Transcript The Properties of Stars

The Properties of Stars
Distance and Intrinsic Brightness
Trigonometric Parallax
To measure the distance to star X, photograph it on two dates separated by 6
months.
In the figure, A and B are the positions of Earth on these two dates.
In 6 months, the angular position of X relative to the
X
background stars changes by an amount q.
Half of q is the parallax angle p.
B
p
q
S
A
Photograph on Date 1
Consider triangle BSX: BS = 1 AU, SX = d = the
unknown distance, and BXS = p = the parallax angle
measured from the photographs.
Photograph on Date 2
BSX can be solved for d. Since the angle p is small for all stars, the small angle formula
can be used. The result is
B
206,265
d
p
1 AU
p
where p is in arcseconds
and d is in AU’s.
d
S
The distances to stars are so large that the AU is not a convenient unit for stellar distances.
The distance unit most commonly used is the parsec.
1 parsec = 1 pc = 206,265 AU = 3.26 light years. In these units, the distance formula is
d
1
p
Example
Procyon has a parallax of 0.29". Calculate the distance to Procyon (a) in parsecs and (b) in
light years..
d
1
 3.47 pc
0.288
d  3.47  3.26 ly  11.3 ly
X
Absolute Magnitude and Luminosity
The apparent magnitude of a star is defined by the equation
mB  mA  2.5log
FA

FB
where mA is the magnitude of star A, mB is the magnitude of star B, FA is the flux from
star A, and FB is the flux from star B. The flux tells how bright a star appears to the
detector.
The luminosity of a star is the total electromagnetic energy it emits in a unit of time. Two
stars can have the same luminosity and very different apparent magnitudes. Alpha Centauri
A and the Sun, for example, are both spectral class G2 stars and have about the same
luminosity L = 3.86×1026 W. On the other hand, the apparent magnitude of Alpha Centauri
A is 0.1 and the apparent magnitude of the Sun is –26.8. The difference is due to the fact
that the Sun is much closer to us than Alpha Centauri A is.
In order to make absolute brightness comparisons among stars, we calculate how bright
they would appear if they were all at the same distance from the Sun. The standard
distance used is 10 pc.
The absolute magnitude M of a star is defined as the apparent magnitude it would have if
it were at the standard distance of 10 parsecs.
The Relation between Apparent Visual Magnitude and Absolute
Visual Magnitude
Due to the inverse square law for radiation, the apparent and absolute magnitudes of a star
are related by the following equation.
mv - M v = 5log
d = 10
d
mv = apparent magnitude, M = absolute magnitude, d = distance in parsecs
10 pc
mv - M v + 5
5
mv = M v + 5log
d
10 pc
M v = mv - 5log
d
10 pc
Because m - M can be used to calculate the distance to a star, it is called the star’s distance
modulus.
Example The distance to Alpha Centauri A is 4.37 ly and its apparent magnitude is 0.1.
Calculate its absolute magnitude.
d  4.37 ly 
4.37
pc  1.34 pc
3.26
M v = 0.1- 5´ (- 0.873)
M v = 0.1- 5log
M v = 0.1- (- 4.36)
1.34
= 0.1- 5log (0.134)
10
M v = 4.46
Radii of Stars
The Stefan-Boltzman Law: E = energy emitted by one square meter in one second = s T 4 .
L = energy emitted from the entire stellar surface in one second at all wavelengths.
Surface area of a sphere = A =4p R 2 .
L = 4p R2s T 4
Le = luminosity of the Sun = 3.826´ 1026W
1 W = 1watt = 1Joule per second
Re = radius of the Sun = 6.960´ 105 km = 6.960´ 08 m
Le = 4p Re 2s Te 4
Te = surface temperature of the Sun = 5800 K
L
4p R 2s T 4
=
Le
4p Re 2s Te 4
2
4
L
RT
=
Le
Re 2Te 4
æR
= çç
ççè R
e
2
ö
æT
÷
çç
÷
÷
÷ èççTe
ø
4
ö
÷
÷
÷
÷
ø
R
=
Re
L
Le
æTe
çç
çè T
4
ö
÷
÷
÷
ø
mbol = apparent bolometric magnitude = magnitude including non-visual
electromangnetic radiation
M bol = absolute bolometric magnitude
M bol ,e = absolute bolometric magnitude of the Sun.
L
M
- M
= 2.512 bol ,e bol
Le
Example
The absolute bolometric magnitude of Vega is 0.20 and its surface temperature is 9900 K.
The Sun’s absolute bolometric magnitude is 4.76. Calculate the luminosity of Vega.
L
4.76- 0.20
= 2.512 ( ) = 2.5124.56 = 66.7
Le
L = 66.7 Le
L = 66.7 Le
Example
Vega’s luminosity is 66.7 times the luminosity of the Sun, and its surface
temperature is 9900 K. What is its radius?
L æ
R
= ççç
Le çè Re
66.7
=
8.488
2
ö æT
÷
çç
÷
÷
÷
ø èççTe
æR
çç
ççè R
e
2
ö
÷
÷
÷
÷
ø
4
ö
÷
÷
÷
÷
ø
æR
66.7 = ççç
çè R
e
2
ö
æ9900 ö
÷
÷
çç
÷
÷
÷
÷
ç
÷ è5800 ø
ø
æR
7.86 = ççç
çè R
e
4
2
ö
÷
÷
÷
÷
ø
æR
66.7 = ççç
çè R
R
=
Re
e
2
ö
4
÷
÷
1.797
(
)
÷
÷
ø
7.86 = 2.80
R = 2.80Re = 2.80´ 6.96´ 105 km = 1.95´ 106 km
æR
66.7 = ççç
çè R
e
2
ö
÷
÷
8.488
÷
÷
ø
3000 K
4500 K
5500 K
7500 K
10,000 K
20,000 K
40,000 K
Dependence of Stellar Spectra on Temperature
(Spectral Classification)
Spectral Classes
Type
Hydrogen
Balmer
Line
Strength
Approximate
Surface
Temperature
Main Characteristics
Examples
Singly ionized helium emission or absorption
10 Lacertra
lines. Strong ultraviolet continuum.
O
Weak
> 25,000 K
B
Medium
11,000 - 25,000
Neutral helium absorption lines .
Rigel
Spica
A
Strong
7,500 - 11,000
Hydrogen lines at maximum strength for A0
stars, decreasing thereafter.
Sirius
Vega
F
Medium
6,000 - 7,500
Metallic lines become noticeable.
Canopus
Procyon
Sun
Capella
G
Weak
5,000 - 6,000
Solar-type spectra. Absorption lines of
neutral metallic atoms and ions (e.g.
singly-ionized calcium) grow in
strength.
K
Very
Weak
3,500 - 5,000
Metallic lines dominate. Weak blue
continuum.
Arcturus
Aldebaran
M
Very
Weak
< 3,500
Molecular bands of titanium oxide
noticeable.
Betelgeuse
T = 50,000 K
T = 27,000 K
T = 11,000 K
T = 7200 K
T = 6000 K
T = 5100 K
T = 3700 K
Composition of the Sun’s Atmosphere
Element
% by Number
of Atoms
% by Mass
Hydrogen
92.5
74.5
Helium
7.4
23.7
Oxygen
0.064
0.82
Carbon
0.039
0.37
Neon
0.012
0.19
Nitrogen
0.008
0.09
Silicon
0.004
0.09
Magnesium
0.003
0.06
Sulfur
0.001
0.04
Iron
0.003
0.16
Others
0.001
0.03
Color Indices
It is useful to measure the magnitude of a star through colored filters. One standard set
(called the UBV system) consists of filters that transmit a narrow range around 350 nm
(ultraviolet), 435 nm (blue), and 555nm (green = visual). The apparent magnitudes
measured through these filters are denoted by U, B, and V respectively. The difference
between B and V, called the B – V color index, is closely related to the temperature of the
star. This can be understood by referring to the continuous spectrum of stars with different
temperatures.
FB
FV
FV
FB
Notice that FV is greater than FB for the cooler star, but FB is greater than FV for the
hotter star. This implies that B > V for the cooler star, but B < V for the hotter star.
3400 K Star:
15,000 K Star:
FV  FB
B V  0
FV
1
FB
F
log V  0
FB
F
B  V  2.5 log V
FB
B V  0
The Hertzsprung-Russel Diagram
The luminosity classes are
Ia bright supergiant
Ib supergiant
II bright giant
III giant
IV subgiant
V main sequence
Absolute Visual Magnitude
Stars of the same spectral class can be distinguished from one another by using the
luminosity effect to determine their size. It is found that, when we consider the data for a
large number of stars, the points on the Hertzsprung-Russel diagram are approximately as
shown below. There are very few stars that don’t have MV and B – V that plot near one of
the lines in the graph. Based on this, we classify stars by luminosity as well as spectral
class.
-10
Bright Supergiants
Supergiants
-5
Bright Giants
Giants
0
Subgiants
5
10
15
-0.5
0
0.5
1
1.5
B-V Color Index
2
2.5
The Luminosity Effect
From “An Atlas of Stellar Spectra” by Morgan, Keenan, and Kellman
•
•
•
•
•
In a dense stellar atmosphere, the atoms collide more frequently than in a low
density stellar atmosphere.
The collisions broaden the absorption lines in the star’s spectrum.
Giant stars are larger and less dense than main sequence stars, so their
absorption lines are narrower than those of main sequence stars.
The spectra shown above illustrate this effect. Notice that the absorption lines
in the spectra of the giant (13 Mono ) and supergiant (HR 1040) stars are
narrower than the absorption lines in the main sequence star a Lyrae (Vega).
This enables us to use the star’s spectrum to determine its luminosity as well
as its temperature.