Transcript Slide 1

A Color-Magnitude diagram
of Galactic Double White
Dwarfs for LISA
Ravikumar Kopparapu
IGPG, PennState
Joel E. Tohline (LSU)
Vayujeet Gokhale (Truman State)
Sep 25, 2007
Outline
Optical and GW sources
 Properties, evolution
 Boundaries on GW amplitude-frequency
plane
 Population synthesis (Joel, vayujeet)
 Future work

Optical and GW sources
Two phases of CE evolution of MS stars.
 Possible progenitors of Type Ia
supernovae (standard candle)
 Candidates for AMCVn systems
 GW sources for LISA


Properties
 Radius proportional to negative power of mass.
R M 


R  M 


1
 3
Max. mass Mch = 1.44 M⊙ (Chandrasekhar mass limit)
Evolution
 Inspiral evolution (angular momentum loss due to GWs
decreasing separation)
 Mass transfer evolution (increasing separation; GW is still
the driving mechanism)
Circular orbit
M2
M1
R2
R1
a
f2
J orb
GM tot
4 2 a 3
(Ga)1 / 2
 M 1M 2
M tot
COM
M
q
tot
 M
M2
M1
1
M
1
2
Evolution : Inspiral

From GR quadruple
radiation formula
h  hnorm cos(2ft)
h  hnorm sin(2ft)
hnorm
4G 2 M 1 M 2

a
rc 4
Angular momentum carried away by gravitationalwaves
1/ 8
J orb

t 

 J 01 


ch 

 ch
a0
(1  q) 2

3
q
256G 3 ( M tot )
5c 5
4
hnorm  ( f 2 / 3 )
Evolution : Mass Transfer
Roche Lobe
L1
M2
M1
Evolution : Mass Transfer

M2 
a 2 J 2( M 2 ) 
1 



a
J
M2 
M1 
Stable and Unstable mass transfer
Webbink & Iben, 1987
Optically observed Detached DWD
(Nelemans, et al. 2000)
Optically Observed Mass-Transferring
Systems
(Nelemans, 2005)
Total Galactic population of DWD ~ 108
DWDs on LISA
Analogy with
Color-Magnitude Diagrams
“Color-Magnitude” versus “Strainfrequency”
Measured
“Color-Magnitude” versus “Strainfrequency”
high
low
Range of Black-body (maximum)
frequencies
high
low
Range of GW frequencies
“Color-Magnitude” versus “Strainfrequency”
Limiting Magnitude
Limiting Detectable Strain
Detection vs. Astrophysics
So, as with a color-(apparent)magnitude
diagram, the strain-frequency diagram is
useful for discussing the detectability of
sources, given a certain instrument design
and observation time.
But to discuss the (astro)physical properties
of individual sources, their evolution, and
various source populations, we really need
to utilize a Color-(Absolute)Magnitude
diagram.
Distances unknown
Distances known
Color-Magnitude Diagram
-10
-8
Absolute Magnitude (M)
-6
-4
-2
0
2
4
6
8
-0.5
0
0.5
1
B-V
1.5
2
Conversion to absolute-magnitude
Measured
Measured
MV  mV  5 log(r )
2.5 log(L) ~ 2.5 log(flux)  5 log(r )
hnorm
4G 2 M 1 M 2

a
rc 4
 4G 2 M 1M 2 
  log(hnorm )  log(r )
log 4
 c
a 

 log(rhnorm )
DWD evolutionary trajectories in
“absolute” strain-frequency domain.
Inspiral evolution
J orb
(Ga)1 / 2
 M 1M 2
M tot
GM tot
f2
4 2 a 3
hnorm  ( f 2 / 3 )
hnorm
4G 2 M 1 M 2

a
rc 4
Inspiral evolution
Less massive WD fills Roche lobe first
Ma = Mch = 1.44 M⊙
Log[ rhnorm (meters) ]
0
Mtot = 2.4
q = 2/3
-0.5
-1
-1.5
Mtot = 1.4
Mtot = 0.8
-2
-2.5
-3
-3.5
-4
-3.5
-3
-2.5
-2
Log[ f (Hz) ]
-1.5
-1
-0.5
0
Termination boundary
(start of mass transfer)
Isochrones
Mass transfer evolution
Log[ rhnorm (meters) ]
0
Mtot = 2.4
-0.5
-1
Mtot = 1.4
Mtot = 0.8
-1.5
-2
-2.5
-3
-3.5
-4
-3.5
-3
-2.5
-2
Log[ f (Hz) ]
-1.5
-1
-0.5
0
Key Population Boundaries
(for given “q”)
Boundaries with respect to
the LISA Noise Curve
DWD systems at r = 10 kpc
Bounds on DWD population
stability curve
Ma = Mch
Type Ia
I
II
Kopparapu & Tohline (2007), ApJ, 655, 1025
DWD Population on “absolute”
GW strain-frequency domain.
Joel Tohline (LSU)
Vayujeet Gokhale (Truman State)
Goals



Generate a population of DWDs using population
synthesis code (Hurley et al. 2002, MNRAS) with
a main-sequence initial mass function (Kroupa et
al.(1993)
Develop an analytic and or/numerical formalism
to describe the evolution of DWDs (Marsh et al.
2004, Gokhale et al. 2007)
Describe on the basis of above points, the
behavior of systems on the GW “C-M” diagram.
Initial DWD population from a
6
single burst of 10 MS binaries.
Evolved DWD systems (~1010 Yrs)
Future work
Dependence on different initial MS
parameters (metallicity, continous star
formation rate etc.)
 Composition dependence on initial
parameters
 Relative no.of systems in different subdomains
 Constructing steady state DWD
populations (independent verification?)

Part II
Goal : Gravitational-wave templates
 Region III systems (unstable mass
transfer).
 Spin of the stars, tidal and rotational
effects
 Mass transfer
 Compare with three-dimensional
hydrodynamic simulations.

Finite-size effects
 Jtot = Jorb
+ J a + Jd
Ja = Ia Ω = ka2 Ma Ra2 Ω
Jd = Id Ω = kd2 Md Rd2 Ω
ki2 is the radius of gyration that includes tidal
and rotational distortions.
Evolution of the system

Part I assumes constant mass transfer rate (dM/dt).
In general, dM/dt has more complex behavior.
Webbink & Iben, 1987
Results: Mass Transfer Rate
Red – Hydrodynamic Simulation
Blue – Model
For q = 0.744
For q = 0.409
Results: Mass Ratio as a Function of Time
For q = 0.744
For q = 0.409
Results: Gravitational-wave amplitude
No. of orbits matched
= 8.4
No. of orbits matched
= 8.0
Summary
DWD population - Important sources of
gravitational waves
 Huge number of galactic binaries
 Evolution - Inspiral and Mass Transfer
 Limits on existence of DWD population in
the amplitude-frequency domain
 Gravitational-wave templates for masstransferring systems
 Illustration by two models
 More exciting work !

Integration time
Assume that a sufficient degree of phase
coherence is maintained if the observed
(“O”) phase minus the theoretically
computed (“C”) phase does not differ by
more than pi/2.
The time duration in which this phase
coherence is maintained is the integration
time (tO-C ).
Definitions
Md
q
1
Ma
Mto t  Ma  Md
1/ 2
(Ga)
Jorb  MaMd
Mtot
orb
2
GM tot
 3
a
hnorm  h  2 h  2
4G 3 ( Mtot ) 5 q 3
hnorm  4
2
rc ( Jorb) (1  q ) 6
f 
G 2 ( Mtot ) 5
q3
 ( Jorb) 3 (1  q) 6
Inspiral
1/ 8
t 

Jorb  J 01  
 ch 
Mass Transfer
a 0 4 (1  q) 2
ch 
3
q
256G 3 ( Mtot )
5c 5
a 2 J 0 2( M d ) 
Md 


1 

a
J
Md 
Ma 
hnorm ( f 2 / 3 )
Overview
Part I
 Double White Dwarf (DWD) binaries (properties,
evolution).
 Boundaries in Gravitational-Wave AmplitudeFrequency domain.
Part II
 Initial Model.
 Mass Transfer Evolution.
 Results.
Summary
Initial model
θ = ρ/ρc = undistorted density
Θ = distorted density
Θ = θ + δ(ξ), where δ(ξ) is a correction
Moment of Inertia
I = ρc A5 ∫Θn ξ4 dξ dφ Sin(θ) dθ
k2 = I/MR2
This initial value of k2 is kept constant
throughout the evolution.
q = 0.744
q= 0.409
 0(t )  2

t
0
 1
2
f 0[1  f ' t ]dt  2 t  f ' t 
 2

q=1
q = 2/3
q = 0.2
Mass-Radius Relation and Roche Lobe
Radius
 M 
R
 0.0114

R⊙
 Mch 
2 / 3
M 


 Mch 
2/3


1/ 2

M 
 1  3.5

 Mp 

RL 
0.49q 2 / 3

a  0.6q 2 / 3  ln 1  q1 / 3

3 

2
2 a

P  4
 GMtot 


2 / 3




1 
M 

 
 Mp  
2 / 3
Binary Star System
M2
M1
R2
R1
COM
a
orb 
2
GMtot
a3
Mtot  M 1  M 2
Double White Dwarf system
M2
M1
R2
R1
COM
a
orb 
2
GMtot
a3
M tot  M 1  M 2
M2
q
1
M1
Evolution
Einstein’s General theory of relativity :


Gravity → Curvature of space-time fabric produced by
matter
Gravitational-waves → Ripples on space-time produced
by accelerated matter
Animation by William Folkner, LISA project, JPL
Evolution : Inspiral of DWD


Image Credit :NASA/CXC/SAO
http://chandra.harvard.edu/photo/2005/j0806/animations.html
LISA (Laser Interferometer Space Antennae)
Image courtesy: http://lisa.nasa.gov/
Optically observed Detached DWD
(Nelemans, et al. 2000)
hnorm
f (Hz)
7.8 x 10-20
3.8 x 10-4
Optically Observed Mass-Transferring
Systems
hnorm ~ 10-21 ,
f ~ 2 x 10-3 Hz
(Nelemans, 2005)
Total Galactic population of DWD ~ 108
Template

A theoretical model of the gravitationalwave signal from the source, which shows
the variation of the amplitude and
frequency with time.
h× = hnorm sin(2πft)
“Color-Magnitude” versus “Strainfrequency”
Nauenberg (1972)
Eggleton (1983)