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PHY2083
ASTRONOMY
Gravity
Multiple Stellar systems: binary stars
Sir William Herschel (1802):
Catalogue of 500 new Nebulae ... and
Clusters of Stars; with Remarks on the
Construction of the Heavens.
“ the union of two stars, that are formed
together in one system, by the laws of
attraction.’’
Binary stars
Modern definition: ‘Two or more stars
in orbit about a common centre of
mass’
~50% of all stars are part of a multiple
system
Historical Digression
Plato (c. 350 B.C.) suggested the need for a framework (e.g.
stars revolve around the Earth which is fixed)
“Geocentric Universe”: fixed relationship between stars
Ptolemy (c. 100 A.D.) refined the system introduced (most
notably) by Hipparchus to explain the observed motions of the
stars and planets.
Copernicus (1473-1543) proposed a heliocentric model of
planetary motion. Simpler and more elegant. Model could order
planets e.g. Mercury and Venus are never seen more than 28
and 47deg. E or W of the Sun => inside orbit of the Earth. This
model, however, could not predict positions.
Enter Tycho Brahe (1546-1601) and Johannes Kepler
Planets placed on rotating epicycles
which in turn moved on a larger deferent.
This explained e.g. the
brightnesses of the planets as
their distances from the Earth
changed.
Equants resulted in the constant
angular speed of the epicycle
about the deferent.
Good agreement!
Fix model with increasing
complexity.
Kepler’s Laws
Tycho Brahe, foremost observer of his time. Measured
celestial objects to an accuracy of better than 4’.
Discovered SN of 1572, and concluded that the heavens
were not unchanging -- a belief propagated by the Church
doctrine.
Kepler sought a geometrical model of the universe that
would be consistent with the best observations available
(i.e. Tycho’s).
Kepler’s Laws
(old version)
As soon as Kepler rejected the idea of purely
circular motion, he was able to construct a model
that was consistent with Tycho’s data.
•A planet orbits the Sun in an ellipse with the Sun at one
focus of the ellipse
• A line connecting a planet to the Sun sweeps out equal
areas in equal time intervals (see later). Orbital speed of a
planet depends on its location in the orbit
• The average orbital distance of a planet from the Sun (a)
is related to its period (P) in the form P2 proportional to a3
Generalise Kepler’s Laws to 2 bodies orbiting each other
Johannes Kepler
(1571-1630)
Isaac Newton
(1643-1727)
How can we find binary
systems of stars?
How can we find binary
systems of stars?
3 common ways:
• Visually
• Spectroscopically
• Astrometrically
What is the difference?
Optical double
Gravitationally bound
VISUAL BINARIES:
Resolvable, generally nearby stars
(parallax likely to be available)
Relative orbital motion
detectable over a number of
years.
Can provide information about
the angular separation of stars
from their common centre of
mass.
ASTROMETRIC BINARIES:
Only one component seen e.g. if one member significantly
brighter than the other.
Deduce existence of unseen star from the motion of the
visible star.
Combination of motion around the centre of mass + proper
motion gives rise to “wobble” on celestial sphere (see Fig.).
Newton’s first law: constant velocity maintained by mass
unless there is a force acting on it.
SPECTROSCOPIC BINARIES
(single-lined, double-lined)
• Unresolved
• If orbital period not too long, and if the orbital
motion has a component along the line-of-sight
then spectral lines will exhibit a periodic shift.
• Double-lined: 2 sets of lines observed
• Single-lined: only 1 set of lines observed e.g. due to
large difference in luminosity.
SPECTROSCOPIC BINARIES
ELLIPSES and ORBITS
An ellipse is defined by the
set of points that satisfies
the following equation:
r + r" = 2a
Relation between!
a,b,e:
a = semi-major axis
b = semi-minor axis
e = eccentricity
0≤e<1
F,F’ = focal points
b 2 = a 2 (1" e 2 )
!
Convenient to express orbit in polar coordinates, with r being the distance from the
principal focus (see fig.)
r"2 = r 2 sin 2 # + (2ae + r cos # )
2
r"2 = r 2 + 4ae( ae + r cos # )
$r=
!
a(1% e 2 )
1+ ecos#
(0 & e < 1)
Kepler’s 1st Law:
orbits are ellipses
Both objects in a binary orbit move about the
centre of mass in ellipses, with the centre of mass
occuping one focus of the ellipse.
Kepler’s 1st Law
Sun
a
θ
r
Kepler s Second Law: equal areas in equal time
Area swept out by a line b/w a planet and the focus is always
the same for a given time interval.
v1t1 = v2t2
v2t2
µ"
!
A2
m1m2
m1 + m2
! Area 1 = Area 2
A1 v1t1
dA 1 L
=
dt 2 µ
(reduced mass)
Rate of change of area swept
! by a line connecting a
body to the focus of its elliptical orbit = constant =
0.5 x orbital angular momentum per unit mass
Kepler’s 2nd Law
δθ
∆A
vt
v
Quick and dirty ‘derivation’ of Kepler’s third law (see
also notes)
v2
F = ma = m
r
GMm
F= 2
r
GM
v2 =
r
4 " 2 r 2 GM
2
v =
=
P2
r
4" 2 r 3
2
#P =
GM
!
Force necessary to keep
mass in a circular orbit
Using P = 2r/v
Kepler’s 3rd law!
Generalised form of Kepler s third law:
4" 2
2
P =
a3
G(m1 + m2 )
P " a3
and
N.B. for P in Earth
years, a in AU, m in
solar masses
4!2/G = 1
P " 1/ M tot
(Tycho s data for solar system only)
!
!
Kepler s laws: applicable to planets, binary systems,
galaxy-galaxy orbits.
!
Period + semi-major axis => mass
Example:
The orbital period of Io, one
of the 4 Galilean moons of
Jupiter, is 1.77 days, and the
semi-major axis of its orbit
is 4.22x108m. Estimate the
mass of Jupiter, stating any
assumptions that you made.
Solution: Apply Kepler’s third law. Answer =
1.9x1027 kg ~ 318 Earth masses. Assumed that
mass of Io is negligible c.f. mass of Jupiter.
G = 6.67300 × 10-11 m3 kg-1 s-2
1.5 x10 11m = 1AU
ans: 1.9 x 10^27 kg ~ 318 M_earth
Mean Orbital Velocity (BB!)
v∝
Earth Mars
Jupiter
�
1
a
Orbital velocity
v=
�
G(m1 + m2 )
�
a
�
GM⊙
a
Basic data of the 8 planets in the solar system
Earth yrs
Kepler’s 3rd Law
Basic data of the 8 planets in the solar system
P2 (Earth years) = a3 (in AU)
Main points from table:
1) The orbits of all planets (except Mercury) are nearly
circular (i.e. eccentricity ~ 0)
2) The further a planet is from the sun, the longer its
orbital period i.e., in agreement with Kepler’s 3rd law.
(The small deviations of P2/a3 for Uranus and Neptune are
due to the gravitational forces between these 2 planets).