Lecture 17 Review Most stars lie on the Main sequence of an H&R diagram including the Sun, Sirius, Procyon, Spica, and Proxima Centauri. This figure is a plot of logl versus logt. The main sequence is explained by the stable burning of hydrogen, the most efficient, lowest temperature energy producer. The radiation produced keeps the star in hydrostatic equilibrium with the Helmholtz gravitational contraction forces.
Remember from Stefan-Boltzmann that 4 2 4 2 L = σat = 4 πσr T, thus log L = log( 4 πσr ) + 4 log T This says that lines of constant radius are straight lines on this H&R diagram. These are shown. Note that over about b of this plot, radius changes very slowly for stars on the main sequence, i.e. increased mass contributes to increased density with little change in radius. While the Main Sequence represents stable hydrogen burning for a variety of different mass stars, there are stars off the main sequence: White Dwarfs: Giants: Super Giants: T 2 T u, hotter than the Sun R R Sun /100 L L Sun /100 T T Sun, same temperature as the Sun R 10R Sun L 100 L Sun T T Sun /2, cooler than the Sun R 1000 R Sun L 10,000 L Sun The Russell-Vogt Theorem (1926) states that the equilibrium structure of a main sequence star is uniquely determined by 1) mass 1) chemical composition This implies that all 1 M Sun stars with the same chemical composition occupy the same spot on the H&R diagram with the same temperature, radius, and luminosity. This is a way to distinguish the Sun from white dwarfs and certain giants. More important, this gives us a way to measure distances beyond 100 pc, the limit of parallax measurements. 1) Measure temperature and spectra to determine where on the H&R plot a star belongs. 2) This determines the absolute magnitude and luminosity. 3) Use 1/r 2 dependence and apparent magnitude to determine distance from the Sun. This is called the method of spectrographic parallax and is good for measurements out to the edge of the Milky Way. The method is unreliable where the interstellar medium dims the star in an unmeasurable way.
Example: 1) Spica has an apparent brightness of m V = 1 and a spectrographic temperature of 20,000 K. 2) From the H&R plot you can determine that L = 2300 L u and M = -4. 3) Correct for 1/r 2 using the following figure and you find that Spica is 84 pc from the Sun. Stars form from large clouds of gas. The process usually involves the formation of clusters of stars, many bound gravitationally together. Binary stars revolve around each other. Time and position measurements tell us the masses of the binary pair. From this sample of stars we find that there is a linear relationship between luminosity and mass to some power. This is the mass-luminosity relation and leads to the following figure showing masses along the main sequence. Masses of the giants and dwarfs can be determined in the same way.
How stars evolve and die depends totally on mass. You might guess that a more massive star would live longer because it has more fuel to burn, but no. energy available Hydrogen burning time = rate of burning mass lu minosity mass burning time For stars more massive than the Sun the mass-luminosity relationship says that 3. 8 L ~ M M 2. 8 t = = MSun time ~ 3. 8 M M t M 2. 8 For M = 10 M u t = tsun billionyears million years 1 1 = 10 16 10 630 This implies that big stars have a much shorter life than the Sun. Also, if you see a big star, it is relatively young. Thus, in the night sky you see a mixture of old dim stars and bright young stars. Sun 2. 8
Relative ages of stars can be inferred from an H&R plot of stars in an open cluster. These stars are presumed to be formed from a common gas cloud and all to be roughly the same age. This plot of stars in the Pleiades is an example. Where do stars come from? Are they still being formed? The evidence for current star formation is the presence of much gas and dust in space. This is called the interstellar medium and where the densities are large the gas and dust formed clouds called nebula. Emission nebula - 100 to 10 4 solar masses of dust and gas, very low density, found near hot stars. Ultraviolet radiation emitted from the stars is absorbed by the gas, ionizing the hydrogen which emits photons characteristic of hydrogen. These are called HII regions. Dark nebula - Dust grains block light. The nebula appears dark in front of a bright region. Reflection nebula - Small dust grains at low concentration scatter light. Like the scattering of sunlight in Earth s sky, the process is most efficient for blue light; therefore, the nebula have a blue cast.
Interstellar reddening - As light from stars passes through the interstellar medium blue light is slowly scattered out of the path, leaving the resultant light with a reddish color. Thus, remote clusters appear dimmer and redder than expected from their distance and age. Interstellar extinction - When enough gas and dust is in the way, far away objects cannot be seen because light is scattered out of the line of sight. Thus, we cannot see the galactic center with visible light. That these clouds are the source of stars follows from several observations: 1) The solar system is only 4.6 billion years old while the Milky Way (with 100 billion stars) is believed to be around 12 billion years old and the Big Bang is on the order of 14 billion years old. 2) Open star clusters like the Pleiades had some hot blue stars but no giants. This implies that the cluster is only about 50 million years old. Very young in Big Bang terms. 3) Short-lived massive stars, 20 to 40 solar masses, live only a few million years but we see them. The process for star formation is thought to involve a large cloud of gas, say 10 35 kg or 10 5 solar masses, cold enough so that gravity can overcome random thermal motion. The cloud contracts gravitationally. Fragmentation and sub-fragmentation produce clusters of stars, some individual stars, others binaries, some other larger groupings. These may disperse with time due to random motion. Clusters 1) Population II clusters - globular clusters - are old stars in groups of 10 4 to l0 5 stars. They form a halo about the galaxy formed when the galaxy was a huge sphere of gas, before gravity and angular momentum had collapsed the gas into a disk. Their paths are typically very elliptical, passing through the center of the galaxy and then out again. They are very old, with many stars off the main sequence. These stars contain few heavy elements. 2) Population I clusters - open clusters - are relatively new stars in groups of approximately 10 3 stars. The are 100,00 to 100 million years old with mainly main sequence stars. The Pleiades is an example. Their spectra are metal rich.
Stellar Evolution 1) As gas collapses towards a single star, it heats up. When the cloud has the dimensions of the current solar system is may have a temperature of 2000 K, highly luminous, but radiating mostly in the infrared. This is called a protostar. 2) As the contracts, it surface area diminishes so it becomes less luminous. 3) As the temperature increases the cloud becomes more opaque, thus, trapping the radiation inside the cloud and increasing the temperature within. 4) At some point the core temperature reaches the ignition point and the protostar becomes a star. The star is still hidden from view by much remaining gas. 5) Magnetic fields sweep the remaining dust away, sometimes huge amounts, to reveal the new star.
The question is, what if the mass is greater than about 50 solar masses? If the forming star is too large, the gas cloud condenses quite fast, is unstable, gets very hot, and either explodes or fragments into smaller clouds which form individual stars. A second question is, can the mass of the gas be too small. The answer is yes. If the mass of the cloud is too small it heats up from gravitational contraction, but never gets hot enough to ignite. The gas ball then reaches some equilibrium and cools off. Jupiter is a case in point. It still radiates more energy than it absorbs from the Sun, but the source of the radiation is not thermonuclear processes. This type of gas giant is called a brown dwarf. The process is shown below.