脉冲星到达时间 Pulsar Timing

23/4/22Pulsar Workshop, 2009, NAOC1

脉冲星到达时间Pulsar Timing


地球的公转：精度 ~0.1秒的小数

平太阳时，恒星时地球自转：精度 ~10 － 8

秒协调世界时

经典计时


古老方法


现代计时

自然界的微观周期运动，以原子 /分子的周期振动制作人工连续计时


脉冲星——精确自转的天体

0.001s 0.01s 0.1s 1s 10s

已知 1850 颗脉冲星，绝大多数在银河系周期～ 1s

毫秒脉冲星性质很不相同 (~ 0.003 s)

多数毫秒脉冲星处于双星系统毫秒脉冲星是再循环的年老脉冲星，自转非常稳定

PSR J0437-4715 周期为 :

5.757451831072007

0.000000000000008 ms

脉冲星简介

23/4/226 Pulsar Workshop, 2009, NAOC

Pulsar Workshop, 2009, NAOC

PSR B0329+54 的脉冲

T (second)

什么是脉冲星脉冲到达时间

The Vela

The Crab

23/4/227

Vela

Crab

Pulsar rotate tens or even hundreds of times every

second. Measure the time-of-arrival (TOA) of pulses. Differences between the observed TOAs and a

model are called timing residuals. The predicted rotation pulse phase given by the

basic timing model:


cos( )cosct A t A: light travel time from

Sun to Eartht: timeλ: ecliptic longitudeβ: ecliptic latitudeω: angular velocity of

Earth位置测量精度的问题：β=90° ： highest angular

accuracy of positionβ=0° ： poor angular accuracy

of position

位置测量误差导致时间延迟：

地球运动造成的脉冲到达时间延迟：

（假设圆运动）

sin( )cos cos( )sinct A t A t


需考虑：•地球自转 21ms•椭圆轨道•太阳系惯性质量中心与太阳并不重合：木星等行星的影响•地球在椭圆轨道上的引力势有周年变化，地面钟相对圆形轨道有周年变化•Doppler效应二阶项，与地球的椭圆轨道相关，∝ V2

地球

Roemer延迟 vs. 视差测量•地球轨道上引力势的变化，时空弯曲 ->时间延迟•群延迟：消色散

2 2 2cos cos

2 2

a a

cd c

视差最大延迟示意图：PSRJ0437-4715:2.7μs


0 2

( , , , 0, ) ( ) ( ,

( , , )

)

, ,R Eb obs C

R b E S A

S

x e P T r

Mt t

s

Dt

参考时刻测站主钟与标准地面时间的差

太阳系内传播效应时间延迟，和相对论时间改正分别对应：“ Roemer”、“ Einstein”、“ Shapiro”效应

双星系统中传播效应时间延迟，和相对论时间改正，分别对应：“ Roemer”、“ Einstein”、“ Shapiro”效应

光行差效应


Roemer 改正ob oe es sb r r r r

ˆ.obct c

r s

oe: observer to earth centeres: earth center to sunsb: sun to solar system barycenterob: observer to solar system barycenter

res：天文单位，由地月质心运动和地球本身的运动决定

rob

res

rsb

roe

Pulsar

s

11

1( )

1sb i iiii

mm

r rrsb：太阳系各行星之和：

roe：观测站的高度

JPL星表

时间改正


广义相对论项改正

1. Time dilation：地球运动及引力红移

S ：随地球椭圆轨道公转的地面原子钟时间t ：距太阳无限远的原子钟协调时

两个时间系统的微分表达：

2

2d 1 11

d 4

GMt

S r a c

r ：日地距离a ：地球轨道半长轴

① r= 常数 a ： 8d

1.48 10d

t

S

② 变化 r ： 10d3.3079 10 cos

d

tf

S

其中 f ：地球实时运动与近日点的角距，非均匀变化：真近点角


原子钟与标准钟之差：

10

0

10 2

3.3079 10 cos d

1 3 13.3079 10 sin sin 2 sin 3 sin

2 8 3

lf l

l e l l e l l

l: 平近点角。，－ el 为常数改正，并入所定义的标准钟，其他变化量表示为相对论改正：

2r

1 1 31.66145 (1 )sin sin 2 sin 3

8 2 8t l e l e l


ms

Sin(i)=0.99975±0.00015Sin(i)=0.99975±0.00015

Shapiro Delay 的残差 23/4/2214 Pulsar Workshop, 2009, NAOC

2. Shapiro delay ：时空弯曲，太阳系时间延迟为：


S 3

2ln(1 cos )

GM

c

θ ：脉冲星 - 太阳 - 地球夹角

太阳附近：最大值 120us木星附近： 200 ns 。行星很小，可忽略

太阳的质量是 1.989133千克，质量使它的四周产生时空扭曲，如果光经过太阳的附近，光发生 1.75“角度的偏转。通过双星系统中的脉冲星也可观测到该现象。

引力作用下的时空扭曲

15 23/4/22Pulsar Workshop, 2009, NAOC

Four General Relativistic Effects—by Andrew Lyne

• 1. Periastron Advance: dω/dt– Due to non-radial force arising from finite speed of

gravity

23/4/22 16Pulsar Workshop, 2009, NAOC

Advance of perihelion of Mercury = 43 arcsec/century

= 0.00012 deg/year

Advance of periastron of 0737-3039 = 16.88 deg/year

Advance of perihelion of Mercury = 43 arcsec/century

= 0.00012 deg/year

Advance of periastron of 0737-3039 = 16.88 deg/year

Four General Relativistic Effects —by Andrew Lyne

• 2. Gravitational Redshift and Time Dilation: γ– Clocks run slow in a gravitational well

e.g. Clocks on Earth run slow and fast by ± 0.0016 sec e.g. Clocks on Earth run slow and fast by ± 0.0016 sec



• 3. Shapiro Delay: “Range” r and “Shape” s– Radiation travels more slowly through a

gravitational well

Small delay

Large delay

Small delay

Large delay

Sin(i)=0.99975±0.00015Sin(i)=0.99975±0.00015



• 4. Orbital Decay: dPb/dt

– Loss of energy through gravitational radiation– Orbital period and size shrink



射电脉冲星实测关键技术 & 数据库脉冲星脉冲到达时间观测是基本观测1 、获得观测起始时刻2 、获得观测时间的视周期3 、数据采样：足够的时间、频率分辨率4 、观测积分折叠5 、消色散（射电观测）提高脉冲轮廓信噪比6 、脉冲轮廓相关，获得 TOA7 、太阳系星历表 TOA归算到惯性系8 、分析，改进的脉冲星自转模型，其他


3200 6

1

2

1)( TTTT

自转模型T 是换算到太阳系质量中心的时间

影响脉冲到达时间的因素 :• 脉冲星自转变化 • 脉冲星位置变化和误差• 色散延迟• 散射• 相对论效应 • 双星系统的轨道运动 • 地球运动• 其他未知因素

1 20 0 0 0 0 0 0 0

1 1( ) ( ) ( ) [ ( )] [ ( )] ......

2 6t t t t t t A t t B t t

残差：观测与模型的差别


如何得到 TOA

脉冲星 PSR B1933+16

Timing Signal

P ， Pdot ， position ， DM

Submitted to MNRAS

周期测量：三次以上的观测周期导数：几个月



位置测量需要至少一年的数据，最高精度达到 0.001arcs自行测量则需要几年的数据，收到达时间噪声影响，若需精确测定PM一般用 VLBI技术

位置位差与自行误差的“ Timing signal”：


PSR J2150+5247 的残差

噪声小的脉冲星，其自转参数、位置、自行等参数的测量也准确


The Shklovsky effect – secular acceleration

对于脉冲星横向速度 V 较大的情形， Doppler 效应造成视周期的变化，甚至可以抵消 slow-down ，时间延迟可表示为：

2 2 2

2 2

a V t

dc dc

观测现象表现为周期导数的变化：21P V

P c d

设 d=1 kpc,V=100km s-1

113.4 10P

P

yr

脉冲星的内禀周期导数与 Shklovsky 效应不易区分，除非自行与距离精确已知对于长周期脉冲星，该效应不明显，毫秒脉冲星则不同

secular acceleration for MSPs：PSR J0437-4715:

12(5.3 0.9) 10bP

a ：轨道半场轴， d ：脉冲星距离

2Obs Int

b b

b b

P P V

P P c d


Gravitational acceleration

obs

aP P

cP

a: 视线方向加速度

考虑银河系引力场、球状星团、伴星中，脉冲星在视线方向被加速，观测到得脉冲星周期导数为：

球状星团中的脉冲星加速明显，甚至表现为脉冲星在加速自转：

球状星团 M15 中的 MSPs PSR B2127+11A 和 PSR B2127+11D ：

位于星团的远端，朝着中心运动，可用来估计星团中心的质量

16obs 2 10P P

脉冲星的摇摆（进动）导致其自转轴随时间的运动轨迹为圆形；就象陀螺的脉冲星的摇摆（进动）导致其自转轴随时间的运动轨迹为圆形；就象陀螺的顶部的运动一样。所以我们就从不同的角度观测锥形辐射束，导致观测到的顶部的运动一样。所以我们就从不同的角度观测锥形辐射束，导致观测到的脉冲形状和脉冲到达时间的变化。脉冲形状和脉冲到达时间的变化。

29

自由进动

23/4/22Pulsar Workshop, 2009, NAOC


自由进动

1. 旋转的物体会表现出自由进动，自转轴与角动量矢量不重合2. 自转轴与磁轴夹角的周期变化 3. 表现为自转速率的长期、准周期变化，可观测到周期导数的变化4. 还会表现为脉冲轮廓的变化5. 进动幅度：对应 0.5—1deg的进动角6. 进动角速度：


Pulsar ages

角频率 Ω ，磁偶极矩 M⊥ 时辐射能量为

磁偶极辐射主导，粒子流。辐射损失角动能21

2I

2 4 32

3M c

2

2 2 4 3

1d( ) 22 sin

d 3

II M c

t

α：磁倾角

19 1 23.2 10 ( ) gaussSB PP 磁赤道表面磁场：

磁极表面磁场： 2Bs2 8 2 1 2

0 ,12( ) ( 3.15 10 )sP t P B t 周期为 P0 的脉冲星周期演化：

假设脉冲星减慢遵循指数变化：n

1 1

( 1) ( 1)

P

n n P

积分上式，得脉冲星特征年龄

n: braking index

磁偶极 n=3 ，星表特征年龄为： Myr2c

P

P

即：

1, in


PSR B0531+21 n=2.515±0.005 (Crab)PSR B1509 － 58 n=2.837±0.001PSR B0540 － 69 n=1.81±0.07PSR J1119 － 6127 n=3.0±0.1PSR B0835 － 45 n=1.4±0.2 (Vela)PSR J1846 － 0258 n=2.65±0.1 (AXP)

The braking index

1. 验证脉冲星磁偶极模型2.脉冲星与超新星遗迹，验证恒星演化3.Crab 的验证： 1054 年超新星爆发

n 求导， 2 2 22

PPn

P

• 仅对六颗年轻脉冲星，测得n ：• 大部分脉冲星受周期噪声影响n 测量值没有物理意义

对

Crab 脉冲星的周期三阶导数可测，并与偶极辐射理论一致！ 3

2

(2 1)n n

Pulsar ages

clock

Matsakis, Taylor & Eubanks, 1997, A&A

• T(1937) and T(1855) are the timescales based on the pulsars PSR B1937+21 and PSR B1855+09, respectively.

• TT96 is a terrestrial atomic timescale. TA(A.1)

• TA(PTB) are free running atomic timescales from the U.S. Naval Observatory and Germany.

Pulsar clock


13 Years of Timing of PSR B1259-63

N. Wang S. Johnston R. N. Manchester


Brief Introduction of PSR B1259-63:

• Discovered in a large-scale high frequency survey of

the Galactic plane (Johnston et. al, 1992a)

• It is the only known pulsar that has a young, massive,

non-degenerate, Be star companion (Johnston et. al,

1992b)

• Characteristic age: ~330 kyr

Period: ~48 ms

Orbital period: ~1237 days -- longest so far

Eccentricity: ~0.87 -- largest known

Periastron: 24 stellar radii (R*)


Companion:

• SS 2883, B2e main sequence star

• 10 solar mass, 6 solar radii

• Equatorial velocity vsin(i) = 280 km/s and runaway

velocity of 80 km/s

• A hot, tenuous polar wind: loss ~10-6 solar mass/yr

• A cooler, high density, equatorial disk:

108—1010cm-3 near the star surface, falls off as power-law

Hα emission at 20 R*, just inside the pulsar orbit (Johnston

et al., 1994 )

Highly tilted with respect to the pulsar orbital plane

PSR B1259-63 eclipse for ~40 days 23/4/2236 Pulsar Workshop, 2009, NAOC

PSR B1259-53

DM

Wex


What’s Pulsar Timing?It measures the pulse arrival time:

PSR B0329+54

T (second)


A rotational model is fitted to data:

3200 6

1

2

1)( TTTT

T is the time at the solar system barycenter

)]([)]([])(6

1)(

2

1)[( 00

20000

100 ttBttAtttttt

Factors that effect pulse arrival time:• pulsar rotation • pulsar position change• dispersion smearing• scattering• relativistic effect • orbital motion in binary system • earth movement

The observed residuals:


Residual of position error Residual of p and pdot error


Interactions of PSR B1259-63 with its surroundings

• Frictional drag from the disk: • Mass accretion: spins-up or slows-down the pulsar, depends on the relative size of Alfven radius and corotation radius • Spin-orbital coupling: spin induced oblateness of the star implies additional 1/r3

gravitational potential (quadrulpole gravitational moment ), introduces precession of the orbital plane

• Tidal effects with the main sequence B2e star: Pb, e, i

ePb ,,

x ,


217105 s

Steps at second periastron19105 s

14 deg102 yr

121015 x

03.0

sx 01.0

6104 e

sPb 30


Earlier Timing Solution by Wex et al. (1998):

Data up to end of 1996, cover 1990 and 1994 periastrons.

Three Solutions:

10

1

10)3(15.0

deg)8(000184.0

x

yrw

10

1

10)4(40.2

deg)9(000396.0

x

yrw

PPPP ,,,

1. Timing noise

2. Spin-orbital coupling: Precession of the orbit


Details of the observation data spans.

Data Preceding T+ T － No. of span Periastron (MJD) (d) (d) TOAs ----------------------------------------------------------------------------------------------------1990.1 － 1990.7 －－ 107 18 1990.7 － 1994.0 48124 171 20 187 1994.0 － 1997.5 49361 24 18 4431997.5 － 2000.9 50597 16 52 237 2000.9 － 2003.5 51834 19 － 146

Altogether 1031 independent observations, obtained on 0.66, 1.4, 2.4, 4.8, 8.4, 13.6 GHz

Observations and Analysis


● Standard profiles

● Pulse profile evolution Differing spectral index

● Alignment of profiles Outer edge


• Dispersion measure variations from 13 years of timing observation.

• The offset is measured relative to the value of 146.8 pc cm-3.

• A mean offset of － 0.2±0.1 pc cm-3 from T+43 of 2000 periastron.

DM variation windISMe DMDMdlnDM


Timing Results

A glitch near MJD 50691 (1997, August 30), 94 days after

1997 periastron.

A glitch is a sudden jump in pulsar spin rate, with relative

amplitude △ν/ν~10-9 － 10-6

Exponential decay with a assumed time scale of 100 days,

we have:

19

19

10)122(

10)167(

s

s

d

g


(a)fitting for and the five Keplerian orbital parameters. Plus the addition of:

(b)

(c)jumps in at each periastron;

(d)jumps in x at each periastron.

Residual:4.3, 4.7, 0.78, 0.46 ms

,,

x ,

,

Keplerian

Spin-orbital coupling

Mass accretion

Orbital jump


Discussion

1. Timing noise: jitter in the pulse arrival times

1.210

)6

||log(

88

3

st

tt

Close to the value of other pulsars with similar frequency derivatives, so the long term timing noise is reasonably well removed

n= － 36.7

Reflect the high level of timing activity in this young pulsar


Problems:

△ν does not have a constant sign at each periastron

Accretion: X-ray obs. show bow shock well outside the

pulsar magnetosphere (Hirayama et al. 1999), require

high density and/or high outflow velocity

Require jumps in frequency derivative

Discussion

2. Steps in Rotational Parameters


Take △x~ 60ms,Kepler’s third law Pb2∝a3 implying large value of

△Pb ~ 0.08d, has to be △i=7″

• Spin-orbital coupling

Pb~3yr, similar to time scale of typical timing noise

Timing noise may mask the true period variation• Frictional drag from the disk

magnitude too small: △Pb =-0.002s, △e=-3X10-13

(Manchester et al. 1995, Wex et al. 1998) The highly inclined disk with respect to the orbit may tilt the orbit with △i=7″• Tidal effects with the main sequence B2e star Depends on the separation of two objects,

PSR J0045-7319: 4R*, PSR B1259-63: 24R*

Enhanced by resonance of the oscillation modes of star

Discussion

3. Changes in Orbital Parameters


sinx a i

Summary

1. We’ve obtained 13 years of timing data for PSR B1259-63, including 4 periastrons.

2. Timing of PSR B1259-63 is complicated: ● Eclipse ● Timing noise ● Glitch ● DM changes ● Interaction with its companion3. Spin-orbital coupling: failed in predicting the latest two orbits 4. Best model: jumps in inclination angle5. Physics implication not clear: tidal effects is the possible

mechanism responsible for the observed period veriation.


脉冲星到达时间 Pulsar Timing

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