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PH507
Multi-wavelength
Professor Michael Smith
1
WEEK 2: LECTURE 4
Standard Candles
1 Spectroscopic Parallax
The term spectroscopic parallax is a misnomer as it actually has
nothing to do with parallax.
Hertzsprung-Russell deduced the main-sequence
nearby stars, relating their luminosity to their colour.
for
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Multi-wavelength
Professor Michael Smith
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Groups of distant stars should also lie along the same mainsequence strip. However they appear very dim, of course
due to their distance. On comparison of fluxes, we
determine the distance. This works out to about 100,000 pc,
beyond which main-sequence stars are too dim.
If we take a spectrum of a star we can determine: its spectral
class.
Using photometry we can measure
the apparent magnitude, mV .
If we use B and V filters we can also determine the blue apparent
magnitude, B and thus determine:
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Multi-wavelength
Professor Michael Smith
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the star's colour index, CI = B - V.
Knowing either the star's spectral class or colour index allows us
to place the star on a vertical line or band along a HertzsprungRussell Diagram.
If we also know its luminosity class we can further constrain its
position along this line, that is we can distinguish between a red
supergiant, giant or main sequence star, for example.
Once we know its position on the HR diagram we can infer what
its absolute magnitude, M V should be by either reading off
across to the vertical scale of the HR diagram or looking it up
from a reference table. A main sequence (luminosity class V)
star with a colour index of 0.0 (ie A0 V) has an absolute
magnitude of +0.9 for example.
Now knowing m from measurement and inferring M we can use
the distance modulus equation:
m - M = 5 log(d/10)
to find the distance to the star, d, in parsecs.
EXAMPLE
Gamma Crucis is an M3.5 III star, a red giant.
Measured values: mV = 1.63 and a colour index of +1.60.
Using its spectral and luminosity classes we can place it where the
red circle is on the HR diagram. Reading across to the vertical
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Multi-wavelength
Professor Michael Smith
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axis this corresponds to an absolute magnitude of about -0.8
….show that the distance is about………?
2 Cepheids as Standard Candles: The Period-Luminosity
Relationship
Cepheids show an important connection between period
and luminosity: the pulsation period of a Cepheid variable
is directly related to its median luminosity.
This relationship was first discovered from a study of the
variables in the Magellanic Clouds, two small nearby
companion galaxies to our Galaxy that are visible in the
night sky of the southern hemisphere.
To a good
approximation, you can consider all stars in each Magellanic Cloud to be at the same distance. Henrietta Leavitt,
working at Harvard in 1912, found that the brighter the
median apparent magnitude (and so the luminosity, since
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Professor Michael Smith
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the stars are the same distance), the longer the period of the
Cepheid variable. A linear relationship was found.
Harlow Shapley recognised the importance of this periodluminosity (P-L) relation-ship and attempted to find the
zero point, for then a knowledge of the period of a Cepheid
would immediately indicate its luminosity (absolute
magnitude).
This calibration was difficult to perform because of the
relative scarcity of Cepheids and their large distances.
None are sufficiently near to allow a trigonometric parallax
to be determined, so Shapley had to depend upon the
relatively inaccurate method of statistical parallaxes. His
zero point was then used to find the distances to many
other galaxies. These distances are revised as new and
accurate data become available. Right now, some 20 stars
whose distances are known reasonably well (because they
are in open clusters) serve as the calibrators for the P-L
relationship.
Further work showed that there are two types of Cepheids,
each with its own separate, almost parallel P-L relationship.
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Professor Michael Smith
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The classical Cepheids are the more luminous, of
Population I, and found in spiral arms. Population II
Cepheids, also known as W Virginis stars after their
prototype, are found in globular clusters and other
Population II systems.
Classical Cepheids have periods ranging from one to 50
days (typically five to ten days) and range from F6 to K2 in
spectral class.
Population II Cepheids vary in period from two to 45 days
(typically 12 to 20 days) and range from F2 to G6 in spectral
class.
Population I and II Cepheids are both regular, or periodic,
variables; their change in luminosity with time follows a
regular cycle. The empirically derived relationship between
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a Type I Cepheid's period P (in days), and its absolute
magnitude Mv is given by
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Cepheids are bright and distant. They can be used to
determine distances to quite distant galaxies, to about 5
Mpc. HST stretched this to 18 Mpc (Virgo cluster).
3 Tully-Fisher Relation
the Tully-Fisher relation, published by astronomers R. Brent
Tully and J. Richard Fisher in 1977, is a standard candle
that measures the distance to rotating spiral galaxies by
the width of the galaxy's spectral lines.
The empirically-derived relation states that the luminosity of
a galaxy is directly proportional to the fourth power of its
rotational velocity, which can be calculated from the width of
the spectral line, and uses the distance modulus to find
distance from luminosity and apparent magnitude.
In a spiral galaxy, the centripetal force of gas and stars
balances the gravitational force:
mV2/R = GmM/R2.
If they have the same surface brightness ( L/R2 is constant)
and the same mass-to-light ratio (M/L is constant), then
L ~ V4. So, provided we can measure the velocity V, certain
galaxies can be used as standard candles. (determine V
through the 21 cm line of atomic hydrogen in the galaxy).
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Professor Michael Smith
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Axes? Brightest Mb = -19, rotation 100 - 500 km/s
4 Type 1a Supernovae.
The peak light output from these supernovae is always
about Mb = -19.33 +- 0.25. Therefore we can infer the
distance from the inverse square law. Being so bright , they
act as standard candles to large distances: to 1000 Mpc.
Why are they standard candles? White dwarfs I binary
systems. Material from a companion red giant is dumped
on the white dwarf surface until the WD reaches a critical
mass (Chandrasekhar mass) of 1.4 solar masses. Explosion
occurs with fixed rise and fall of luminosity.
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Multi-wavelength
Professor Michael Smith
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QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
5 Other methods include:
1. time delay of light rays due to gravitational lensing,
2. cluster size influences Compton scattering of CMB
radiation and bremsstrahlung emission (X-rays).
Combining, yields the size estimate (SunyaevZeldovich effect).
New Method?
Reverse argument: knowing the Hubble constant is 72
km/s/Mpc, (WMAP result), distances can be found directly
from the redshift! ( Nearby: Distance = velocity / Hubble
Constant )
PH507
Multi-wavelength
Professor Michael Smith
Questions
How do we scale the solar system?
How do we find the distance to gas clouds?
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Multi-wavelength
Professor Michael Smith
MASS MEASUREMENT??
PLANET REVIEW
The Solar System
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Professor Michael Smith
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In the picture above we see the positions of the asteroid belt
(green) and other near-earth objects
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The material in the plane of the Solar System is known as
the Kuiper Belt (50 – 1000AU) or Trans-Neptunian
Objects. Surrounding this is a much larger region known
as the Oort Cloud (3000 – 100,000 AU), that contains
material that occasionally falls in, under the influence of
gravity, towards the Sun as comets.
The Sun
over 1.4 million kilometers (869,919 miles) wide.
contains 99.86 percent of the mass of the entire solar
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system: well over a million Earths could fit inside its bulk.
total energy radiated by the Sun averages 383 billion
trillion kilowatts, the equivalent of the energy generated
by 100 billion tons of TNT exploding each and every
second…………. 3.83 1026 W
Planetary configurations
• Some of the definitions below make the assumption of
coplanar circular orbits. True planetary orbits are
ellipses with low eccentricity and inclinations are small
so the concepts are applicable in real cases.
• Copernicus correctly stated that the farther a planet lies
from the Sun, the slower it moves around the Sun.
When the Earth and another planet pass each other on
the same side of the Sun, the apparent retrograde loop
occurs from the relative motions of the other planet and
the Earth.
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Professor Michael Smith
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 As we view the planet from the moving Earth, our
line of sight reverses its angular motion twice, and
the three-dimensional aspect of the loop comes about
because the orbits of the two planets are not coplanar.
This passing situation is the same for inferior or
superior planets.
 A Retrograde loop occurs when a superior planet moves
through opposition, and occurs as the earth's motion about
its orbit causes it to overtake the slower moving superior
planet. Thus close to opposition, the planet's motion relative
to fixed background stars, follows a small loop.
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Multi-wavelength
Professor Michael Smith
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Lecture 5: Measuring Mass
Mass can be measured in two ways.
1 We could count up the atoms, or count up the
molecules and grains of dust and infer the number
of atoms. This method can be used if the object is
optically thin and we have good tracer: a radiation
or scattering mechanism in which the number of
photons is related to the number of particles.
2 Otherwise, measuring the mass of an object relies
upon its gravitational influence….on nearby
bodies or on itself (self-gravity).
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Multi-wavelength
Professor Michael Smith
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Newton’s second law states: F = m a, while the first
law relates the acceleration to a change in speed or
direction (law of inertia). (Third law; action –
reaction = 0)
Kepler’s empirical laws for orbital motion describe
the nature of the acceleration from which masses
can be derived.
Kepler's Laws
First Law: The orbit of each planet is an ellipse
with the Sun at one focus
p
b
F
C
Q
r
S f q
a
S = Sun, F = other focus, p = planet.
r = HELIOCENTRIC DISTANCE.
f = TRUE ANOMALY
a = SEMI-MAJOR AXIS = mean heliocentric
distance), which defines the size of the orbit.
b = SEMI-MINOR AXIS.
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Multi-wavelength
Professor Michael Smith
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e = ECCENTRICITY,
defines shape of orbit.
Ellipse, by definition:
SP + PF = 2a
(1)
e = CS / a
(2)
Therefore
b2 = a2(1-e2)
(3)
•When CS = 0,
When CS = ,
e = 0,
e = 1,
b = a, the orbit is a circle.
the orbit is a parabola.
• q = PERIHELION DISTANCE = a - CS = a –
ae
q = a(1-e)
(4)
• Q = APHELION DISTANCE = a + CS = a + ae
Q = a(1+e)
(5)
Second Law: For any planet, the radius vector
sweeps out equal areas in equal time intervals
• Time interval t for planet to travel from p to p1 is
the same as time taken for planet to get from p2 to
p3. Shaded areas are equal.
• Let the time interval t be very small. Then the arc
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from p to p1 can be regarded as a straight line
and the area swept out is the area of the triangle S
p p1. If f1 is the angle to p1, and f is the angle to p:
p1
p2
p3
r1
p
r
f
S
Area = 1/2 r r1 Sin (f1-f).
Since t is very small,
(f1-f) = f
r ~ r1 and
Sin (f1-f) ~
Area = 1/2 r2 f
The rate this area is swept out is constant
according to Kepler's second law, so
r2 df/dt = h
(6)
where h, a constant, is twice the rate of
description of area by the radius vector. It is the
orbital angular momentum (per unit mass. The
orbital angular momentum is conserved.
The total area of the ellipse is πab which is swept
out in the orbital period P, so using eqn (6)
2ab/P = h.
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The average angular rate of motion is n = 2/P,
so
n a2(1-e2)1/2 = h
(7)
Kepler’s Third Law
Kepler's third law took another ten years to develop after
the first two. This law relates the period a planet takes to
travel around the sun to its average distance from the
Sun. This is sometimes called the semi major axis of an
elliptical orbit.
P 2 = ka3
where P is the period and a is the average distance from
the Sun. Or, if P is in years and a is in AU:
P 2 = a3
Kepler’s Third Law follows from the central inverse
square nature of the law of gravitation. First look at
Newton's law of gravitation - stated
mathematically this is
Gm1 m2
r2
Newton actually found that the focus of the elliptical orbits for
two bodies of masses m1 and m2 is at the centre of mass.
F
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The centripetal forces of a circular orbit are
F1
r1
v2
X
Centre of M ass
m1
m2
v1
F2
r2
2
F1
m1 v1
4 2 m1 r1


r1
P2
and
2
F2
m2 v2
4 2 m2 r2


r2
P2
where
v
2r
P
and since they are orbiting each other (Newton’s 2nd
law)
r1
m2

r2
m1
Let's call the separation a = r1 + r2. Then
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m1 r1
 m1

 r1 
 1 and multiplying both sides by m2 , am2  m1 r1 
m2
m2
r1 
or, solving for r1 ,
am2
m1  m2 
Now, since we know that the mutual gravitational
force is
Fgrav  F1  F2 
Gm1 m2
2
a
then substituting for r1,
a3 
G m1  m2  P2
42
Solving for P:
P  2
a3
G M1  M2 
Third Law is therefore: The cubes of the semimajor axes of the planetary orbits are
proportional to the squares of the planets'
periods of revolution.
Example
Europa, one of the Jovian moons, orbits at a distance of
671,000 km from the centre of Jupiter, and has an
orbital period of 3.55 days. Assuming that the mass of
Jupiter is very much greater than that of Europa, use
Kepler's third law to estimate the mass of Jupiter.
Using Kepler's third law:
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m jupiter  meuropa
4 2 a 3

GP 2
The semi-major axis, a = 6.71 x 105 km = 6.71 x 108 m,
and
the period, P = 3.55 x 3600 x 24 = 3.07 x 105 seconds
Thus:
m jupiter  meuropa 
c
4 2 6.71  108
h
3
 19
.  1027 kg
c6.67  10 hc3.07  10 h
 11
5 2
and since mjupiter >> meuropa, then mjupiter ~ 1.9 x 1027
kg.
Summary of Kepler’s Laws
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Summary: Measuring the mass of a planet
• Kepler’s third law gives G(M+m) =  a3/P2
(To remember, do a dimensional analysis: GM/R2
and R/P2 are both accelerations)
.
Since M >> m for all planets, it isn't possible to make
precise enough determinations of P and a to
determine the masses m of the planets.
However, if satellites of planets are observed, then
Kepler's law can be used.
• Let mp = mass of planet
satellite
ms = mass of satellite
Ps = orbital period of
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as = semi-major axis of
satellite's orbit about the
planet.
Then:
G(mp+ms) = 42 as3/Ps2
If the mass of the satellite is small compared with
the mass of the planet then
mp = 42 as3/(G Ps2)
• All the major planets have satellites except
Mercury and Venus.
Their masses were
determined from orbital perturbations on other
bodies and later, more accurately from changes in
the orbits of spacecraft.
So: we can determine the masses of massive objects if we
can detect and follow the motion of very low mass
satellites. That doesn’t lead very far. How can we
determine the masses of distant stars and exoplanets?
BASIC STELLAR PROPERTIES - BINARY STARS
• For solar type stars, single:double:triple:quadruple
system ratios are 45:46 : 8 : 1.
• Binary nature of stars deduced in a number of
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ways:
1. VISUAL BINARIES:
- Resolvable, generally nearby stars (parallax
likely to be available)
- Relative orbital motion detectable over a
number of years
- Not possible for exoplanets!
2. ASTROMETRIC BINARY: only one component
detected
3. SPECTRUM/PECTROSCOPIC BINARIES:
- Unresolved
- Periodic oscillations of spectral lines (due to
Doppler shift)
- In some cases only one spectrum seen
4. ECLIPSING BINARY:
- Unresolved
- Stars are orbiting in plane close to line of sight
giving eclipses observable as a change in the
combined brightness with time (light curves).
Some stars may be a combination of these.
*************************
DETAILS
***************************************************
1. Visual Binaries
• Angular separation ≥ 0.5 arcsec (close to Sun, long
orbital periods - years – remember: at 1 parsec, 1
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arcsec corresponds to 1 AU)
EXAMPLE: Sirius:
Also known as Alpha Canis Majoris, Sirius is the fifth
closest system to Sol, at 8.6 light-years. Sirius is
composed of a main-sequence star and a white
dwarf stellar remnant. They form a close binary,
Alpha Canis Majoris A and B, that is separated "on
average" by only about 20 times the distance from
the Earth to the Sun -- 19.8 astronomical units
(AUs) of an orbital semi-major axis -- which is about
the same as the distance between Uranus and our
Sun ("Sol").
The companion star, is so dim that it cannot be
perceived with the naked eye. After analyzing the
motions of Sirius from 1833 to 1844, Friedrich
Wilhelm Bessel (1784-1846) concluded that it had
an unseen companion
1. Visual Binaries
• Angular separation ≥ 0.5 arcsec (close to Sun, long
orbital periods - years) – example Sirius:
HST:
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• Observations:
Relative positions:
 = angular separation
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 = position angle
Absolute positions: Harder to measure orbits of
more massive star A and less massive star B about
centre of mass C which has proper motion µ.
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Declination
N
M otion of centre of mass
= proper motion µ
Secondary

E

Right Ascension
B
Primary
C
A
NB parallax and aberration must also be accounted
for.
• RELATIVE ORBITS:
- TRUE orbit:
q = peri-astron distance (arcsec or km)
Q = apo-astron distance (arcsec or km)
a = semi-major axis (arcsec or km)
a = (q + Q)/2
- APPARENT orbits are projected on the celestial
sphere
Inclination i to plane of sky defines relation between
true orbit and apparent orbit.
If i≠0° then the centre of mass (e.g. primary) is not at
the focus of the elliptical orbit.
Measurement of the displacement of the primary
gives inclination and true semi-major axis in
arcseconds a".
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i
i
Incline by 45°
Apparent orbit
True orbit
• If the parallax p in arcseconds is observable then a
can be derived from a".
Earth
B
radius
of Earth's
orbit
a
a"
Sun
p
r = distance of binary star
For i=0°, a = a"/p AU
A
(In general correction for i≠0 requ
Now lets go back to Kepler’s Law …
• From Kepler's Law, the Period P is given by
2 3
4 a
P =
G (mA + mB)
2
For the Earth-Sun system P=1 year, a=1 A.U.,
mA+mB~msun so 4π2/G = 1
3
a
P =
(mA + mB)
2
provided P is in years, a in AU, mA, mB in solar
masses.
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The total system mass is determined:
mA + mB = (
a" 3 1
)
2
p
P
• ABSOLUTE ORBITS:
d
c
rA
*
rB
B
f
e
B
q
A
Q
A
Semi-major axes aA = (c+e)/2
aB = (d+f)/2
Maximum separation = Q = c + f
Minimum separation = q = d + e
So aA + aB = (c+d+e+f)/2 = (q + Q)/2 = a
a = aA + aB
(1)
(and clearly r = rA + rB)
From the definition of centre of mass, mA rA = mB
rB ( mA aA = mB aB)
mA/mB = aB/aA = rB/rA
So from Kepler’s Third Law, which gives the sum of
the masses, and the equation above, we get the
ratio of masses, ==> mA, mB. Therefore, with both,
we can solve for the individual masses of the two
stars.
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3 Spectroscopic Binaries
To measure the periodic line shifts requires
• Orbital period relatively short (hours - months)
and i≠0°.
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• Doppler shift of spectral lines by component of
orbital velocity in line of sight (nominal position is
radial velocity of system):
wavelength
Time
wavelength
Time
2 Stars observable
1 Star observable
See:
http://instruct1.cit.cornell.edu/courses/astro101/j
ava/binary/binary.htm
• Data plotted as RADIAL VELOCITY CURVE:
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Professor Glenn White
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the line of sight (i<90°),
the shape is unchanged
but velocities are reduced
by a factor sin i.
• Take a circular orbit
with i=90°
a = rA + rB
v = v A + vB
Orbital velocities:
vA = 2π rA / P
vB = 2π rB / P
v = 2πr/P
Since mA rA = mB rB
Ratio of masses:
mA/mB = rB/rA
vB/vA (2)
=
rB
vA
•Shape of radial velocity
curves depends on orbital
eccentricity and
orientation
r
v
=
rA
v
B
• If the orbit is tilted to
• In general, measured velocities are vB sin i and vA
sin i, so sin i terms cancel: so we measure the mass
ratio accurately.
• From Kepler's law mA + mB = a3/P2 (in solar units).
PH507
Astrophysics
Dr. S.F. Green
Observed quantities:
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vA sin i => rA sin i
vB sin i => rB sin i
} a sin i
So can only deduce
(mA + mB) sin3 i = (a sin i)3/P2
For a spectroscopic binary, only lower limits to
each mass can be derived, unless the inclination i
is known independently.
DETAILED DERIVATION when only one radial
velocity is known (eliminate VB using mass ratio)
****************************************************
1. Assume a planet and star, both of considerable
mass, are in circular orbits around their centre of
mass. Given the period P, the star’s orbital speed
v* and mass M*, the mass of the planet, Mp is
given by
Mp3/ (M* + Mp)2
=
v*3 P / (2  G)
Note that there are 9 unknowns: P, a*, ap, M*,
Mp, v*, vp, a, M - 9 variables
However,
a = a* + ap
M = M*+ Mp
Centre of mass;
M* a* = mp ap
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Dr. S.F. Green
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Kepler’s law relates: P, a, M
P = 2  a*/v*
P = 2  ap/vp
….so that is 6 equations.
Note: we usually only know vr* = v* sin I and
we assume the planet mass is small.
4 Eclipsing Binaries
• Since stars eclipse i ~90°
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Professor Glenn White
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• For a circular orbit:
1, 1' FIRST CONTACT
2,
2'
SECOND
CONTACT
3, 3' THIRD CONTACT
4,
4'
FOURTH
CONTACT
v
4' 3'
2' 1'
1 2
3 4
Observer in plane
• Variation in brightness with time is LIGHT
CURVE.
• Timing of events gives information on sizes of
stars and orbital elements.
• Shape of events gives information on properties of
stars and relative temperatures. If smaller star is
hotter, then:
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Case 1
Smaller star is hotter
Case 2
Larger star is hotter
F
or
magnitude
Secondary minimum
Primary minimum
time
Case 1 t'1
Case 2 t 1
t'2
t2
t'3 t'
4
t3 t4
t1
t'1
t2
t'2
t3
t'3
t4
t'4
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• If orbits are circular:
minima are symmetrical ie t2-t1 = t4-t3 = t2'-t1' =
t4'-t3';
minima are half a period apart; eclipses are of
same duration.
Asymetrical and/or unevenly spaced minima
indicate eccentricity and orientation of orbit.
•
For a circular orbit: DistanceR=S/R
velocity
x time
=
L
t3 v(t2
t 4 – t1)
2RS
=t 1 t 2
and
2RS + 2RL
=
v (t42R-L t1) => 2RL = v(t4 - t2)
2RS
• Light curves are also
affected by:
Non-total eclises
No flat minimum
Limb darkening
(non-uniform
brightness)
"rounds off"
eclipses
Ellipsoidal stars
(due to
proximity)
"rounds off"
maxima
Reflection effect
(if one star is
very bright)
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Professor Glenn White
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5 Eclipsing-Spectroscopic Binaries
• For eclipsing binaries i ≥ 70°
(sin3i > 0.9)
• If stars are spectroscopic binaries then radial velocities
are known.
So: masses are derived,
radii are derived,
ratio of temperatures is derived
Examine spectra and light curve to determine which
radius corresponds with which mass and temperature:
• Since Luminosity L = 4 R2 T4, (Stefan-Boltzmann
equation) the ratio of luminosities is derived from
LA
LB
=
RA
RB
2
TA
TB
4
PH507
Astrophysics
Professor Glenn White
How else to determine mass?
1. Mass-Luminosity relation
binaries.
2. Lensing/micro-lensing.
3. Virial expression
42
derived
from