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ASTROPHYSICS
SUMMARY
• Constellation is a group of stars that form a pattern as seen from the Earth,
but not bound by gravitation
• Stellar cluster is a group of stars held together by gravitation in same region
of space, created roughly at the same time.
• Galaxy is a huge group of stars, dust, and gas held together by gravity,
often containing billions of stars, measuring many light years across.
• Clusters: Gravitationally bound system of galaxies/stars.
• Star is a massive body of gas held together by gravity, with fusion going
on at its center, giving off electromagnetic radiation. There is an
equilibrium between radiation/gas pressure and gravitational pressure
• Comet: A small body composed of mainly ice and dust that orbits the sun
in an elliptical orbit.
Planet
Mass/kg
Radius/m
Orbit radius/m
Orbital period
(avg.)
Mercury
3.30 x 1023
2.44 x 106
5.79 x 1010
88.0 days
Venus
4.87 x 1024
6.05 x 106
1.08 x 1011
224.7 days
Earth
5.98 x 1024
6.38 x 106
1.50 x 1011
365.3 days
Mars
6.42 x 1023
3.40 x 106
2.28 x 1011
687.0 days
Jupiter
1.90 x 1027
6.91 x 107
7.78 x 1011
11.86 years
Saturn
5.69 x 1026
6.03 x 107
1.43 x 1012
29.42 years
Uranus
8.66 x 1025
2.56 x 107
2.88 x 1012
83.75 years
Neptune
1.03 x 1026
2.48 x 107
4.50 x 1012
163.7 years
SPECTRA
1. A hot solid, liquid or gas at high pressure produces a continuous spectrum – all λ.
2. A hot, low-density / low pressure gas produces an emission-line spectrum –
energy only at specific λ.
3. A continuous spectrum source viewed through a cool, low-density gas
produces an absorption-line spectrum – missing λ – dark lines.
Thus when we see a
spectrum we can tell
what type of source
we are seeing.
Explain how atomic spectra may be used to deduce chemical and physical data for stars.
•surface temperature of a star is determined by measuring the wavelength
at which most of the radiation is emitted.
2.9×10-3
max (m) 
T(K)
•Most stars essentially have the same chemical composition, yet show
different absorption spectra as they have different temperatures.
• Absorption spectra gives information about the temperature of the star
and its chemical composition.
• Doppler shift information of speed relative to earth (red shift → longer
wavelength, blue shift → shorter wavelength)
Astronomical Unit
The average distance between Earth and the sun, about 150 million kilometers
D = 1 AU = 149597870691 m ≈ 1.5x1011 m
• A black body is a hypothetical object which absorbs
all incident electromagnetic radiation
• Black bodies in thermal equilibrium emit energy to balance the energy
they absorb and remain at a constant temperature.
The wavelengths at which an ideal blackbody emits electromagnetic
radiation depends on the temperature and absolutely nothing else.
That is why the radiation emitted by an blackbody is often called
thermal radiation.
A radiating blackbody emits energy at all wavelengths and the hotter
a blackbody is, the more total energy it emits.
Black body emits energy according
to Planck’s and Wien’s law
Stars’ and planets’ radiation spectrum is approximately the same as
black-body radiation/ Plank’s law.
Intensity as a function of wavelength depends upon its temperature
Wien’s law: Wavelength at which the intensity
of the radiation is a maximum λmax, is:
2.9×10-3
max (m) 
T(K)
Luminosity (of a star) is the total power (total energy per second) radiated
by an object (star). If we regard stars as black body, then luminosity is
L = A σT4 = 4πR2σT4
(Watts)
Stefan-Boltzmann’s law
A is surface area of the star, R is the radius of the star,
T surface temperature (K), σ is Stefan-Boltzmann constant.
(Apparent) brightness (b) is the power from the star received per
square meter of the Earth’s surface
L
b = 4π𝑑2
(W/m2)
L is luminosity of the star; d its distance from the Earth
Suppose I observe with my telescope two red stars A and B that are part of a binary
star system. Star A is 9 times brighter than star B.
What can we say about their relative sizes and temperatures?
Since both are red (the same color), the spectra peak at the same wavelength.
By Wien's law
2.9×10-3
max (m) 
T(K)
L = 4π R2 σ T4
then they both have the
same temperature.
(W)
Star A is 9 times brighter and as they are the same distance away from Earth.
Star A is 9 times more luminous:
LA 4RA2TA4

LB 4RB2TB4
RA2
 9 2
RB
 RA  3 RB
So, Star A is three times
bigger than star B.
Suppose I observe with my telescope two stars, C and D, that form a binary star pair.
▪ Star C has a spectral peak at 350 nm - deep violet
▪ Star D has a spectral peak at 700 nm - deep red
What are the temperatures of the stars?
By Wien's law
 peak
3
2.9

10

T
 peak in m
T in K
Thus we have for star C,
3
3
2.9

10
2.9

10
TC 

 8300 K
 peak
350 10 9
and for star D
3
3
2.9

10
2.9

10
TD 

 4150 K
9
 peak
350 10
If both stars are equally bright (which means in this case they
have equal luminosities since the stars are part of a pair the
same distance away), what are the relative sizes of stars C and D?
LC 4RC2 TC4

LD 4RD2 TD4
RC2 83004
RC2
4
 1  2

2


RD 41504
RD2
RD2  16 RC2  RD  4 RC
So that stars C is 4 times smaller than star D.
Magnitude Scale •
Magnitudes are a way of assigning a number to a star so we know how bright it is
Apparent magnitude (m) of a celestial body is a measure of its brightness as
seen from Earth. The brighter the object appears the lower its apparent
magnitude. Greeks ordered the stars in the sky from brightest to faintest…
Later, astronomers accepted and quantified this system.
• Every one step in magnitude corresponds to a factor of 2.51 change
in brightness. Ex: m1 = 6 and m2 = 9, then b1 = (2.51)3 b2
Absolute magnitude (M) of a star is the apparent magnitude
that a star would have if it were at distance of 10 pc from Earth.
It is the true measurement of a star’s brightness seen from a set distance.
m – M = 5 log
d
10
m – apparent magnitude
M – absolute magnitude of the star
d – its distance from the Earth measured in parsecs.
• If two stars have the same absolute magnitude but different apparent
magnitude they would have the same brightness if they were both
at distance of 10 pc from Earth, so we conclude they have the same
luminosity, but are at different distances from Earth !!!!!!!!!!!!!!
• Every one step in absolute magnitude corresponds to a factor of 2.51
change in luminosity. Ex: M1 = – 2 and M2 = 5, then L1 / L2 = (2.51)7
The table shows some data for Procyon A and Procyon B.
Apparent
magnitude
Absolute
magnitude
Apparent brightness
/ W m–2
Procyon A (PA )
+ 0.400
+ 2.68
2.06 x 10–8
Procyon B (PB )
+ 10.7
+ 13.0
1.46 x 10–12
(a) Explain, using data from the table, why
(i) as viewed from Earth, PA is much brighter than PB.
(ii) the luminosity of PA is much greater than that of PB.
(i) • the apparent magnitude of PA is (much) smaller than that of PB;
• the smaller apparent magnitude, the brighter the star
or
• apparent brightness of PA is greater than PB;
• apparent brightness is intensity at surface of Earth;
[2]
(ii) • the absolute magnitude of PA is smaller than that of PB;
• the absolute magnitude is the apparent magnitude at a distance of 10 pc from the Earth;
• so at the same distances from Earth PA is much brighter than PB so must be more luminous;
Or
[3]
• absolute magnitude of PA is less than absolute magnitude of PB;
• absolute magnitude is a measure of luminosity;
• lower values of absolute magnitude refer to brighter/more luminous star;
or
• Accept answer based on answer to (c).
• distances are the same from (c);
• since L = 4πd 2b PA is brighter than PB;
Apparent
magnitude
Absolute
magnitude
Apparent brightness
/ W m–2
Procyon A (PA )
+ 0.400
+ 2.68
2.06 x 10–8
Procyon B (PB )
+ 10.7
+ 13.0
1.46 x 10–12
(b) Deduce, using data from the table, that PA and PB are approximately
the same distance from Earth.
[2]
(b) • for PA
m – M = – 2.28 for PB
m – M = – 2.30
𝑑
• since m – M = 5 lg
then d for each star is very nearly same;
10
or
• d = 10 x 10
𝑚−𝑀
5
• 𝑑𝐴 = 10 × 10−0.456 ≈ 3.5 pc
𝑑𝐵 = 10 × 10−0.45 ≈ 3.5 pc
(c) State why PA and PB might be binary stars.
[1]
• same distance from Earth and in the same region of space
(d) Calculate, using data from the table, the ratio
𝐿𝐴
𝐿𝐵
=
2
4𝜋𝑑𝐴
𝑏𝐴
2𝑏
4𝜋𝑑𝐵
𝐵
=
2.06×10−8
1.46×10−12
= 1.41 × 104
𝐿𝐴
.
𝐿𝐵
Binary star is a stellar system consisting of two stars
orbiting around their common center of mass.
The ONLY way to find mass of the stars is when they are the part of binary
stars. Knowing the period of the binary and the separation of the stars
the total mass of the binary system can be calculated (not here).
Visual binary: a system of stars that can be seen as two separate
stars with a telescope and sometimes with the unaided eye
They are sufficiently close to Earth and the stars are well enough separated.
Sirius A, brightest star in the night sky and its companion first white dwarf star
to be discovered Sirius B.
Spectroscopic binary: A binary-star system which from Earth
appears as a single star, but whose light spectrum (spectral
lines) shows periodic splitting and shifting of spectral lines due
to Doppler effect as two stars orbit one another.
Eclipsing binary: (Rare) binary-star system in
which the two stars are too close to be seen
separately but is aligned in such a way that
from Earth we periodically observe changes in
brightness as each star successively passes in
front of the other, that is, eclipses the other
Stellar Spectra classification system
Class
Temperature
Colour
O
30 000 - 60 000
Blue
B
10 000 - 30 000
Blue-white
A
7 500 - 10 000
White
F
6000 - 7500
Yellow-white
G
5000 - 6000
Yellow
K
3500 - 5000
Orange
M
2000 - 3500
Red
The Hertzsprung–Russell (H – R) diagram(family portrait) is a scatter graph of stars
showing the relationship between the stars' absolute magnitudes / luminosities versus
their spectral types(color) /classifications or surface temperature. It shows stars of
different ages and in different stages, all at the same time.
main sequence stars:
fusing hydrogen into helium, the
difference between them is in mass
left upper corner more massive than right
lower corner.
white dwarf compared to a main sequence
star: • has smaller radius • more dense
LQ = sun luminosity = 3.839 × 1026 W
• higher surface temperature
• energy not produced by nuclear fusion
Techniques for determining stellar distances:
• stellar parallax,
• spectroscopic parallax
• Cepheid variables.
Parallax Method relies on the apparent movement of the nearby star against
the background of further stars as the earth orbits the sun.
• It can be used for the nearest stars to about 100 parsec.
• two apparent positions of the star as seen by an observer from two widely
separated points are compared and recorded to find angle p
• knowing the distance of Earth to the sun, the distance to the star
can be calculated using geometry
sin p  D
d
For small angles:
p D
d
→
sin p  tan p  p
dD
p
1 pc = 3.09 X 1016 m
Parallaxes are expressed in seconds
if p = 1 sec of arc,
d = 3.08x1016 m defined as 1 pc
‘One parsec is a distance corresponding to a parallax of one arc second'
d  pc  
1
p  sec 
Spectroscopic parallax: no parallax at all!!!! (a lot of uncertainty in calculations)
• light from star analyzed (relative amplitudes of the absorption spectrum lines) to
give indication of stellar class/temperature
• HR diagram used to estimate the luminosity
• distance away calculated from apparent brightness
• limit: d ≤ 10 Mpc
Spectroscopic parallax is only accurate enough to measure stellar distances of up to
about 10 Mpc. This is because a star has to be sufficiently bright to be able to measure
the spectrum, which can be obscured by matter between the star and the observer.
Even once the spectrum is measured and the star is classified according to its spectral
type there can still be uncertainty in determining its luminosity, and this uncertainty
increases as the stellar distance increases. This is because one spectral type can
correspond to different types of stars and these will have different luminosities.
1 - Spica
• Apparent magnitude, m = 0.98
• Spectral type is B1
• From H-R diagram this indicates an absolute magnitude, M, in the range: -3.2 to -5.0
m – M = 5 log
d
10
d = 10 (m-M+5)/5
M= – 3.2,
d = 10 (0.98 - (-3.2) +5)/5 = 68.54 pc
M= – 5.0,
d = 10 (0.98 - (-5.0) +5)/5 = 157.05 pc
The Hipparcos measurements give d = 80.38 pc
2 - Tau Ceti
• Apparent magnitude, m = 3.49
• Spectral type is G2
• From H-R diagram this indicates an absolute magnitude, M, in the range: +5.0 to +6.5
m – M = 5 log
d
10
d = 10 (m-M+5)/5
M= +5.0,
d = 10 (3.49 -5.0 +5)/5 = 5.00 pc
M= +6.5,
d = 10 (3.49 -6.5 +5)/5 = 2.50 pc
The Hipparcos measurements give d = 3.64 pc
Cepheid variables are stars with regular variation in absolute magnitude (luminosity)
(rapid brightening, gradual dimming) which is caused by periodic expansion and
contraction of outer surface (brighter as it expands). This is to do with the balance
between the nuclear and gravitational forces within the star. In most stars these forces are
balanced over long periods but in Cepheid variables they seem to take turns, a bit like a
mass bouncing up and down on a spring.
Left: graph shows how the apparent
magnitude (the brightness) changes,
getting brighter and dimmer again
with a fixed, measurable period for a
particular Cepheid variable.
There is a clear relationship between the period of a Cepheid variable and its absolute
magnitude. The greater the period then the greater the maximum luminosity of the
star. Cepheids typically vary in brightness over a period of about 7 days.
Left is general luminosity – period graph.
So, to find out how far away Cepheid is:
•
•
•
•
•
Measure brightness to get period
Use graph absolute magnitude M vs. period to find absolute magnitude M
Measure maximum brightness
Calculate d from b = L/4πd2
Distances to galaxies are then known if the Cepheid can be ascertained to be within a specific galaxy.
Newton assumptions about the nature of the universe:
• universe is infinite in extent
• contains an infinite number of stars uniformly distributed
• is static and exists forever
• these assumptions led to Olber’s paradox
Olber’s paradox: Why in the world is the sky dark???
According to these assumptions the sky should not be dark
• density of stars n = N/V = number of stars per unit volume
• divide the whole universe into concentric shells around Earth
of constant thickness t
• look at one shell of thickness t at distance d from the Earth
• since stars are uniformly distributed the number of stars
seen from Earth increases as d2:
number of stars in shell = density x volume = n 4πd2 t
• brightness of one star decreases as 1/d2 ; b = L/4 πd2
• brightness of shell is constant; assuming that luminosity L is
the same for all stars, the received energy per sec per unit
area (brightness) from all stars in the thin shell is:
L
4π d2
× 4π d2 nt = Lnt = const.
• amount of light we receive from shell does not depend upon how far away
the shell is
• adding all shells to infinity; each contributing a constant amount of energy
• the total energy is infinite
• sky would be uniformly bright
• but it’s dark in night
Possible solutions to Olber’s paradox
• Perhaps the Universe is not infinite.
But current model of the Universe is that it is infinite.
• Perhaps the light is absorbed before it gets to us. But then Universe would
warm up and eventually reradiate energy.
Real help: the Big Bang model leads to the idea that the observable universe is not infinite and
to the idea of the expansion of the universe;
• Universe is not static, it is expanding, hence the most distant stars/ galaxies
are strongly red - shifted, out of the visible part of the spectrum.
• There is a finite time since the Big Bang. Some 12 to 15 billion years.
That means we can only see the part of it that lies within 12 to 15 billion
light-years from us. And the observable part of the universe contains too
few stars to fill up the sky with light.
Calculation shows that the helium produced by nuclear fusion within stars cannot
account for the real amount of helium in Universe (24%). In 1960 it was proposed that
sometime during the early history of the Universe, long before any star, Universe was at a
sufficiently high temperature to produce helium by fusion. In this process many high
energy photons would be produced. The CMB (Cosmic Microwave Background Radiation)
radiation was emitted only a few hundred thousand years after the Big Bang, long before
stars or galaxies ever existed. The photons would have a black body spectrum
corresponding to the then temperature of the Universe. As the Universe expanded and
cooled the photon spectrum would also change with their maximum wavelength shifting
in accordance with Wien’s law. It is estimated that at the present time the photons
should have a maximum wavelength corresponding to a black body spectrum of an
extremely cold object of temperature of 2.7 K.
Cosmological background radiation / Cosmic microwave background radiation
(CMB) is microwave radiation - left over from the Big Bang that fills the universe
roughly uniformly in all directions. The Big Bang predicts an expanding universe that
had a very high temperature at the beginning; during the expansion the universe is
cooling down and the temperature of the radiation should fall to its present low value
of about 2.7 K.
• That radiation corresponds to a black body spectrum of about 2.7 K.
The other way of explaining CMB is:
• Big Bang producing initially produced very short wavelength photons /EM radiation.
As the universe expands, the wavelengths become red shifted to reach current value.
█ Explain how knowledge of the spectrum of a black body and the existence of cosmological
background radiation is consistent with the “Big Bang” model of the universe.
• Big Bang predicts a low temperature radiation at 2.7 K (i.e. CMB radiation).
The Big Bang theory also predicts an expanded universe which we observe through red- shifting
of the galaxies and the lowering of CMB radiation temperature. This expanding universe is the
result of the initial energy released in the Big Bang
█ State one piece of evidence that indicates that the Universe is expanding.
• light from distant galaxies/stars is red-shifted
(which means they move away from us – as the red-shifting occurs in all direction,
the universe must be expanding)
• existence of CMB
• the helium abundance in the universe which is about 25 % and is consistent with
a hot beginning of the universe;
The eventual fate of the Universe is determined by the amount of mass in the Universe.
• Critical density is the density of the Universe which produces a flat universe,
i.e. it would take an infinite amount of time to stop expansion of it.
• Critical density is the density of the Universe that would be necessary to stop
the expansion after an infinite amount of time.
• Closed Universe A model of the universe in which density of the Universe is such
that gravity will stop the universe expanding and then cause it to contract.
Eventually the contraction will result in a ‘Big Crunch’ after which the whole creation
process could start again.
• Open Universe A model of the universe in which density is such that gravity is too
weak to stop the Universe expanding forever.
• Flat Universe means that the density is at a critical value whereby the Universe will
only start to contract after an infinite amount of time.
Distinguish between the terms open, flat and closed when used to describe the development
of the universe.
OPEN: keeps expanding
FLAT: the rate of expansion tends to zero at infinite time
CLOSED: stops expanding and starts to contract
• non-coincident starts (not at beginning);
• correct shapes and correctly labelled;
• coincident at appropriate place;
Dark matter is the matter that makes up for most of the mass in the universe, but
cannot easily be detected because it does not emit radiation .
Its existence is inferred from techniques rather then direct
visual contact (gravitational effects on visible matter, radiation
and the large-scale structure of the cosmos).
Examples of dark matter.
► two of Neutrinos / WIMPS / MACHOS / black holes / exotic super symmetric
particles / grand unified predicted particles / magnetic monopoles
etc.; or maybe our current theory of gravity is again not correct
VOCABULARY:
http://en.wikibooks.org/wiki/IB_Physics/Astro
physics_SL#F.1.2_Bodies_within_the_Universe
EVERYTHING IN NICE DETAILS
http://sciencevault.net/ibphysics/astrophysics/
astrophysicsindex.html
Arc Second “
One degree split up into seconds (1/3600 of a degree).
Light Year ly
The distance light travels during one year
Absolute Luminosity L
The total power radiated from a star in watts
Apparent Brightness b
Power of star light reaching Earth in W m-2
Absolute Magnitude M
Apparent Magnitude of a star at a standard distance of 10 parsec.
Apparent Magnitude m
How bright a star looks (scale 6 to 1, 1 being the brightest).
Black Body
Hot body who’s emission spectrum depends on its temperature only (stars are
considered black bodies because they emit all energy produced).
Black Body Radiation
Radiation from a theoretical “perfect” emitter.
Cepheid Variable Star
A star that varies periodically in size (not mass!!) and hence luminosity,
which is caused by periodic expansion and contraction of outer surface.
Luminosity increases sharply and falls off gently with a well-defined period.
The period is related to the absolute luminosity of the star and so can be used
to estimate the distance to the star (greater than 10 Mpc).
Binary Star
Two (or more) stars orbiting a common center of mass.
Planetary Order
Mercury, Venus, Earth, Mars, Asteroid Belt, Jupiter, Saturn, Neptune, Uranus,
Pluto
Eclipsing Binaries
Two stars that regularly eclipse each one another causing a periodic variation in
brightness.
Red Giant
A large and relative cool star in one of the later stages of its life. The
source of its energy is fusion of some other element than hydrogen.
Red Super Giant
A very large mass and relative cool star in one of the later stages of
its life. The source of its energy is fusion of some other element than
hydrogen.
White Dwarf
A small and relative cool star at one of the possible final stages of a
star’s life. Fusion is no longer taking place and hence it is cooling
down.
Parsec
1 parsec is the distance which will give a parallax angle of exactly 1
arc second, with 1 AU as the baseline (3.26 light years).
1 pc = 3.08x1016 m
Astronomical Unit
The radius of Earth’s orbit around the Sun
1 AU = 1.5 x 1011 m
Red Shift
Increase in wavelength emitted from galaxies and stars moving away
from Earth. This can either be due to the Doppler Effect or
expansion of space (which expands the wave).
Critical Density
The theoretical density of the universe necessary to create a “flat”
universe after an infinite amount of time
Dark Matter
Matter in galaxies that is too cold to radiate. Its existence is inferred
from techniques rather then direct visual contact (gravitational effects
on visible matter, radiation and the large-scale structure of the
cosmos).
MACHOs
Black holes, high-mass planets or/and failed stars – all who produce
very little or no light (hence not seeable).
WIMPs
A new type of particle that can prove the existence of dark matter.
Newton's First
Postulate about the
Universe
The universe is infinite.
Newton's Second
Postulate about the
Universe
Mass is uniformly distributed in space.
Obler's Paradox
Why is not the sky bright at night if there is a bright star at every point
in the sky? (Inverse Square Law)
Wein's Displacement
Law
Wavelength of radiation with maximum intensity (peak wavelength) is
inversely proportional to the temperature of the black body.
Stefan-Boltzman Law
Links the total power output radiated by a black body to its
temperature.