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Stellar Spectra
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Flux and Luminosity
Brightness of stars
Temperature vs heat
Temperature vs color
Colors/spectra of stars
Classifying stars: H-R diagram
Stefan-Boltzman law, stellar radii
Measuring star masses
Mass-luminosity relation
Flux and luminosity
• Luminosity - A star produces light – the total
amount of energy that a star puts out as light
each second is called its Luminosity.
• Flux - If we have a light detector (eye, camera,
telescope) we can measure the light produced
by the star – the total amount of energy
intercepted by the detector divided by the area
of the detector is called the Flux.
Flux and luminosity
• To find the luminosity, we take a shell
which completely encloses the star and
measure all the light passing through the
shell
• To find the flux, we take our detector at
some particular distance from the star and
measure the light passing only through the
detector. How bright a star looks to us is
determined by its flux, not its luminosity.
Brightness = Flux.
Flux and luminosity
• Flux decreases as we get farther from the star
– like 1/distance2
L
F
2
4D
Brightness of stars
• The brightness of a star is a measure of its flux.
• Ptolemy (150 A.D.) grouped stars into 6
`magnitude’ groups according to how bright they
looked to his eye.
• Herschel (1800s) first measured the brightness
of stars quantitatively and matched his
measurements onto Ptolemy’s magnitude
groups and assigned a number for the
magnitude of each star.
Brightness of stars
• In Herschel’s system, if a star is 1/100 as
bright as another then the dimmer star has
a magnitude 5 higher than the brighter one.
• Note that dimmer objects have higher
magnitudes
Apparent Magnitude
Consider two stars, 1 and 2, with apparent magnitudes
m1 and m2 and fluxes F1 and F2. The relation between
apparent magnitude and flux is:
 F1 
m1  m2  2.5 log 10  
 F2 
F1
m2  m1  / 2.5
 10
F2
For m2 - m1 = 5, F1/F2 = 100.
Flux, luminosity, and magnitude
L
F
2
4D
 F1 
m2  m1  2.5 log 10  
 F2 
 L1 4D22 

m2  m1  2.5 log 10 
2
 4D1 L2 
L2
D2
m2  m1  2.5 log 10
 5 log 10
L1
D1
Distance-Luminosity relation:
Which star appears brighter to the
observer?
Star 1
10L
L
Star 2
d
10d
Flux and luminosity
L2
 10
L1
D2
 10
D1
2
F2
L2 4D
L2  D1 
1
  10   0.1

 
2
F1 4D2 L1
L1  D2 
 10 
2
1
2
Star 2 is dimmer and has a higher magnitude.
Flux and luminosity
L2
 10
L1
m2  m1  2.5 log 10
D2
 10
D1
L2
D2
 5 log 10
L1
D1
m2  m1  2.5 log 10 10  5 log 10 10
m2  m1  2.5  5  2.5
Star 2 is dimmer and has a higher magnitude.
Absolute magnitude
• The magnitude of a star gives it brightness
or flux when observed from Earth.
• To talk about the properties of star,
independent of how far they happen to be
from Earth, we use “absolute magnitude”.
• Absolute magnitude is the magnitude that a
star would have viewed from a distance of
10 parsecs.
• Absolute magnitude is directly related to
the luminosity of the star.
Absolute Magnitude
Absolute magnitude, M, is defined as
M  m  5 log 10 D  5
where D is the distance to the star measured in parsecs.
For a star at D = 10 parsecs, 5log10 = 5, so M = m.
Absolute Magnitude and
Luminosity
The absolute magnitude of the Sun is M = 4.83.
The luminosity of the Sun is L
L  L 10
4.83 M  / 2.5
 L 
M  4.83  2.5  log 10  
 L 
Note the M includes only light in the visible band, so this is
accurate only for stars with the same spectrum as the Sun.
Absolute Bolometric Magnitude
and Luminosity
The bolometric magnitude includes radiation at all
wavelengths.
The absolute bolometric magnitude of the Sun is
Mbol = +4.74.
M bol
 L 
 4.74  2.5  log 10  
 L 
Is Sirius brighter or fainter than Spica:
(a) as observed from Earth [apparent magnitude]
(b) Intrinsically [luminosity]?
Sun
Little Dipper
(Ursa Minor)
Guide to naked-eye
magnitudes
Which star would have the highest
magnitude?
1.
2.
3.
4.
Star A - 10 pc away, 1 solar luminosity
Star B - 30 pc away, 3 solar luminosities
Star C - 5 pc away, 0.5 solar luminosities
Charlize Theron
Temperature
lower T
higher T
• Temperature is proportional to the average kinetic
energy per molecule
1 2 3
K  mv  kT
2
2
k = Boltzmann constant =
1.3810-23 J/K = 8.6210-5 eV/K
Temperature vs. Heat
lower T
higher T
• Temperature is proportional
to the average kinetic
energy per molecule
• Heat (thermal energy) is
proportional to the total
kinetic energy in box
less heat
same T
more heat
Wien’s law
• Cooler objects produce radiation which peaks
at lower energies = longer wavelengths =
redder colors.
• Hotter objects produce radiation which peaks
at higher energies = shorter wavelengths =
bluer colors.
• Wavelength of peak radiation:
Wien Law max = 2.9 x 106 / T(K) [nm]
A object’s color depends on its
surface temperature
• Wavelength of peak radiation:
Wien Law max = 2.9 x 106 / T(K) [nm]
What can we learn from a
star’s color?
The color indicates the temperature of the
surface of the star.
Observationally, we measure colors by
comparing the brightness of the star in two
(or more) wavelength bands.
U
B
V
This is the same way your eye determines color, but
the bands are different.
Use UVRI filters to
determine apparent
magnitude at each color
Stars are assigned a `spectral
type’ based on their spectra
• The spectral classification essentially
sorts stars according to their surface
temperature.
• The spectral classification can also use
spectral lines.
Spectral type
• Sequence is: O B A F G K M
• O type is hottest (~25,000K), M type is coolest
(~2500K)
• Star Colors: O blue to M red
• Sequence subdivided by attaching one numerical
digit, for example: F0, F1, F2, F3 … F9 where F1
is hotter than F3 . Sequence is O … O9, B0, B1,
…, B9, A0, A1, … A9, F0, …
• Useful mnemonics to remember OBAFGKM:
– Our Best Astronomers Feel Good Knowing More
– Oh Boy, An F Grade Kills Me
– (Traditional) Oh, Be a Fine Girl (or Guy), Kiss Me
Very
faint
The spectrum of a star is
primarily determined by
1.
2.
3.
4.
The temperature of the star’s surface
The star’s distance from Earth
The density of the star’s core
The luminosity of the star
Classifying stars
• We now have two properties of stars
that we can measure:
– Luminosity
– Color/surface temperature
• Using these two characteristics has
proved extraordinarily effective in
understanding the properties of stars –
the Hertzsprung-Russell (HR) diagram
HR diagram
HR diagram
• Originally, the HR diagram was made by
plotting absolute magnitude versus
spectral type
• But, it’s better to think of the HR
diagram in terms of physical quantities:
luminosity and surface temperature
If we plot lots of stars on the HR
diagram, they fall into groups
These groups indicate types of stars,
or stages in the evolution of stars
Luminosity of a ‘Black Body’ Radiator
Stephan-Boltzmann Law: an opaque object at a
given temperature will radiate, per unit
surface area, at a rate proportional to the
surface temperature to the fourth power :
P/m2
= T
4
 = Stephan-Boltzman constant
= 5.6710-8 W/m2 ·K4
T = surface temperature
P/m2 = power radiated per square meter
Luminosity of a ‘Black Body’ Radiator
For the spherical object, the total power radiated
= the total luminosity is:
L=
4
2
4R T
T = temperature
 = Stephan-Boltzman constant
= 5.6710-8 W/m2 ·K4
R = radius
Luminosity of a ‘Black Body’ Radiator
Suppose the radius of the Sun increased by a
factor of 4 but the rate of power generated by
fusion remained the same, how would the
surface temperature of the Sun change?
L  4R T  4R T
2
1
4
1
1
2
2
2
1
2
4
2
 R1 
1
1
T2    T1    T1  T1
2
4
 R2 
Stars come in a variety of sizes
• If we know luminosity and temperature,
then we can find the radius:
L = 4R2T4
• Small stars will have low luminosities
unless they are very hot.
• Stars with low surface temperatures must
be very large in order to have large
luminosities.
Sizes of Stars on an HR Diagram
• We can calculate R from
L and T.
• Main sequence stars are
found in a band from the
upper left to the lower
right.
• Giant and supergiant
stars are found in the
upper right corner.
• Tiny white dwarf stars
are found in the lower left
corner of the HR
diagram.
Hertzsprung-Russell (H-R) diagram
• Main sequence stars
– Stable stars found on a
line from the upper left to
the lower right.
– Hotter is brighter
– Cooler is dimmer
• Red giant stars
– Upper right hand corner
(big, bright, and cool)
• White dwarf stars
– Lower left hand corner
(small, dim, and hot)
Luminosity
classes
•
•
•
•
•
Class Ia,b : Supergiant
Class II: Bright giant
Class III: Giant
Class IV: Sub-giant
Class V: Dwarf
The Sun is a G2 V star
‘Spectroscopic Parallax’
Measuring a star’s distance by
inferring its absolute magnitude (M)
from the HR diagram
1.
If a star is on the main-sequence, there is a
definite relationship between spectral type
and absolute magnitude. Therefore, one can
determine absolute magnitude by observing
the spectral type M.
2.
Observe the apparent magnitude m.
3.
With m and M, calculate distance
Take spectrum of star, find it is F2V, absolute
magnitude is then M = +4.0.
Observe star brightness, find apparent
magnitude m = 9.5.
Calculate distance:
m  M  5 log 10 D  5  D  10
( 5.5 5 ) / 5
pc  126 pc
Masses of stars
• Spectral lines also allow us to measure the
velocities of stars via the Doppler shift that we
discussed in searching for extra-solar
planets. Doppler shift measurements are
usually done on spectral lines.
• Essentially all of the mass measurements that
we have for stars are for stars in binary
systems – two stars orbiting each other.
• The mass of the stars can be measured from
their velocities and the distance between the
stars.
Binary star systems Classifications
• Double star – a pair of stars located at nearly the
same position in the night sky.
– Optical double stars – stars that appear close together,
but are not physically conected.
– Binary stars, or binaries – stars that are gravitationally
bound and orbit one another.
• Visual binaries – true binaries that can be
observed as 2 distinct stars
• Spectroscopic binaries
– binaries that can only be detected by seeing two sets of
lines in their spectra
– They appear as one star in telescopes (so close
together)
• Eclipsing binaries – binaries that cross one in front
of the other.
Visual Binary Star Krüger 60
(upper left hand corner)
About half of the stars visible in the night sky
are part of multiple-star systems.
Kepler’s 3rd Law applied to Binary Stars
4
3
2
a P
G (m1  m2 )
2
Where:
• G is gravitational constant
• G = 6.67·10-11 m3/kg-s2 in SI units
• m1, m2 are masses (kg)
• P is binary period (sec)
• A is semi-major axis (m)
Simplified form of Kepler’s 3rd law
using convenient units
3
a
M1  M 2  2
P
Where M in solar masses
a in AU
P in Earth years
Example: a = 0.05 AU, P = 1 day = 1/365 yr, M1 + M2 = 16.6 Msun
Mizer-Alcor : A double-double-double system!
10 arcmin
Alcor
Mizar A
Mizar A+B
Mizar B
Note: Mizar B is
also a binary
with period of 6
months!
Mizar A
(Binary, P = 20.5 days)
Historical Notes on Mizar-Alcor discoveries
•
Romans (c. 200BC): Used Mizar-Alcor (11 arcmin
separation) as test of eyesight for soldiers
•
Benedi Castelli (c. 1613, student and friend of Galileo)
discovers Mizar is a double star (separation 15")
– "It's one of the beautiful things in the sky and I don't
believe that in our pursuit one could desire better",
remarked Castelli in letter to Galileo
–
•
Both Galileo and Castelli were interested in ‘optical doubles’
to prove the heliocentric view of solar system (nearer star
would move w.r.t more distant star annually)
Johann Liebknecht (1722) announced that the 8thmag
star SW of Mizar was a new planet! (Incorrect
observation of motion)
–
He named it Sidus Ludoviciana (Ludwig’s Star) in honor of
his local monarch King Ludwig.
•
1887: Pickering at Harvard announces Mizar A is a
spectroscopic binary, 20.5 day period
•
1996: NPOI directly images the Mizar A binary
(separation 0.008 arcsec)
Alcor
Sidus Ludoviciana
Mizar
A+B
Mizar A: A Spectroscopic Binary
• 1887 Spectroscopy of Mizar A shows periodic
doubling of spectral lines, with 20.5 day period
Note: Asymmetric
light curves
indicated ellipical
orbits
Mizar observations using the NPOI
(Naval Prototype Optical Interferometer, near Flagstaff Arizona)
Determining masses of Mizar-A binary stars from
observations of period, angular separation, distance
1. Distance (from parallax) d = 24 pc (88ly)
2. Max. angular separation (NPOI meas.) Θ = 0.008"
3. Physical separation D = θ·d=0.19 AU
4. Sum of masses (Kepler’s 3rd law)
a3
0.193
M1  M 2  2 
 2.2
2
P
0.056
5. Orbit shows a1 ~ a2 (NPOI meas.) so:
M 1a1  M 2 a2
M 1  M 2  1.1M 
Spectroscopy makes it possible to
study binary systems in which the two
stars are very close together.
Determining component masses of eclipsing
binaries using velocity curves
1. Determine semi-major axis using
observed velocity (V), period (P)
2a1
v1 
P
a  a1  a2
2a2
v2 
P
2. Determine sum of masses using
Kepler’s 3rd law
3
a
M1  M 2  2
P
3. Determine mass ratio using a1, a2
M1a1  M 2 a2 or M1v1  M 2v2
4. Use sum, ratio to determine component masses
a1
a2
a = a 1 + a2
Tilt of Binary Orbits
We have been assuming that we see the binary system face on
when imaging the orbit and edge-on when measuring the velocity.
In general, the orbit is tilted relative to our line of sight. The tilt, or
inclination i, will affect the observed orbit trajectory and the
observed velocities. In general, one needs both the trajectory and
the velocity to completely determine the orbit or some
independent means of determining the inclination.
Light curves of eclipsing binaries provide
detailed information about the two stars.
Light curves of eclipsing binaries provide
detailed information about the two stars.
Eclipsing Binary EQ Tau
Light curve from
Astronomical
Laboratory Course
Fall 2003
Java-animation of
binary stars
Eclipse of an Exo-planet
(HD209458)
“Vogt-Russell” theorem for
spheres of water
• Spheres of water have several properties:
mass, volume, radius, surface area …
• We can make a “Vogt-Russell” theorem for
balls of water that says that all of the other
properties of a ball of water are determined
by just the mass and even write down
equations, i.e.
volume =
mass/(density of water).
• The basic idea is that there is only one way
to make a sphere of water with a given mass.
“Vogt-Russell” theorem
• The idea of the “Vogt-Russell” theorem for
stars is that there is only one way to make a
star with a given mass and chemical
composition – if we start with a just formed
protostar of a given mass and chemical
composition, we can calculate how that star
will evolve over its entire life.
• This is extremely useful because it greatly
simplifies the study of stars and is the basic
reason why the HR diagram is useful.
Mass in
units of
Sun’s mass
Mass - Luminosity
relation for mainsequence stars
Mass-Luminosity relation on
the main sequence
L  M 

 
L  M  
3.5
Mass-Lifetime relation
• The lifetime of a star (on the main sequence) is
longer if more fuel is available and shorter if that
fuel is burned more rapidly
• The available fuel is (roughly) proportional to the
mass of the star
• From the previous, we known that luminosity is
much higher for higher masses
• We conclude that higher mass star live shorter lives
M
M
1
t
 3.5  2.5
L
M
M
A ten solar mass star has about ten times the
sun's supply of nuclear energy. Its luminosity is
3000 times that of the sun. How does the
lifetime of the star compare with that of the
sun?
1.
2.
3.
4.
10 times as long
the same
1/300 as long
1/3000 as long
M
10
1
t


L 3000 300
Mass-Lifetime relation
Mass/mass of Sun
Lifetime (years)
60
400,000
10
30,000,000
3
600,000,000
1
10,000,000,000
0.3
200,000,000,000
0.1
3,000,000,000,000
Stellar properties on main
sequence
• Other properties of stars can be calculated
such as radius (we already did this).
• The mass of a star also affects its internal
structure
(solar masses)
Evolution of stars
• We have been focusing on the properties of
stars on the main sequence, but the chemical
composition of stars change with time as the
star burns hydrogen into helium.
• This causes the other properties to change
with time and we can track these changes via
motion of the star in the HR diagram.
HW diagram for people
• The Height-Weight diagram was for one
person who we followed over their entire life.
• How could we study the height-weight
evolution of people if we had to acquire all of
the data from people living right now (no
questions about the past)?
• We could fill in a single HW diagram using lots of
different people. We should see a similar path.
• We can also estimate how long people spend on
particular parts of the path by how many people we
find on each part of the path.
HR diagram