Transcript The Sun and the Stars

The Sun and the Stars
The Sun and the Stars
Dr Matt Burleigh
The Sun and the Stars
Binary stars:
Most stars are found in binary or multiple systems.
Binary star systems consist of 2 stars which are gravitationally bound with each star
orbiting a common centre of mass. We can distinguish between several different types






Apparent – chance alignment – not true binaries
Visual – resolved binaries (individual components can be separated visually)
>1” ,generally long orbital periods
Astrometric – unresolved, companion identified by stellar wobble
Spectroscopic – unresolved, other component revealed by period shift in spectral lines
Spectrum – unresolved – spectral decomposition reveals two stellar components
Eclipsing- systems which show periodic dips in their apparent brightness
(systems may also be visual, astrometric or spectroscopic)
[check out eclipsing binary simulator at http://astro.unl.edu/naap/ebs/animations/ebs.html]
The most important of these are visual, spectroscopic and eclipsing binaries
Why are binaries important?

because analysis of their orbits allow us to determine the masses of the individual stars,
their radii and shape (particularly in eclipsing systems), and the physical characteristics of
the systems (separations, periods)
.
Dr Matt Burleigh
The Sun and the Stars
Visual binaries – in the optical, require separations of > 1” from the ground,
otherwise components are unresolved.
Examples – alpha-Cen A and B, Sirius A and B
The angular separations and orbital paths are only apparent
because in general the orbit is inclined to the plane of the sky,
so we see the orbit in projection
Measuring the displacement of the primary relative to the apparent focus, yields the orbital inclination, i,
the true ellipticity e, and the true semi-major axis a”
Dr Matt Burleigh
The Sun and the Stars
e.g. consider the following:
m2
r2
a1
a2
r1
m1
From Kepler’s III law we have
(a1  a 2 ) 3
G (m1  m2 )  4
P2
2
where m1 and m2 are the masses of the 2 components,
and a1 ,a2 are the semi-major axes of their orbits
In the case of the Earth-Sun system, mSun>>mEarth, and the common centre of mass is located
within the stellar radius, i.e. a1>>a2
Expressing the masses in solar masses and orbital radii in AU, then P has units of years, and
thus the general form of Kepler’s third law can be written:
m1  m2 
a3
P2
If we express the separation between the binary components in seconds of arc, ”, then a   "
"
q "3
m1 + m2 = "3 2
p P
Dr Matt Burleigh
The Sun and the Stars
Since
m1r1  m2 r2
,
where r1+r2=a
to determine the individual masses, we must find
the relative distance of each star from the centre
of mass of the system.
NB In proper motion, the centre of mass travels
along a straight line relative to background
stars (see e.g. Sirius A and B)
Proper motion of the visual binary Sirius A and B
relative to background stars
Dr Matt Burleigh
The Sun and the Stars
Spectroscopic binaries
Two unresolved stars, separation 1AU, Period ~ hours to months, inclination i>0.
Binaries exhibit lines (in absorption or emission) that show periodic variations.
Systems may be:
(i) single-lined (only one component displays lines) or
(ii) double-lined (both components display lines)
Lines are shifted in wavelength by an amount 
relative to the rest-wavelength 0, ,blueward (star
approaching), and redward (star receding)
[doppler effect], such that

0

  0 
0

vr
c
Detection of shifts limited by spectral resolution
for two stars  vr ~ km/s
for a planet/star vr ~ m/s
Dr Matt Burleigh
Spectroscopic Binaries
Dr Matt Burleigh
The Sun and the Stars
Radial Velocity curves
Constructed by converting wavelength shifts to velocity shifts as a function of time, folded
on the orbital period e.g.
Radial velocity curves for nearby binary stars
**Radial velocity curve for a hot Jupiter**
Dr Matt Burleigh
The Sun and the Stars
The simplest radial velocity curves are from those systems viewed edge-on (i=90 degrees).
They appear sinusoidal with opposite phases e.g.
In this case, each star orbits around the centre of mass with orbital period P,
so
r1 
v1 P
2
and r2  v2 P
2
The ratio of the stellar masses is given by
The relative semi-major axis is
and
m1  m2 
m1 r2 v2
 
m2 r1 v1
a  r1  r2
a3
P2
The system is completely determined!!
Dr Matt Burleigh
The Sun and the Stars
This rarely ever happens because
(i) The system may be single lined (can only determine P and r1)
(ii) Unless the system is also eclipsing we don’t know the inclination
3
m 2 sin 3 i
( m1  m 2 ) 2
If (i) is true then we can only quote the mass function
Why?
Recall
then
So
m1  m2 
a3
P2
and
m1r1  m2 r2

mr
(m1  m2 ) P  (r1  r2 )   r1  1 1

m2

2
3
3


m
  r13 1  1

 m2




3
 m  m1 

 r1  2
m
2


3
3

r1 '3 (m 1  m2 ) 3
sin 3 im 2
3
r1' = r1 sini
i is the inclination
m3 sin3 i
(r1' )3
f (m1, m2 ) = 2
=
(m1 + m2 )2 P 2
NB if the primary mass m1 can be obtained from the spectral type, the system can be solved.
More generally the system will be inclined. If the radial velocity curve is sinusoidal, we know
we are dealing with circular orbits in which case we measure the projected velocity Vr sini
for each component.
In the case of elliptical orbits, the velocity curves are no longer sinusoidal.
Although radial velocity curves are mirror images, they may have differing amplitudes
Dr Matt Burleigh
The Sun and the Stars
Eclipsing binaries
Close binary systems (small separations and short periods) in which one star passes in front of
the other periodically blocking some of the light. For each orbit there will be two eclipses, a
primary eclipse (when the primary star is eclipsed by the secondary and a secondary eclipse
wherein the primary passes in front of the secondary (by convention, the hotter star is
designated the primary, the cooler star the secondary).
Eclipses can be either total or partial e.g. SV Cam
HIP 59683
Dr Matt Burleigh
Eclipsing Binaries
Dr Matt Burleigh
The Sun and the Stars
Note that the type of eclipse observed, depends upon
the orbital eccentricity and inclination and the stellar
radii and surface temperatures.
Rc
Rp
Conditions for eclipse:
(i)
(ii)
 cos i  R p  Rc
no eclipse

R p  Rc   cos i  R p  Rc partial eclipse
(iii)  cos i  R p  Rc
Rp+Rc

total and annular eclipse
NB =90-i
From timing the points of contact we can estimate the relative stellar radii Rp /a, and Rc /a
From the relative depths of the eclipses we can estimate the relative effective surface temperatures
Tp /Tc
Dr Matt Burleigh
Accreting binaries
• Cataclysmic variables consist of a
white dwarf and a cool secondary
(usually an M dwarf)
• Periods of 1.5 to a few hours
• Material is accreted via Roche Lobe
Overflow into a disc surrounding the
white dwarf
• Occasionally the disc suffers a
thermonuclear detonation when too
much material has accumulated
• Observed as novae
• See also Xray Binaries (accretion
onto a neutron star or a black hole,
eg Sco X-1)
Dr Matt Burleigh
Accretion in Binaries
Dr Matt Burleigh
The Sun and the Stars
Additional notes – derivation of Keplers III law
2
Gm1 m2
mv
mv
 1 1  2 2
2
r1
r2
(r1  r2 )
2
Balance between gravity and centripetal force
Relocate to frame of one of the masses and replace mass with reduced mass
Since r  r1  r2
,then
m1 m2
m1  m2
Gm1 m2 v 2

r
r2
The period of the orbit T, is
So,

T
2r
v
Gm1 m2  (2r ) 2

r2
rT 2
and
T2 
4 2 r 3
Gm1 m2
therefore
T2 
4 2 r 3
G (m1  m2 )
Dr Matt Burleigh