Stellar Properties

1
Stellar Properties
How We Determine The Distances, Motions, Sizes,
Temperatures, and Masses of the Stars
2
Trigonometric Parallax
A star is observed against a background of
distant stars at 6-month intervals
t1
Trig Parallax p
p
p
Earths orbit
Nearby Star
t2
Very Distant Background Stars
2 x p
t2
t1
On photographic plates, the nearby star appears
to shift back and forth with respect to the
distant background stars
3
Trigonometric Parallax
A star is observed against a background of
distant stars at 6-month intervals
t1
Trig Parallax p
p
Earths orbit
More Distant Star
t2
Very Distant Background Stars
2 x p
t2
t1
A somewhat more distant star presents a smaller
back and forth with respect to the distant
background stars
4
Parallax and Distance
The distance to a star is inversely related to
its parallax, i.e. nearer stars have larger
parallaxes. The relation between parallax p and
distance d is d 1/p Where, if p is measured
in seconds of arc, d is then determined in
parsecs. Thus, a star whose parallax is 1.0
arcseconds has a distance of 1.0 parsecs one
with p 0.01 seconds of arc has a distance of
1/0.01 100 parsecs, etc. One parsec equals
3.26 light years (the distance light travels in
one year) or about 2x1013 miles! That is more
than 200,000 AU. The nearest star to the sun is
Proxima Centauri whose parallax is 0.76 seconds
of arc, the angle subtended by a dime seen from a
distance of nearly 3 miles! The distance to this
star is approximately 270,000 AU or 6,750 times
further from the sun than Pluto. This is the
nearest star! Although parallax was postulated
long before Copernicus and Galileo attempted to
measure this shift, it was not until 1835 by
Friedrich Bessel measured the parallax of the
star 61 Cygni.
5
The Five Nearest Stars

• Star Parallax
  Distance
• 
  arcsec
  pc
• a Centauri 0.76 1.31
• Barnards Star 0.54 1.83
• Wolf 359 0.43 2.35
• Lalande 21185 0.40 2.49
• Sirius 0.37 2.67

Some of the Brightest Stars

• Star Parallax
  Distance
• 
  arcsec
  pc
• Sirius 0.37 2.67
• Canopus 0.03 30
• a Centauri 0.76 1.31
• Arcturus 0.09 11
• Vega 0.13 8
• Capella 0.07 14
• Betelgeuse 0.006 150
• Deneb 0.002 430

Note that the brightest stars are not necessarily
the nearest. What does that mean?
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Stellar Motions I. Proper Motion
All stars, including the sun, move through our
home galaxy, the Milky Way, at speeds of hundreds
of kilometers per second with respect to the
center of the Galaxy. Stars move relative to
each other at speeds of a few tens of km/sec to
more than 100 km/sec, depending on their motions
in the Galaxy. While these seem to be large
speeds, the distances to stars are so great that
the effect on the apparent motion of a star is
quite small. This causes stars to appear to us
to slowly drift in angular motion on the sky.
This angular drift is known as proper motion.
This quantity is often abbreviated by the Greek
letter m.

• Star Proper
  Motion Distance
• 
  arcsec/year
  pc
• Sirius 1.32 2.67
• a Centauri 3.68 1.31
• Capella 0.44 14
• Betelgeuse 0.03 150
• Deneb 0.00 430

Are m and d related in this table?
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Stellar Motions II. Radial Velocity
The Doppler Effect causes the spectrum of a star
to be slightly shifted towards longer wavelengths
if the star is moving away from us and shorter
wavelengths if it is approaching us. This effect
can be accurately measured using the technique of
spectroscopy.
VR
Star A
Sun
VR
Star B
Star A has a component of motion away from us
while star B is approaching us. These line of
sight components of the actual motion, shown here
as dotted lines, are the radial velocities of the
two stars, shown as VR in the diagram.
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Stellar Motions III. Tangential Velocity
The motion of a star perpendicular to the line of
site is called the tangential velocity shown as
VT in the diagram below. This can only be
measured if the proper motion and the distance
are known. The actual velocity of a star through
space measured relative to the sun, i.e. as if
the sun were stationary, is called the space
velocity, VS .
VR
Star
Sun
m
VS
VT
Once the radial and tangential velocities are
measured, the space velocity can be determined
from the Pythagorean Theorem as VS (VR2
VT2)1/2.
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Proper Motions of the Big Dipper
Recall that the Big Dipper is an asterism of
the constellation Ursa Major.
Positions of the Big Dipper Stars in the Year
4000 AD
Positions of the Big Dipper Stars in the Year
10,000 AD
Positions of the Big Dipper Stars in the Year
50,000 AD
Positions of the Big Dipper Stars in the Year
100,000 AD
Positions of the Big Dipper Stars in the Year
250,000 AD
a Dubhe
d Megrez
b Merak
e Alioth
80 Alcor
z Mizar
g Phecda
h Alkaid
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Proper Motions of the Big Dipper
Recall that the Big Dipper is an asterism of
the constellation Ursa Major.
Stars of the Big Dipper as seen today
a
d
b
e
80
z
g
Proper Motion Effect After 250,000 years
h
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Motions Distances of Big Dipper Stars

• mX mY
  VR d
• (milliarcseconds)
  (km/sec) (parsecs)
• a UMa Dubhe -136.5 -35.3 -8.9 37.9
• b UMa Merak 81.7 33.7 -12.0 24.3
• g UMa Phecda 107.8 11.2 -12.6 25.6
• d UMa Megrez 103.6 7.8 -13.4 24.9
• e UMa Alioth 111.7 -9.0 -9.3 24.8
• z UMa Mizar 121.2 -22.0 -5.6 23.9
• h UMa Alkaid -121.2 -15.6 -10.9 30.9
• 80 UMa Alcor 120.4 -16.9 -8.9 24.8

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The Spectum of a Star
Red
Orange
Star Light Collected by a Telescope
l
Yellow
Green
Blue
Glass Prism
Indigo
Violet
Intensity
Focus of a spectrograph
l
The spectrum is a plot of the intensity of light
as a function of wavelength.
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Classical Astronomical Spectrograph
Mildly off-axis collimator
Fork-Mounted Equatorial with Coude Relay Optics
Polished Slit
Photographic Plate or Digital Detector
Diffraction Grating
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Black Body Spectra I. Appearance
A black body is an idealized perfect radiator of
electromagnetic energy whose spectrum depends
only on the temperature of the black body. Stars
radiate approximately as such objects. (Dont
confuse these with black holes! Thats
something else altogether.)
Black body spectra of stars with temperatures of
4500, 5000, and 5500 Kelvins
Energy Emitted
Wavelength l
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Black Body Spectra II. Properties
Note the following from the spectra corresponding
to three values of temperature 1. Hotter BBs
radiate more energy at all ls than do cooler
BBs. 2. Hotter BBs are brightest at shorter
ls than cooler BBs. 3. BBs radiate at all ls
. Emitted energy asymptotically goes to zero.
Black body spectra of stars with temperatures of
4500, 5000, and 5500 Kelvins
Energy Emitted
Wavelength l
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Black Body Spectra III. Wiens Law
This law of radiation from a black body states
that the wavelength at which a black body is
brightest, lmax , is inversely proportional to
the temperature, T, of the black body. Or, lmax
0.2897 T gives the wavelength in centimeters
at which a black body is brightest when T is
measured in degrees Kelvin
This law lets you determine the temperature of a
star simply by measuring the wavelength in its
spectrum where the star is brightest. For
example, the suns spectrum peaks in brightness
at l 5.0x10-5 cm, indicating that its
temperature is about 5800 K
Wavelength (cm)
5800
5x10-5
Temperature (degrees K)
17
Black Body Spectra IV. Stefan-Boltzmann Law
If the radiation at every wavelength is added
together, this law states that the total amount
of energy emitted over all wavelengths from each
square centimeter of the surface of a blackbody
is proportional to the temperature T raised to
the fourth power. Or, E 5.67 x 10-5
T4 gives the energy in (ergs per cm2) when T is
measured in degrees Kelvin
This law lets you see how the energy emitted from
a black body increases or decreases as the
temperature increases or decreases. In the case
of the sun with T 5800 K, if its temperature
were doubled to 11,600 K, the amount of energy
coming from every cm2 of the suns surface would
increase by a factor of 24 16.
16
Energy Relative to Sun
5800
1
11,600
Temperature (degrees K)
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The Colors of Stars
Compare the brightness of a star at two different
wavelengths. The difference in brightness is
defined as the color of a star, and this quantity
is related to the stars temperature.
Hotter Star
The hotter star is equally bright at the two
wavelengths, but the cooler star is brighter at
the longer wavelength.
Energy Emitted
Cooler Star
Wavelength l
This is another method for measuring stellar
temperature in addition to the application of
Wiens Law.
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Stellar Luminosity
The luminosity, L, of a star is the total amount
of energy emitted from the entire surface of a
star at all wavelengths. It it thus the amount of
energy emitted from every square centimeter
multiplied by the total number of square
centimeters on the surface of a spherical star of
radius R. Or, L (5.67 x 10-5 T4) x
(4pR2) gives the total energy (in ergs) emitted
at all wavelengths from the entire star.
Think about what this means Basically, the
luminosity is proportional to R2T4 so you can
easily calculate what happens to L if you change
R or T by some multiplicative factor. For
example, what would be the effect on the sun if
its temperature is changed from 5,800 K to 11,600
K and the sun is shrunk to half its present
radius?
Well, in this example The temperature would
double, resulting in a factor of 24 16 increase
in luminosity, but the radius would halve,
resulting in a factor 0.52 0.25 decrease in
luminosity. The net result would be that the
suns luminosity would be changed by the product
of these two factors, or by 16 x 0.25 4. So,
the suns luminosity would become four times its
present value. (Enough to exterminate life on
Earth, incidentally.)
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Spectral Classes of Stars
On the basis of the appearance of the spectra of
stars, astronomers in the early 20th century
assigned letters to a variety of categories based
upon the nature of spectral features that we now
know are the result of absorption and emission of
light by the chemical elements in the atmospheres
of stars. It turns out that the sequence of
spectral classes is essentially another indicator
of a stars temperature.
The modern sequence of spectral classes (or
types) is Spectral Class
Spectrum Shows Features of O
Strongly ionized elements B A
Strong neutral hydrogen F G
Many heavier elements (Fe in particular) K M
Molecules (TiO in particular)
Temperature
These are further subdivided for higher accuracy
by assigned numbers from 0 to 9 to indicate small
increases in temperature. For example a B2 star
is hotter than a B8 star.
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Luminosity Classes of Stars
Again, on the basis of the appearance of the
spectra of stars, astronomers discovered that the
density of gas and the strength of gravity at the
surface of a star indicate that some stars are
much larger than other, even if their
temperatures are the same. This difference is
denoted by the luminosity class of a star.
The sequence of luminosity classes is
Luminosity Class Name assigned to class I
Supergiants II
Bright Giants III Giants IV
Sub-giants V
Dwarfs VI Sub-dwarfs
The complete classification of a star is based
upon the spectral type and luminosity class of a
star. Thus, it turns out that the sun is
classified as a G2V star. Our old friend
Betelgeuse is an M1I star.
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The H-R Diagram I.
The Hertzsrpung-Russell Diagram is named for
the Danish and American astronomers who
discovered relationships between the spectral
types and luminosities of stars. It is one of the
most important tools in astronomy.
Supergiant Branch (I)
Giant Branch (III)
Main Sequence (V)
.
Henry Norris Russell first described his
discovery of this relationship at a meeting of
the American Astronomical Society in Atlanta in
December 1913. The AAS met here last in January
2004.
Luminosity
Sun
White Dwarfs (WD)
O B A F G K M
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The H-R Diagram II. Luminosity Temperature
Temperature
Because spectral type is a proxy for temperature,
the HR Diagram shows fundamental relationships
between temperature and luminosity.
50,000 20,000 10,000 5,800 5000 3500
Supergiant Branch (I)
10,000
Giant Branch (III)
100
Main Sequence (V)
.
Consider three K0 stars of luminosity classes I,
III, and V. They all have the same temperature.
Why is there such a difference in
luminosity? Think about the two things that
determine a stars luminosity.
Luminosity (times that of sun)
1.0
Sun
0.01
White Dwarfs (WD)
0.0001
K0
O B A F G K M
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The H-R Diagram II. You Plot Em
Temperature
Look at the following information for some stars
and plot them on the HR Diagram.
50,000 20,000 10,000 5,800 5000 3500
Supergiant Branch (I)
10,000
Look at the following information for some stars
and plot them on the HR Diagram. Proxima
Centauri M5V Rigel Kentaurus (a Cen)
G2V Barnards Star M4V Sirius A (a CMa)
A1V Sirius B DA Arcturus (a Boo)
K1III Vega (a Lyr) A0V Betelgeuse (a Ori)
M1I Deneb (a Cyg) A2I Rigel (b Ori)
B8I Spica (a Vir) B1V
Giant Branch (III)
100
Main Sequence (V)
.
Luminosity (times that of sun)
1.0
Sun
0.01
White Dwarfs (DA)
0.0001
O B A F G K M
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The H-R Diagram III. You Figure it Out
Temperature

• For these stars, which is
• the hottest?
• the most luminous?
• the least luminous?
• the smallest?
• the largest?
• most like the sun?
• Proxima Centauri M5V
• Rigel Kentaurus (a Cen) G2V
• Barnards Star M4V
• Sirius A (a CMa) A1V
• Sirius B DA
• Arcturus (a Boo) K1III
• Vega (a Lyr) A0V
• Betelgeuse (a Ori) M1I
• Deneb (a Cyg) A2I
• Rigel (b Ori) B8I
• Spica (a Vir) B1V

50,000 20,000 10,000 5,800 5000 3500
Supergiant Branch (I)
10,000
Giant Branch (III)
100
Main Sequence (V)
3
.
6
Luminosity (times that of sun)
1.0
Sun
1
4,
0.01
5
White Dwarfs (DA)
2
0.0001
O B A F G K M
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Binary Stars
System of two stars gravitationally bound in an
orbit around a common center of gravity such
that M1/M2 r2/r1
M2
r2
r1

M1
where M1 gt M2.
Keplers Third Law MA MB a3/P2
where a orbital semimajor axis (in
AU) P orbital
period (in years) Permits the determination of
the mass of the two stars. This is the only means
we have for measuring stellar masses.


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Types of Binary Stars

• Visual Binaries The individual two stars can be
  separately seen by the eye or by other techniques.

• Spectroscopic Binaries Duplicity detected
  through variable radial velocity using the
  Doppler effect.

• Eclipsing Binaries Duplicity detected through
  variations in brightness arising from eclipses
  when the orbital plane is nearly edge on.

• Composite Spectrum Binaries These objects
  spectra show dual types that can only be
  explained by the presence of two stars of
  different spectral types that are too close
  together to be seen as visual binaries.

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The Visual Binary x Ursae Majoris
From Chris Dolans website at the University of
Wisconsin
Orbit from 1,223 measurements Orbital Period of
59 years
Alula Borealis Alula Australis
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Light Curve of an Eclipsing Binary
1st contact
4th contact
Intensity
mid-eclipse
2nd contact
3rd contact
Time
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Stellar Radii from Eclipsing Binaries
With P Period a Semi- major axis
Ra
Rb
(t2 t1)/P (t4 t3)/P 2 Ra/2pa
and
(t3 t2)/P 2 Rb/2pa
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Mass-Luminosity Relation for Stars (As determined
from Binary Star Orbits)
At the simplest level, it shows that more massive
stars are more luminous, indicating that the
amount of energy produced in the interiors of
stars is related to their mass.
Updated by W.I. Hartkopf (1999)