The amount of energy emitted from stars is determined by measuring their brightness or the amount of light they emit. This is called photometry. However, two major developments expanded our understanding of the chemical make-up of stars. They were:
Three leaders in this field were:
Fraunhofer who, in the early 1800's, magnified the Sun's spectrum and discovers spectral lines
Kirchhoff who, in the mid-1800's, developed the three laws of spectroscopic analysis which, in turn, is used to determine the chemical composition of the Sun and stars.
Lockyer who, in the late-1800's, discovered an unknown element in the Sun, later named helium.
Kirchhoff's Laws:
Kirchhoff showed that there are three types of spectra emitted by objects:
1) Continuous spectrum - a solid or liquid body radiates an uninterrupted, smooth spectrum (called a Planck curve)
2) Emission spectrum - a radiating gas produces a spectrum of discrete spectral lines
3) Absorption spectrum - a continuous spectrum that passes through a cool gas has specific spectral lines removed (inverse of an emission spectrum)
Planck's curve:
One of the primary results from the field of spectroscopy was the discovery of how the spectrum of an object changes with temperature. In particular, was the formulation of the two laws of radiation:
Stefan-Boltzmann's law relates the energy output of a luminous object, in ergs, with its temperature, in Kelvins (note that Kelvin is a measure of temperature from absolute zero), such that:
where σ is Stefan-Boltzmann's constant of 5.67x10-5 erg cm-2 sec-1 Kelvins-4.
Again, it is often easier to talk in terms of ratios for two objects of temperatures, T1 and T2, and their energy output is E1 and E2 such that:
notice that the σ's cancel out and we have
Wien's law determines the peak wavelength emitted by an object, and is given by:
So, for example, the Sun, which has a surface temperature of 5500K, emits its peak energy at
where yellow light is about 5.2x10-5 cm.
Quantum Physics :
The word quantum derives from quantity and refers to a small packet of action or process, the smallest unit of either that can be associated with a single event in the microscopic world.
Changes in energy, such as the transition of an electron from one orbit to another, are done in discrete quanta. Quanta are not divisible. The term `quantum leap' refers to the abrupt movement from one discrete energy level to another, with no smooth transition. There is no `inbetween'.
The quantization, or `jumpiness' of action, as depicted in quantum physics, differs sharply from classical physics which represented motion as smooth, continuous change.
The field of quantum mechanics concerns the description of phenomenon on small scales where classical physics breaks down.
The quantum world can be not be perceived directly, but rather through the use of instruments. The question of the reality of quantum properties remains unsolved. All quantum mechanical principles must reduce to Newtonian principles at the macroscopic level (there is a continuity between quantum and Newtonian mechanics).
Emission and Absorption of Light with the Bohr atom:
Bohr developed a different model of the atom to account for quantum physics, and the spectra of elements. The Bohr atom is similar to Rutherford atom, except the electrons moved in fixed or quantized orbits.
The quantized orbits of the electrons allows for a simple explanation of the origin of photons, and the spectrum of light. Photons are produced by the transition of electrons downward in their orbits. A downward transition releases potential energy in the form of a light particle, a photon. Likewise, photons could be absorbed by electrons, and they move upward in their orbits.
Stellar Color:
Stars have a range of colors which represent their surface temperatures due to Wien's law (which states that the peak emission of light from an object goes as the inverse of temperature). The color of a star is determined by that part of the visible spectrum where the peak amount of radiation is emitted.
Blue stars are extremely hot, red stars are relatively cool. Temperature here is a relative thing; cool means temperatures near 2,000 to 3,000K, about 15 times hotter than your oven. Blue stars have temperatures near 20,000K. The Sun is an intermediate yellow star with a surface temperature of 6,000K. The color of a star is determined by measuring its color index.
It is important to remember temperature and luminosity for a star are not strictly related. Stefan-Boltzmann's law states that the amount of energy emitted goes as the temperature to the 4th power; but, this relation is only strictly true for an object that is a point source (i.e. it has no size). The temperature of a normal object is proportional to its surface area (for example, things cool faster if you spread them out = increase their surface area).
So, it is possible for a star to be very bright (emit alot of energy) yet, be cool and red. We will see below that this means the star must be very large to be both bright and cool.
Stellar Spectral Type:
Stars are divided into a series of spectral types based on the appearance of their absorption spectra. Some stars have a strong signature of hydrogen (O and B stars), others have weak hydrogen lines, but strong lines of calcium and magnesium (G and K stars). After years of cataloging stars, they were divided into 7 basic classes: O, B, A, F, G, K and M. Note that the spectra classes are also divisions of temperature such that O stars are hot, M stars are cool.
Between the classes there were 10 subdivisions numbered 0 to 9. For example, our Sun is a G2 star. Sirius, a hot blue star, is type B3.
Why do some stars have strong lines of hydrogen, others strong lines of calcium? The answer was not composition (all stars are 95% hydrogen) but rather surface temperature.
As temperature increases, electrons are kicked up to higher levels (remember the Bohr model) by collisions with other atoms. Large atoms have more kinetic energy, and their electrons are excited first, followed by lower mass atoms.
If the collision is strong enough (high temperatures) then the electron is knocked off the atom and we say the atom is ionized. So as we go from low temperatures in stars (couple 1,000K) we see heavy atoms, like calcium and magnesium, in the stars spectrum. As the temperature increases, we see lighter atoms, such as hydrogen (the heavier atoms are all ionized by this point and have no electrons to produce absorption lines).
As we will see later, hotter stars are also more massive stars (more energy burned in the core). So the spectral classes of stars is actually a range of masses, temperatures, sizes and luminosity. For normal stars (called main sequence stars) the following table gives their properties:
type Mass Temp Radius Lum (Sun=1)
-------------------------------------------
O 60.0 50,000 15.0 1,400,000
B 18.0 28,000 7.0 20,000
A 3.2 10,000 2.5 80
F 1.7 7,400 1.3 6
G 1.1 6,000 1.1 1.2
K 0.8 4,900 0.9 0.4
M 0.3 3,000 0.4 0.04
-------------------------------------------
So our Sun is a fairly middle-of-the-road G2 star:type Mass Temp Radius Lum (Sun=1) ------------------------------------------- O 60.0 50,000 15.0 1,400,000 B 18.0 28,000 7.0 20,000 A 3.2 10,000 2.5 80 F 1.7 7,400 1.3 6 G 1.1 6,000 1.1 1.2 K 0.8 4,900 0.9 0.4 M 0.3 3,000 0.4 0.04 -------------------------------------------
A B star is much larger, brighter and hotter. An example is HD93129A shown below:
Luminosity Classes:
Closer examination of the spectra of stars shows that there are small changes in the patterns of the atoms that indicate that stars can be separated by size called luminosity classes.
The strength of a spectra line is determined by what percentage of that element is ionized. An atom that is ionized has had all its electrons stripped off and can produce no absorption of photons. At low densities, collisions between atoms are rare and they are not ionized. At higher densities, more and more of the atoms of a particular element become ionized, and the spectral lines become weak.
One way to increase density at the surface of a star is by increasing surface gravity. The strength of gravity at the surface of a star is determined by its mass and its radius (remember escape velocity). For two stars of the same mass, but different sizes, the larger star has a lower surface gravity = lower density = less ionization = stronger spectral lines.
This was applied to all stars and it was found that stars divide into five luminosity classes: I, II, III, IV and V. Stars of type I and II are called supergiants, being very large (low surface gravity), stars of type III and IV are called giant stars. Stars of type V are called dwarfs. The Sun is a G2 V type stars.
So now we have a range of stellar colors and sizes. For example, Aldebaran is a red supergiant star:
Arcturus is an orange giant star:
HST imaging found that Betelgeuse is one of the largest stars, almost the size of our whole solar system.
The other extreme was also found, that there exist a class of very small stars called white and brown dwarfs, with sizes close to the size of the Earth:
Red and blue supergiant stars, as well as giant stars exist. The following is a comparison of these types.
Luminosity Function:
Surveying the skies for stars is a very biased method of doing science since clearly the brightest stars are the easiest to observe. But are the brightest stars typical of the stellar population? To determine what a typical star is like we construct a luminosity function, the number of stars as a function of absolute magnitude in the form of a histogram.
A luminosity function is constructed by sampling a volume of space and counting all the stars in that volume. The resulting plot will look like the diagram below:
Notice that the most common type of star is actually small, low luminosity stars. Bright stars are quite rare (although they can be seen from great distances). Since luminosity is correlated with mass, then this means that high mass stars are rare.
Russell-Vogt Theorem:
Despite the range of stellar luminosities, temperatures and luminosities, there is one unifying physical parameter. And that is the mass of the star. Hot, bright stars are typically high in mass. Faint, cool stars are typically low in mass. This sole dependence on mass is so strong that it is given a special name, the Russell-Vogt Theorem.
The Russell-Vogt Theorem states that all the parameters of a star (its spectral type, luminosity, size, radius and temperature) are determined primarily by its mass. The emphasis on `primarily' is important since we will soon see that this only applies during the `normal' or hydrogen burning phase of a star's life. A star can evolve, and change its size and temperature. But, for most of the lifetime of a star, the Russell-Vogt Theorem is correct, mass determines everything.
Binary Stars:
Planet's revolve around stars because of gravity. However, gravity is not restricted to between large and small bodies, stars can revolve around stars as well. In fact, 85% of the stars in the Milky Way galaxy are not single stars, like the Sun, but multiple star systems, binaries or triplets.
If two stars orbit each other at large separations, they evolve independently and are called a wide pair. If the two stars are close enough to transfer matter by tidal forces, then they are called a close or contact pair.
Binary stars obey Kepler's Laws of Planetary Motion, of which there are three.
- 1st law (law of elliptic orbits): Each star or planet moves in an
elliptical orbit with the center of mass at one focus.
- 2nd law (law of equal areas): a line between one star and the
other (called the radius vector) sweeps out equal areas in equal
times
- 3rd law (law of harmonics): The square of a star or planet's orbital period is proportional to its mean distance from the center of mass cubed
Visual Binaries:
Any two stars seen close to one another is a double star, the most famous being Mizar and Alcor in the Big Dipper. Odds are, though, that a double star is probably a foreground and background star pair that only looks near each other. With the invention of the telescope may such pairs were found. Herschel, in 1780, measured the separation and orientations of over 700 double stars and found that only about 50 pairs changed orientation over 2 decades of observation.
One such example is Sirius A and B shown below. Their motion through the sky is a complex, twisted path which takes decades to map and plot.
The observations made relative to center of mass of the two stars shows their respective elliptical orbits.
Eclipsing Binaries:
In the late 1600's, Italian astronomers noticed that some stars occasionally drop in their brightness up to 1/3 their peak luminosity. Later measurements showed that these declines were periodic, ranging from hours to days. It is now recognized that these brightness changes are due to the eclipsing of one star by another (as they pass in front of each other).
Eclipsing binaries are studied by monitoring their light curves (shown below), the changes in brightness with time. When the smaller, dimmer star passes in front of the brighter star, there is a deep minimum. When the dimmer star passes behind the bright star there is a second, less deep, minimum. Notice the transition zone at the start and end of each eclipse.
Eclipsing binaries are very rare since the orbits of the stars must be edge-on to our solar system. Notice that an eclipsing binary is the only direct method to measure the radius of a star, both the primary and the secondary from the time for the light curve to reach and rise from minimum.
Spectrum Binary:
Often a system of binary stars are too close (or too far away) to be resolved into an optical pair. However, a spectrum of such an object will display the spectral fingerprints of two different stellar types (if the stars are different in spectral type).
Of course, the problem with this method is that since faint, cool stars are more common than brighter stars, the odds are that the companion is too faint to be detected in a spectrum. Also, just detecting two spectrum will not determine their masses since relative velocities are needed.
Spectroscopic Binary:
Another avenue to determine the masses of stars is to measure their relative velocities via the Doppler shift of their spectral lines. This is used when the pair can not be resolved as an visual binary, but motion is seen in the spectra of one star.
Notice that you do not need to see two spectra, only the motion of one of the stars is needed to deduce the existence of the binary system (why would one star be moving on its own?). Most binary stars are too close to separate the components, yet their existence can be deduced from Doppler shifts.
Typical velocities between binaries are 3 to 5 km/sec, so very high resolution, Coude spectra must be taken to observe this phenomenon.
Contact Binaries:
When two stars are close in separation it is possible for tidal forces to come into play. Since stars are not solid bodies, rather made of gases, then gravity can strip material and transfer it from one star to the other. Thus we say the binaries are in contact, even if their surfaces are not touching directly.
How stars exchange material is similar to the way a ball rounds over and down a hill. The ball must have enough kinetic energy to exceed the potential energy of the hill. Around two stars there are lines of equipotential. Imagine two nearby lakes. If the water rises it takes on the shape of the contours of the land, the equipotential contours. If the water level rises too high, the lakes merge.
In the same way, there exist lines of equipotential around stars, where the gravitational pull from one star exceeds that of another. This line where the forces or energies balance is called the Roche lobe. When the star's radii exceed the Roche lobe, the gases are free to transfer from one star to the other. Usually in the form of a tube or stream.
In some binary stars, such as Phi Persei, one of the binary stars evolves and expands (see stellar evolution lecture). Its surface exceeds the Roche lobe and material is streamed from one star to the other.
Some contact systems, such as the Algol system require sophisticated supercomputer simulations to understand the mass exchange.
