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GENERAL RELATIVITY AND PRECISE MEASUREMENTS
OF PULSAR MASSES
D.G. Yakovlev
Ioffe Physical Technical Institute, St.-Petersburg, Russia
• Introduction
• X-ray binaries
• Double neutron star binaries
• Pulsar – white dwarf binaries
• Summary
FFC, Pulkovo Observatory, October 10, 2013
INTRODUCION
Galaxy, stars and the Sun
Galaxy: more than 1011 stars
Luminosity: L~1046 erg/s
Sun: M=2x1033 g, R=700,000 km,
L=3.83x1033 erg/s,
mean density of matter = 1.4 g/cm3,
surface temperature ~6,000 К,
internal temperature 15.7 MК.
Composition: rarefied plasma,
pressure P=nkT ~1017 dyn/cm2.
Supported by thermonuclear reactions
in central region
SCHEME!
WD : M ~ 0.6 M SUN ,
M<8 MSUN
Quiet removal of outer shell,
birth of white dwarf (WD)
R ~ 5000 km,
 ~ 10 g/cm
6
i=isolated
b=binary
WD
i, b
WD
SN Ia
3
b
M=(8—25 ) MSUN
Core-collapsed supernova (SN II)
birth of neutron star
NS
Normal
star
Giant star
i, b
NS
b
BH
NS : M ~ 1.4 M SUN ,
R ~ 10 km,
 ~ 1015 g/cm 3
BH
BH : R  2GM / c 
3 M / M SUN km
2
M>25 MSUN
collapse into black
hole (BH)
WD, NS, BH =
graveyard
Extreme Physics Problem: EOS, High B, High Tc
M ~ 1.4M SUN ,
R ~ 10 km
Main mystery:
EOS of super-dense core –
longstanding fundamental
problem of physics and
astrophysics
complicated by high B and Tc
U ~ GM 2 / R ~ 5 1053 erg ~ 0.2 Mc 2
g ~ GM / R 2 ~ 2 1014 cm/s 2
Main practical problem:
How to relate EOS to
observables
  3M /(4 R3 )  7 1014 g/cm3 ~ (2  3) 0
0  2.8 1014 g/cm3  standard density of nuclear matter
Nb ~ M / mN ~ 1057 = the number of baryons
In our Galaxy:
there are ~ 108  109 neutron stars
observed ~ 2000 neutron stars
MOTIVES TO ACCURATELY MEASURE NS MASSES
•
Мass – most important parameter of any star
•
To find critical mass which separates NSs and BHs
•
To constrain EOS of superdense matter in NS core
Most massive NSs are most important!
X-ray binaries
Companion in
binary system
NS
Riccardo Giacconi
Nobel Prize: 2002
Kepler Orbits
M 1 , M 2 , a1 , a2 , e
M  M 1  M 2 , a  a1  a2
a1  aM 2 / M , a2  aM 1 / M
Integrals of motion:
E  GM1M 2 /( 2a), J 2  GM12 M 22 a(1  e 2 ) / M
Orbital period:
Pb  2 / b , b2  GM / a 3
Measuring radial velocities of companion 1:
Pb , e, K1 
b x1
1 e
2
 x1  a1 sin i,
Need more
parameters:
( M 2 sin i)3 x13b2
f1 

2
M
G
2
Measuring radial velocities of companion 2:
K2,
f2
1
Vela X-1
Vela X-1 (=4U 0900--40)
GP Vel (=HD 77581, B0.5 Ib supergiant)
Pspin=283 s, Pb=8.96 d, e=0.09
a=50 Rsun, i>70o, R2=30 Rsun
Discovery: Chodil et al. (1967)
GP Vel: Brucato & Kristian (1972),
Hiltner et al. (1972)
K2 for GP Vel: Hiltner et al. (1972)
P for Vela X-1: McClintock
et al. (1976)
K1 for Vela X-1: Rappaport
et al. (1976)
Quaintrell et al. (2003):
M 1 (1 )  2.27  0.17 M
for i  70
M 1 (1 )  1.88  0.13 M
for i  90
Masses of Neutron Stars in X-ray Binaries
SUMMARY: NEUTRON STAR MASSES IN X-RAY BINARIES
(1) There is a wide spectrum of
neutron star masses in XRBs
(2) XRBs almost certainly
contain massive neutron
stars
(3) The best candidates are
Vela X-1 (M>1.62 MSUN)
Cyg X-2
4U 1700—37
(4) The prospects to accurately
measure M are poor
Spin
axis
Radio Pulsars in
Compact Binaries
L
Relativistic Objects: Radio Pulsar – Compact Companion
Advantages:
(1) Very precise timing P(t)
(2) Point-like masses
(3) GR effects
da
64G 3 M 1M 2 M  73 2 37 4 
 5 3
1 e  e 
2 7/2 
dt
5c a (1  e )
96 
 24
Peters & Mathews (1963), Peters (1963)
Energy and orbital momentum:
de
304eG 3 M 1M 2 M  121 2 

e 
1 
dt
15c 5 a 4 (1  e 2 ) 5 / 2  304 
dE
32G 4 M 12 M 22 M  73 2 37 4 
 5 5
1  e  e ,
dt
5c a (1  e 2 ) 7 / 2  24
96 
dPb
3 da
 Pb
,
dt
2a dt
7/2
2
1
2
2
1/ 2
dJ
32G M M M

dt
5c 5 a 7 / 2 (1  e 2 ) 2
 7 2
1  e .
 8 
Evolution of orbital parameters:
3 b GM
3 5b / 3 (GM ) 2 / 3
d


2
2
dt a (1  e )c
(1  e 2 )c 2
Example: Timing of pulsars and NS mass measurements
Stage 1: Measurements of Keplerian parameters
Pb , K1 , e, x1 ,  , f1
: 2 extra equations are required
Stage 2: Measurements of relativistic parameters
d / dt
(a) Pereastron advance:
(e  0)  M  M 1  M 2 ; M 1MAX ; M 1MIN
(b) Transverse Doppler effect + gravitational dilation of signals by М2:
v 2 GM 2

2c 2 r12c 2
  
eGM 2 ( M 1  2M 2 )
(e  0)
 b c 2 aM
(c) Shapiro parameters:
 b2 / 3 M 2 / 3 x1
GM 2
s  sin i 
,
r

G1/ 3 M 2
c3
(d) Orbital decay:
(i  90 )
dPb / dt
Up to 5 extra equations can be obtained !
.
Russel Hulse and Joseph Taylor
The Arecibo 305-m radio telescope
(NAIC-Arecibo Observatory, NSF)
The Hulse-Taylor Pulsar (PSR B1913+16)
Discovery: 2 June 1974 (ApJ Lett, January 15, 1975)
5083 observations from 1981 to 2001
Nobel Prize: 1993
Orbit:
e  0.617, a  2 106 km, i  470
vmax  400 km / s, P  59 ms, Pb  7.75 hrs
Relativistic effects (Weisberg & Taylor, 2010) :
.
(a)
d / dt  4.226598  0.000005 deg/ year
Rotation by 125о in 30 years (Mercury: 43’’ in 100 yrs)
(b)
(c)
  0.0042992  0.0000009 s
Observations:
Theoretical
prediction:
dPb / dt  (2.398  0.005)  1012 s / s
dPb / dt  (2.402531  0.000014) 1012 s / s
The mass of the Hulse-Taylor Pulsar (PSR B1913+16)
MASSES OF
PSR B1913+16
& COMPANION
(Weisberg, Nice, Taylor, 2010)
M1 (2 )  (1.4398  0.0004) M SUN
M 2 (2 )  (1.3886  0.0004) M SUN
In M SUN !!!
Evolution of the Hulse-Taylor pulsar
.
t PSR  Pspin / 2 Pspin  100 Myrs;
At birth:
Now:
tdeath  300 Myrs (1640 Myrs if e  0)
e  0.666, a  2.3 1011 cm, Pb  9.93 hr , d / dt  3.12 deg/ yr
e  0.617, a  2.0 1011 cm, Pb  7.75 hr , d / dt  4.23 deg/ yr ,
LG  7.77 1031 erg / s
In 200 Myr:
e  0.439, a  1.2 1011 cm, Pb  3.64 hr , d / dt  11.5 deg/ yr
The last 10 Years of the Hulse-Taylor Pulsar
Time to merging = 300 Myr
M31
10 years before death:
e  0.00081, a  17300 km, Pb  23 s, d / dt  39.6 deg/ hr,
LG  1.2 1041 erg / s
1 ms before death :
a  40 km, Pb  1 ms, LG  1055 erg / s
Geodetic precession of the Hulse-Taylor pulsar
Barker & O’Connell (1975):
 prec   b
 prec  1.21deg/ yr , Pprec  300 yrs
ton  1940; tout  2025; tout  240 yr
 ( spin , prec )  22 ;  ( spin , B)  27 
3GM 2 
M1 
1


2
2 
ac (1  e )  3M 
Ideal Wolszczan Pulsar (PSR B1534+12)
Discovery: Wolszczan (1991)
P  37.9 ms, Pb  10.1 hr , e  0.274, d / dt  1.76 deg/ yr
i  770 
All 5 GR parameters measured:
d / dt ,  , dPb / dt , s, r
Neutron star masses (Stairs et al. 2003):
M1 (2 )  (1.3332  0.0020) M SUN
M 2 (2 )  (1.3452  0.0020) M SUN
J0737-3039 A and B: Double Pulsar Binary
Burgay et al. (2003)
PulsarА
Observation:
4.5 min in August 2001 + systematic observations since 2003 (5 months)
P  22.7 ms, Pb  2.45 hr , e  0.0878, d / dt  17 deg/ yr
M  (2.58  0.02) M Sun
Pulsar B
Lyne et al. (2004)
Systematic observations since May 2003 (7 months)
P  2.773 s,
f 2   ; r , s  i  87 
M 1 (1 )  (1.337  0.005) M Sun , M 2 (1 )  (1.250  0.005) M Sun
Results:
t death  86 Myrs 
Fifth binary with short lifetime
t prec1  75 yrs, t prec2  71 yrs
Radio eclipses

Double Neutron Star Binaries
MASSES OF DOUBLE NEUTRON STAR BINARIES
• 5 DNSB = 10 neutron star
masses accurately measured
• All masses are in narrow range
• HT pulsar is most massive
among them
• No recent progress with these
objects
RADIO PULSARS AND WHITE DWARFS
(or other compact companions)
Advantages:
• Compact stars – point-like masses
• Often – recycled millisecond pulsars:
pulsars can be massive,
short periods – good timing,
weak magnetic fields – no glitches or pulsar noise
Disadvantages:
• Underwent active accretion phase – as a rule, almost circular orbits =
difficult to measure periastron advance and gamma-parameter
• Low-mass companions – difficult to measure Shapiro effect
and dPb/dt
Specific feature:
• Often observed in globular clusters
Neutron Stars and White Dwarfs
White dwarfs: M2—Pb
Neutron Stars and White Dwarfs
Ideal System
Radio Pulsar—White Dwarf (PSR J1141—6545)
Discovery: Kaspi et al. (2000)
P  394 ms, Pb  4.75 hr , e  0.172, d / dt  5.3 deg/ yr
i ~ 760
Three GR parameters measured:
d / dt ,  , dPb / dt
Masses (Bailes et al. 2003):
PSR: M1 (2 )  (1.30  0.04) M SUN
WD: M 2 (2 )  (0.99  0.04) M SUN
Ideal Binary
Radio Pulsar—White Dwarf (PSR J1909—3744)
Discovery: Jacoby et al. (2003)
P  2.9 ms, Pb  1.53 d , e ~ 107 , i  86.6
Two relativistic parameters measures: s, r
Masses of stars (Jacoby et al. 2005):
PSR: M1 (1 )  (1.438  0.024) M SUN
WD: M 2 (1 )  (0.2038  0.022) M SUN
Fallen Down Angel
Radio Pulsar—White Dwarf (PSR J0751+1807)
Discovery: Lundgren et al. (1995)
P  3.48 ms, Pb  6.3 hr , e  0.000003
One relativistic parameter measured: dPb/dt
Shapiro effect is poorly pronounced: i~65-850
Masses of companions (Nice, Splaver, Stairs 2004, 2005):
PSR: M1 (2 )  2.10.4
0.5 M SUN
WD: M 2 (2 )  (0.19  0.03) M SUN
After 2007 (Nice, Stairs, Kasian 2008):
PSR: M1 (2 )  (1.26  0.28) M SUN
WD: M 2 ~ 0.2 M SUN
Radio Pulsar—White Dwarf (PSR J1911—5958A)
Discovery: D’Amico et al. (2001)
P  3.26 ms, Pb  0.84 d , e  0.000003
No relativistic parameters measured
Bassa et al. (2006), Cocozza et al. (2006) – radial velocity curve
and mass of white dwarf are measured in optical observations
PSR: M1 (1 )  1.400.16
0.10 M SUN
WD: M 2 (1 )  (0.18  0.02) M SUN
PSR J1903+0327 (2009)
Discovery: Cordes et al. (2006)
P  2.15 ms, Pb  95 d , e  0.44
The first eccentric binary MCP in the galactic disk
Companion: MS star, M~1 MSUN
Evolutionary scenario: unclear
Measured: periastron advance + s, r
PSR: M1 (1 )  1.67  0.01 M SUN
MS: M 2 (1 )  1.028  0.004 M SUN
Problem: large size of companion can affect periastron advance
Perspective: timing, refined measurements of
periastron advance, s, r
Most Massive Known Neutron Star
PSR J1614-2230 + WD
28 0ct. 2010, Nature 467, 1081
Discovery: 2002 (Hessels et al. 2005)
P  3.15 ms, Pb  8.69 d , e  1.3 106 , i  89.17o
Measured: Shapiro effect, s, r
PSR: M1 (1 )  1.97  0.04 M SUN
WD: M 2 (1 )  0.500  0.006 M SUN
Most massive neutron star currently known
Most Massive Known Neutron Star
Time residual, microseconds
Shapiro delay in PSR J1614-2230 + WD
0
0.5
Orbital phase
Demorest et al. (2010)
1.0
THE SECOND MOST MASSIVE NEUTRON STAR
PSR J0348+0432 + WD
Science, 26 April 2013, Vol. 340, Issue 6131, 448
Radio observations:
Green Bank (USA) 2007
Publication: Lynch et al. (2013)
P  39 ms, Pb  2.46 h,
i  40.2 , d  2.1 kpc
Pulsar: moderately spun up by accretion
WD: low-massive, He core
Age of the system: about 3 Gyrs
Measured: radial velocities of PSR and WD and
spectroscopic WD mass
o
THE SECOND MOST MASSIVE NEUTRON STAR
PSR J0348+0432 + WD
PSR: M1 (1 )  2.01  0.04 M SUN
WD: M 2 (1 )  0.172  0.003 M SUN
Measured without
GR effects
Checked by orbital decay:
Theory
13
dPb / dt  2.580.07

10
0.11
Observations
dPb / dt  ( 2.73  0.45)  1013
Time to merging: 400 Myr
Ideal binary for checking GR!
Summary of NS-WD and
NS-NS binaries
Kiziltan et al. (2013)
MOST MASSIVE NEUTRON STAR VERSUS TIME
PSR J0751+1807
PSR J1614—2230
PSR J0348+0432
PSR J1903+0327
PSR B1913+16
Mass—Radius Diagram for Exploring EOS of Superdense
General
Relativity
Causality
PSR J1614-2230
PSR J0348+0432
HT pulsar
RESULTS
•
General Relativity Theory was tested
Gravitational radiation discovered
Geodetic precession discovered
Double neutron star mergers were discovered
Gravitational observatories of new generation are built
General Relativity has become useful tool
Masses of some neutron stars accurately measured
Currently: Mmax>2 MSUN
=> soft and moderate EOSs are ruled out
=> EOS is stiff => little room for exotic matter
•
•
•
•
Main feature at present: Rapid progress!
Unsolved Problems
•
MMAX = ?
•
Stiff EOS = just stiff or superstiff?