The properties of stars can be grouped into two categories:
Apparent Properties such as:
The brightness (apparent magnitude)
The distance to a star
Absolute (or Intrinsic)
Properties such as:
The
temperature (or color) of the star
The
luminosity (intrinsic brightness; absolute magnitude)
The size (or diameter) of the star
The
mass of the star
The lifetime of the star
In order to determine some of the
intrinsic properties of the star, We must first estimate the
distance to the star - one of the most, if not the most,
difficult measurements in Astronomy.
Distances to Stars:
Distances to nearby stars
measured by a method called parallax. The distance to the star in "parsecs"
is:
d=1/p
Here the parallax angle is
expressed in arc-sec (1 degree = 3,600 arc-sec), and 1 parsec = 3.26 light years.
The European Space agency
satellite "Hipparchus" measures distances up to 1,500 ly = 460
pc using the parallax method.
Intrinsic Brightness of Stars:
We know that Sirius with an apparent magnitude of m=-1.47 is the brightest star in the night sky. However, if we could place all stars at the same distance - say place all stars at exactly 10 parsecs from us - will Sirius still be the brightest? The absolute magnitude (M) of a star is defined as its apparent magnitude if we could place it at exactly ten parsecs or 32 ly from us. Our sun - with a large apparent magnitude m=-27 and located at only 8 light-minutes from us - will look much dimmer if placed at a distance of 10 pc. Indeed, the absolute magnitude of our Sun is of only M=+5. We may compute the absolute magnitude of any star from its apparent magnitude (m) and its distance (in pc)
M = m + 5 -5 log(d) [where the logarithm is in base ten]
Brightness
B = L/(12.6 d2), here L -
Luminosity, and d - distance
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Luminosity of Stars:
The Luminosity (L) of a star indicates how much energy the star emits per second. The Stefan-Boltzmann Law indicates that the amount of energy emitted by the star per second per unit area is: E=sT4. Hence, to obtain the luminosity of the star we just need to multiply the above equation by the total surface area of the star, which equals 4pR2. So - up to universal constants - the luminosity of the star depends on its temperature and radius:
L/Lsun=(R/Rsun)2(T/Tsun)4
| Example:
Suppose that a star has a radius four times as large as the Sun but is only half as hot. What is its relative luminosity? L/Lsun = (4)2*(1/2)4 = 1 Even though the stars are quite different, they have the same exact luminosity. Luminosity of the Sun = 3.83 x 1026 watts ! |
Diameter of Stars:
For a few nearby stars the
diameter can be measured directly. Betelgeuse, a "nearby"
(520-ly) red giant star has a diameter of 500 times that of
the Sun and has been measured directly. Most stars, however,
are either too far or too small to have the diameter measured
directly by photographic methods. Yet, if you can find the
temperature and luminosity of the star you can use the above
equation to obtain the radius (or the diameter) of the star.
For eclipsing binary stars (about 2,000) the diameter of the
stars can be computed by measuring the length of time the
eclipse occurs times the velocity of the star as measured by
Doppler shift. Look
at a Lonely Neutron Star:
Masses of Stars:
About half the stars are in groups of two (binary) or more. In binary stars the mass of each star can be computed by measuring the period and the distance between them. Hence, by measuring the distance between the stars (a in AU) and their period (p in years) you can obtain the total mass (in solar masses) of the binary system: MA + MB = a3/p2. Note that the equation is a relatively simple generalization of Kepler's third law of planetary motion. If A is the Sun and B is any planet, then the sum of the masses is essentially the mass of the Sun - which, of course, equals one in units of solar masses. Hence, for the solar system the above equation becomes a3=p2 - which you should recognize as Kepler's third law of planetary motion. The individual star masses can be computed by measuring the distance to the common center of mass of the two star system.
Binary stars: visual binary where each star is visible in a telescope.
| Example:
Suppose that we observe a binary star
with a period of 8 years MA + MB = a3/p2 = (4)3/(8)2 = 1 The combined mass of the binary star is
exactly one solar mass. |
Spectroscopic binary where the Doppler spectral shift is observed.
The Doppler effect can be
used to determine the radial velocity of a moving object -
such as a star or planet. The laboratory wavelength L0
is the wavelength that a certain spectral line would
have in the "laboratory" where the source is not moving (for
example, the red line in Hydrogen L0=656nm).
However, if the source (ambulance) is moving, the wavelength
of the spectral line will change; we say that the line will be "Doppler
shifted" by a small amount DL.
If the wavelength is increased DL is positive and the source
is moving away from us; we call this a RED shift. If on the other
hand the wavelength decreases, DL will be negative and the
source will be moving towards us; we call this a BLUE shift. By measuring
the Doppler shift in the spectral lines of a well-known
element - such as hydrogen - we can compute the velocity of
the source by using the formula:
v/c=Dl/l0.
By measuring If we could also determine how long does it take the star to complete one orbit we could compute their separation. By having the period and the separation we can extract the mass from Kepler's third law.
When the luminosity of the star is plotted against its temperature, most of the stars (about 90%) lie in a narrow band called the Main Sequence.There are also other stars that lie away from the main sequence due to their stage in their evolution. Bright cool stars are called RED giants and small hot stars called White Dwarfs.
For main sequence stars -
but only for them - the larger the mass, the larger the
luminosity, the higher the surface temperature and the larger
the diameter. Luminosity L is proportional to M3.5.
The Power of the H-R Diagram:
Consider the following information on:
SIRIUS ("scorching")
Other Names
Canicula; Dog
Star; Aschere.
Alpha Canis
Majoris
HR 2491
HD 48915
Data
RA 06 45 08.9
Dec -16 42 58
V -1.46
B-V 0.00
Spectral Type
A1Vm
Distance = 8.6 ly
If you look under Spectral Type
you will discover that Sirius is an A1 - Main-Sequence Star (V). The A1 classification
tells you by looking at the H-R diagram that Sirius
has approximately:
Luminosity = 60
Absolute
Magnitude = +1.5
Surface
Temperature = 9,000 K
Since we can measure Sirius
apparent visual magnitude relatively easy to be m = -1.5 we can
extract a wealth of information about Sirius:
| Sirius (Approximate)
Properties:
From Wien's Law: Lmax=3,000,000/T
=
333 nm The H-R diagram is
extremely powerful, indeed!! |