSUN, s L, tating away.400 MSUN, s L, tating twa.657 MSUN, s L, tating away.657 MSUN, s L, tating twa.000 MSUN, s L, tating away.000 MSUN, s L, tating twa

21 Pcnt Incas in Flux as Rtatin Incass Pcnt Incas in Flux fm Sphical Sta MSUN, sa.400 MSUN, sl.657 MSUN, sl.000 MSUN, sl Fquncy f Rtatin (Hz

22 Pcnt Incas f Flux p Aa as Rtatin Incass Pcnt Incas f Flux p Aa fm Sphical Sta Fquncy f Rtatin (Hz.400 MSUN, sa.400 MSUN, sl.657 MSUN, sl.000 MSUN, sl

23 Dtmining th Raius f a Sta fm its Flux A sta s luminsity aius is fin as R L F σt In th Schwazschil mtic, th flux is shift. In this cas th luminsity aius, whn th flux is viw fm infinity, is RL R. M R 4.

24 Slving f R givs th stimat aius. R R M L M Hw s this valu cmpa with th actual quatial an pla aii f apily tating nutn stas?

25 Cmpaisn f Obsv Raii with Actual Raii Stlla Raius (km Rtatin Fquncy (Hz R_,.4 MSUN, s A R_p,.4 MSUN, s A R_inf,.4 MSUN, s A R_,.4 MSUN, s L R_p,.4 MSUN, s L R_inf,.4 MSUN, s L R_,.657 MSUN, s L R_p,.657 MSUN, s L R_inf,.657 MSUN, s L R_,.000 MSUN, s L R_p,.000 MSUN, s L R_inf,.000 MSUN, s L

26 Pcnt Diffnc Btwn R_ an R_inf Pcnt Diffnc MSUN, s A.400 MSUN, s L.657 MSUN, s L.000 MSUN, s L Fquncy f Rtatin (Hz

27 Futu Dictin f this Rsach Implmnt bsvs at any angl in th sta s sky. Calculat th stlla spctum civ by th bsv. Cnsi th tatinal baning f spctal lins.

28 I wul lik t thank: Shan Msink f intucing m t this pblm an guiing m thugh it. NSERC f its suppt f this sach.

The angle between L and the z-axis is found from

The angle between L and the z-axis is found from Poblm 6 This is not a ifficult poblm but it is a al pain to tansf it fom pap into Mathca I won't giv it to you on th quiz, but know how to o it fo th xam Poblm 6 S Figu 6 Th magnitu of L is L an th z-componnt