Download Properties of Stars

Stars come in all sizes, small, large
and even larger.
Stellar Properties:
There are 6 important properties of
stars.:
(1) Distance – measure parallax for
closer stars
(2) Effective Temperature (T) - determined
from continuous spectrum (blackbody curve),
Wien’s Law.
(3) Luminosity (L) - determined from
apparent magnitude and distance, or from
spectrum (luminosity class).
(4) Chemical
composition - determined
from line spectrum.
(5) Radius (R) - determined from
luminosity and temperature, or from
distance .
(6) Mass (M) - determined from binary
stars
We can get some accurate distances by Stellar
Parallax
1
• The distance to a star in parsecs is: d  p
1 parsec = 3.26 light-years =
13km
3.09x10
One arcsecond = 1’’and, is the angular size of a
dime seen from 2 miles or a hair width from 60 feet.
•The nearest star, aside from the Sun, is called
Proxima Centauri with a parallax of 0.77
arcsecond. Its distance is therefore: 1.3 pc
1
d
1.3pc
0.77
Inverse Square Law of Brightness
•The apparent brightness of a
light source varies inversely
with the square of the
distance
•Brightness = 1/D2 = 1/D2
The Apparent Brightness of a source is inversely proportional to
the square of its distance:
2-times Closer or 1/2 distance = 4-times Brighter
2-times Farther = 1/4-times as Bright
Black Body Radiation
A Blackbody is a perfect absorber, an object
that absorbs light at all wavelengths, and it
heats up.
It is also the perfect radiator: emits at all
wavelengths (continuous spectrum)
characterized by its Temperature.
Energy emitted depends strongly on Temp.
Stars act nearly like Black Bodies.
Radiation Laws
Three Laws Characterize Continuous Spectra
Planck
Wien
Stefan-Bolzman
With these laws we can determine the temperature &
other characteristics of stars.
1. The Planck radiation law assumes that the object observed is a
perfect radiator and absorber of energy (black body).
2. Stars, although not perfect black bodies, are close enough so that
Planck curves are useful descriptions of their radiation.
Wien’s Law for Black Bodies : The peak of a black body curve
shifts toward shorter wavelength if the temperature is increased.
Black Body Temperature
According black-body radiation, the
spectrum of a hotter star will have a
higher, sharper peak closer to the
blue end of the spectrum.
• Star A is
hotter
• Star B is
cooler
• A cooler star will have a
lower, flatter peak
closer to the red end.
• Our
Sun
Continuum & Lines
Real stars usually have a blackbody-like continuous
spectrum, upon which absorption lines are superimposed
A spectrum can be converted to a trace spectrum.
Flux
Hydrogen
Continuum
Absorption Lines
4000
5000
6000
Wavelength
7000
Wien’s Law
The peak wavelength of a Black Body depends upon the
Temperature.
The higher the temperature, the shorter the wavelength of
the peak radiation.

Is measured in nm which is 10 9 m
 Needs to be in meters (m)
3
  2.9 X 10
or
T  2.9 X 10
3
T

So, we can get the temperature of a star form its
spectrum.
Problem
The sun’s max intensity is at a wavelength of
9
about 500nm or 500 X 10 m
Using Wien’s Law, calculate the sun’s
surface temperature.
T  2.9 X 10
T  2.9 X
3
3

/ 500 X 10
9
T  .0058 X 10  5800 K
6
F


T
If you know the temperature of a
Stefan-Boltzmann Law
Black Body then the total energy
emitted from from each square
meter, called Energy Flux (F),
can be calculated.
4
F
R
It only depends upon the temperature of the object
and a constant. It’s the rate of heat flow/sec.
This is only for a square meter and stars are
different sizes, so to find the total energy, which is
called Luminosity, we change the formula to :
L  FA  F 4R  4R  T
2
2
4
Luminosity 
Temperature
Surface Area
T
4
(how hot)
4R
2
(how big)
So,Luminosity depends upon Radius(R) & Temperature(T)
Luminosity is, the total amount of energy per
second emitted. The Star’s total Wattage!
2
The area of a sphere is , A= 4R
So, the total energy emitted by the object each
second is called the Luminosity (L).
L  FA  F 4R
2
 4R  T
2
4
Brightness: How bright something appears to us,depends
on temperature, size, and the distance.
Same Size
Greater Temperature
Greater Luminosity
These 2 will put out the same energy per square unit because:
same temperature
This one is much
bigger (R)
So the total L is
much more.
L=4R T
2
4
This formula relates a Star’s Temperature
and Surface Area (its size) to its Luminosity.
 is a constant that you will not have to use.
L=4 R T
2
4
looks rather messy
It’s more natural to compare an object to a
known object. Comparing a star to the Sun
would be easier and more helpful, since we
know a lot about the Sun and it is a star.
Let’s get rid of the constants !
Since 4  and  are constants , in the next
formula they cancel each other when
compared to the sun.
2
L
4   R   T 


 

L
4   R   T 
Giving
 R 
L




L
 R 
2
 T

T
 




4
4
Example Problem
Betelgeuse has a Luminosity of 60,000 L and
a surface temperature of 3500 K. Find the
radius compared to the Sun.
 R 
L

 
L  R 
2
T

 T



4
60, 000  R
60,000  R 
 
1
1
2
2
 3500 


 5800 
.1322 
453, 857.8  R
673.7 R  R
2
4
Suppose a star is 10 times the Sun’s radius,
but only ½ as hot.Find the luminosity of the
star compared to the Sun.
2
 R  T 
L

 

L  R   T 
4
4
1

2
L  10   2 
1
      (100)( )  6.25L
L  1   1 
16
 
The star is 6.25 times the Sun’s Luminosity
The next two formulas are for Main
Sequence stars only !
1
Life Time 
3.5
2.5
LM
M
M is the mass of the star in solar mass.
L is the Luminosity of the star in solar Luminosity.
Life Time is the approximate life time of a MS star
in solar life times.
What is the Luminosity of a MS star that has a
mass 4 times the sun ?
LM
3.5
L4
3.5
 128 Lsun
How long can a 4 solar mass MS star live ?
1
T 
2.5
M
1
T  2.5  1/ 32  .031
4
Solar life times
Or, since the sun will live for 10 billion years
T   3.1X 10
2
1X 10 
T  3.1X 10 years
8
10
Over half of the stars in the sky have stellar
companions, bound together by gravity and in orbit
around each other.
Types of Binaries
Visual Binaries
Optical Binaries- are chance superpositions,
where two stars appear close together but do not
actually orbit one another. (Like Mizar & Alcor)
Physical Binaries- where one star orbits
another, and each star can be seen in the
telescope.
OPTICAL DOUBLES
• Not a true binary system
• Stars only appear close together in the sky
• Mizar & Alcor in the Big Dipper
While Alcor and Mizar are Optical Double stars
and only appear to be near each other, Mizar is
actually a Physical Binary star.
Types of Physical Binaries
Eclipsing Binary –(If the angle is good ) two
stars that regularly eclipse one another
causing a periodic variation in brightness.
Spectroscopic Binary - two stars that are
found to orbit one another through
observations of the Doppler effect in their
spectral lines .
At least half of the stars in the sky are
binaries. Eclipsing Binary stars are also
referred to as Extrinsic Variable Stars.
Orbits and Masses of Visual Binaries
The primary importance of binaries is that they
allow us to measure stellar parameters
(especially mass). The center of mass is the
location where a fulcrum would be placed to
balance the stars on a seesaw.
Masses of Binary stars
Newton’s Modification of Kepler’s Law
P must be in years, a in AU
M in solar mass, where Sun = 1
Problem - Ignoring the mass of one object.
A nearby star Epsilon Eridani has a planet circling the
star at a distance of 3.4 AU. The period of the planet is
7.1 years. Find the mass of the star, assuming the mass
of the planet to be negligible.
M1  M 2 )P  a
2
3
M star
M star
M star
a
 2
P
3
(3.4)

2
(7.1)
 0.78 M sun
3
When dealing with binary stars, the mass of the two stars
are similar, and cannot be simply ignored. Sirius b is a
white dwarf, and its orbital period around Sirius takes 50
years.If the distance between the the stars is 20 AU, find
the mass of the stars.
(M1  M 2 ) P  a
2
3
3
( M sirius
( M sirius
( M sirius
a
 M siriusB )  2
P
3
(20)
 M siriusB ) 
2
(50)
 M siriusB )  3.2 M sun
M SiriusA  M SiriusB  3.2M Sun
But we happen to know that Sirius’ mass 1.99 Msun
So,
and so,
1.99M sun  M SiriusB  3.2M Sun
M SiriusB  1.2M sun
Sometimes we might be able to get
information about one of the stars from the HR diagram to help us determine the mass of
both stars.
Eclipsing Binaries
Sometimes the orbital plane is lined up so
that the stars pass in front of each other as
seen from the Earth. Each eclipse will cause
the total light from the system to decrease.
The amount of the decrease
will depend on how much of
each star is covered up (they
can have different sizes) and
on the surface brightness
of each star.
Spectroscopic Binaries
Some binaries are too close together to be resolved,
you may still be able to detect the binary through
the Doppler shift (in one or both stars). They must
be relatively close to each other (short orbital
period).
If you can see both stars’ spectrums, you may
be able to use Doppler shifts to measure the
radial velocities of both stars.
This gives you the mass ratio, regardless of the
viewing angle (e.g. nearly face-on, nearly edgeon, etc.). This is usually useful information.
Thank goodness, my brain is full