6/28 Based on the reduction of the intensity of the light from a star with distance. It drops off with the inverse square of the distance. Intensity is power per unit area of electromagnetic radiation. Relation between absolute magnitude, apparent magnitude, and the distance: Note that we can use this relation to calculate the absolute magnitude of those stars whose distances can be measured via parallax. Suppose that we could deduce the absolute magnitude of a star without having to know the distance to the star. We can always measure the apparent magnitude. Inverting this relation will allow us to measure the distance to the star: Examples: 1. We can develop a classification scheme that will help us deduce the absolute magnitudes of stars. Spectral Classifications Current Scheme: O B A F G K M mnemonic: Oh be a fine guy or girl, kiss me. O blue-violet 30,000 K to 50,000 K B blue-white 11,000 K to 30,000 K A white 7000 K to 11,000 K F yellow-white 5900 K to 7000 K G yellow 5200 K to 5900 K K orange 3900 K to 5200 K M red-orange 2500 K to 3900 K In addition, each major classification is divided into 10 minor classifications from 0 to 9 from high temp to low. The Sun, at 5800 K, is a G2 star.
Now that we have this classification scheme, what can we do with it? Hertzsprung-Russell diagram HR diagram or color magnitude diagram. Plot the absolute magnitude (and/or the luminosity) of stars on the vertical axis with temperature (and/or spectral class) on the horizontal axis. When we do this we find that stars are not randomly scattered on the diagram the congregate in certain regions. Most stars were found to lie on a diagonal from the upper left to the lower right called the main sequence. Some stars were found in regions above the main sequence giant stars and supergiant stars. Some stars were found in a region below the main sequence white dwarfs. Once the HR diagram has been plotted with stars whose distances can be measured via parallax and whose absolute magnitudes can be calculated, we can use the diagram to calculate distances to more distant stars. 1. Measure the temperature of the star. 2. If a main sequence star, go to the portion of the main sequence with that temperature. 3. Move horizontally over to the absolute magnitude axis and read off its absolute magnitude. 4. Use the distance formula to calculate the distance. How do we know if a star lies on the main sequence it could be a giant. We look at luminosity classes due to the fact that there are subtle differences between the main sequence spectra and spectra from giant stars. Giant stars, while much larger than main sequence stars, are no more massive. They are much less dense, and the effect of the low density on the spectrum can be measured. Luminosity Classes: Ia. Bright Supergiants
Ib. Supergiants II. Bright Giants III. Giants IV. Subgiants V. Main Sequence We can use the HR diagram to measure the distances to all the stars. This procedure is known as spectroscopic parallax. Finding the masses of stars use binary star systems. Binary star systems are classified by how they are detected: 1. Visual Binary System one in which we can see both stars in the system. 2. Astrometric Binary System only one star in the system can be seen visually; the presence of the second is inferred from a wobbly motion of the visible star. 3. Spectroscopic Binary System So far away, the two stars cannot be resolved individually; we infer the presence of the two stars by two separate spectra undergoing opposite Doppler shifts. 4. Eclipsing Binary System We detect the existence of two stars because the alternately eclipse one another. We can use Kepler s third law as modified by Newton to measure the masses of stars in a binary system. 1. Visual and Astrometric Systems: Plot the orbit and from the orbital parameters, we calculate the total mass of the two stars. We can get the individual masses of the two stars by using the center of mass. 2. Spectroscopic Binary Systems: Can t measure the distance between the stars because they are imaged as a point. But we can rewrite Kepler s third law in terms of speed and period rather than distance and period. We can use the Doppler shift to measure the speeds. But we don t measure the actual speed, only the part that is along our line of sight get only estimates of the masses in the system. 3. Eclipsing Binary Systems: Analyzed like spectroscopic systems but, since we are looking in the plane of the orbit, we measure their actual speeds and get the actual masses. We have found: Lowest mass stars 8% the mass of the Sun = 0.08 M S.
Objects less massive than this are either brown dwarfs or planets. The most massive stars found are around 100 solar masses. By putting masses on the HR diagram, we can find a rough relation between the mass of a star and its luminosity. E xample: Range of Luminosities of Stars: Lowest mass: Greatest mass: We note that there is a factor of 100 billion between the dimmest and brightest stars. Sizes of Stars Three ways to find 1. Eclipsing by the Moon Carefully watch a star as the Moon eclipses it and measure the time to go from full brightness to zero. We know the speed of the Moon in its orbit and multiplying by the time gives the distance the Moon travels as it eclipses the star. If we know the distance to the star: the ratio of the distance to the star to the distance to the Moon is equal to the ratio of the diameter of the star to the distance the Moon travels while eclipsing the star. Disadvantages: must know the distance to the star and only those stars along the Moon s path through the sky are ever eclipsed. In addition, the smaller the star, the more diffraction effects mask the time to be measured.
2. Eclipsing Binary System We can measure the speeds of these stars using the Doppler effect. We measure the time for one star to pass in front of the other star (system drops in brightness). Multiplying the speed of one star by this time gives the diameter of the other star. In the picture at right, we measure the time when the brightness begins to dim (point A) and the time at which it stops dimming (point B). Multiply the difference between these two times by the speed of the star to get its diameter d. To get the diameter D of the large star, use the time at A and the time at C, where system starts to brighten again. Don t need to know the distance to the star, but very few eclipsing binary systems. 3. Use the Stefan-Boltzmann Law: Written in solar units Solve for the diameter: Measure luminosity and temperature to get size. Example: This method applies to any star in the sky. Density of Stars divide the mass of star by its volume (from the size) to get its density. We find: Main Sequence Stars all have a density about that of 3 water (1 g/cm ). Giant Stars about the same mass as main sequence stars but much larger small densities range form a hundredth to a tenth of the density of water. Supergiant Stars still no more massive than main sequence stars but much much larger 0.00000001 to 0.001 times the density of water. White Dwarfs 2 million times the density of water. Chapter 9 The Lives of Stars from Birth through Middle Age Stars are born as a result of the gravitational contraction of the interstellar medium. Today there are four types of interstellar medium: 1. HII Regions hydrogen, helium, and dust Hot bright stars in the region ionize the gas emit light when electrons recombine with ions hot and low density we observe them as
emission nebulae and reflection nebulae. 2. HI Regions Just like an HII region except no hot stars to ionize the gas. Low density but cool because of the absence of hot stars. 3. Giant Molecular Clouds denser and much colder than either HI or HII regions. Contain molecules near their centers protected from radiation from outside. 4. Coronal Gas very low density, extremely hot, and much like the Sun s corona probably due to supernova explosions. By weight, about 50% of the interstellar medium is in the form of giant molecular clouds, HI and HII regions account for nearly 25% apiece, and coronal gas about 2 %. Stars form from the contraction of this medium, but there are mechanisms that impede the contraction: 1. Thermal energy hard to make a hot gas contract. 2. Magnetic fields will only act on charged particles such as found in HII regions. 3. Rotation contraction impeded perpendicular to the axis of rotation. Stars will form in regions that are coolest and densest giant molecular clouds. Still, some mechanism must trigger the star formation: 1. Supernova explosion sends matter through the giant molecular cloud causing density fluctuations. Where the density fluctuates upward, gravitational contraction into a star can begin. 2. If star formation has already begun in the cloud, a star entering the main sequence can inject matter into the cloud to produce additional star formation. 3. If a giant molecular cloud crosses the boundary between spiral arms it can generate density fluctuations that can cause star formation. 4. If two giant molecular clouds collide, density fluctuations will be produced to cause star formation how the first stars were born. How a star is formed: 1. Density fluctuations are triggered in a giant molecular cloud. 2. A region of high density begins to contract. 3. The region is almost certainly rotating as it contracts, the rotation speeds up to form a disk as we described in talking about the formation of the solar system. Planets form in the disk.
4. In the center, the gas continues to contract and as it contracts it gets hotter. 5. Eventually the gas at the center gets hot enough to glow converting gravitational energy into thermal energy. 6. This glowing object is called a protostar. 7. At this stage, the protostar is much like a giant star but we can t see it because it is enclosed in a cocoon of gas and dust. Can be viewed in IR. 8. The protostar continues to contract and heat until its center gets hot enough to fuse hydrogen (H) into helium (He) at this point the star enters the main sequence. 9. The outgoing stellar wind blows away the cocoon of gas and dust and the star becomes visible to the universe at large. Once the star reaches the main sequence, it becomes relatively stable and evolves only slowly. We can understand the structure of a main sequence star using four principles: 1. Hydrostatic Equilibrium outflow of energy or the radiation pressure prevents further gravitational contraction radiation pressure balances gravity. 2. Energy Transport imagine breaking the star into spherical shells and energy is driven across the shell because the inside of the shell is hotter than the outside. 3. Conservation of Mass the sum of the masses of the shells is equal to the total mass of the star. 4. Conservation of Energy summing the energy production in each shell gives the total luminosity of the star. What happens to a star depends almost completely on the mass of the star. Mass Categories: Low-Mass Stars 0.2 solar masses and less Medium-Mass Stars 0.2 solar masses up to between 2 and 3 solar masses. High-Mass stars from 2 to 3 solar masses on up Main Sequence Lifetime of a Star
The lifetime of a star is given the energy E available to the star divided by the luminosity L of the star: Put this in terms of solar units the energy available to the star is proportional to its mass 2 3.5 (recall E = mc ) E = M. Recall that, in solar units, L = M, so we get Example: Range of lifetimes of stars. Lowest Mass Star: M = 0.08 M S Highest Mass Star: M = 100 M S Since a star s evolution depends so much on its mass, we should note that the mass of a star changes over time. For instance, due to the solar wind, the Sun will lose about 0.1% of its mass over its main sequence existence. Once a star evolves off the main sequence, its mass changes more drastically. Some stars will lose as much as 90% of their mass over their total lifetimes.