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Chapter 11. Stellar Brightness, Magnitudes, the Distance Modulus and the HR Diagram Please read the Wikipedia articles linked on the ”Links” section of the Homepage, as well. They will give you additional important insights into this chapter and the. 1. The Magnitude Scale The brightness of a star in physical units is its flux which has the units of Watts per square meter in mks. However, astronomers almost never use flux to describe brightness of objects, they use magnitude. Magnitudes have no units, as they are always on a relative scale. We only give the magnitude of a star relative to another star (often a standard star). If the flux we are referring to is the flux from a star measured at the Earth (f: technically, just above the Earth’s atmosphere, so that the influence of the atmosphere in reducing the brightness of a star is eliminated) then the associated magnitude is referred to by astronomers as a star’s apparent magnitude (m). The relationship between magnitude and flux is: m1 − m2 = −2.5log10 f 1 f2 Note that a difference in flux of a factor of 10 corresponds to a difference in magnitude of 2.5 mag. Also note that brighter objects have smaller magnitudes. See the Wikipedia article for some apparent magnitudes of common objects. The brightest star, Sirius, has an apparent magnitude of about m = -1.4, while the faintest star visible with the naked eye on the Wesleyan campus is about m = 4.5. Hubble Space Telescope can record objects as faint as about m = 27. 2. Stellar Distances by Parallax An obvious factor in determining m for a star is its distance. The closer a star is to the Earth, the brighter it will appear. For relatively nearby stars we can determine their distance using the parallax method. Recall that one requires two vantage points to observe a nearby object and a set of background objects against which to measure the apparent shift in position as one shifts vantage points. In the case of the distance to Mars, the two vantage points were in opposite hemispheres on Earth. In the case of stars, they are too far away to show any substantial shift in position when viewed from any two locations on Earth – we need a longer baseline. The longer baseline is provided by the orbit of the Earth around –2– the Sun. If we view the same star field six months apart in time, then we view from two different points in the Earth’s orbit separated by approximately the diameter of the orbit (not exactly, of course, because the orbit is elliptical – but lets ignore that small effect here). The diameter of the Earth’s orbit is, of course, approximately 2 AU. As in the case of Mars, there is a shift angle that the star moves through. The observed shift angle can be converted to the parallax angle (p) that a star would shift through as we shift our vantage point by precisely 1 AU. Since p is always a very small angle (even for the closest star it is only 0.7 arc-seconds) we can use the small angle approximation to relate the distance to the star (d) to p. If p is in radians, then p= 1AU d(inAU) Usually, we prefer to measure p in arc-seconds. Since there are 206,265 arc-seconds in one radian, astronomers define a new and more useful unit for measuring stellar distances, namely the parsec (pc) as 206,265 AU. With this definition, the parallax equation becomes: p(in arcseconds) = 1 d(in pc) Note that distance measurement by parallax only works for stars that are quite close to the Sun. More distant stars than about 100 pc have parallax angles so small they just cannot be measured even by modern techniques. Astronomers use the Kiloparsec (Kpc) and megaparsec (Mpc) to measure even larger distances in the Universe. The size of the Galaxy is about 20 Kpc, while the size of the Universe is about 4600 Mpc. Obviously it is only a tiny fraction of all stars out there for which we can get direct distances by parallax – only those stars within about 100 pc. To get distances to other stars, we use the fact that there is a one-to-one correlation between a star’s luminosity and its spectral type, as described in the last chapter. If we can get the distance to at least one star of each spectral type, then we can calibrate its luminosity and assume that all other stars in the Universe, regardless of their distance, would have the same luminosity if they are the same spectral class. (Note that I am simplifying this a little – there could be other factors besides temperature and radius affecting a spectral class, such as chemical composition, rotation, etc. But, in general, the other factors do not vary much and/or have little influence on a star’s spectral appearance in the relevant wavelength range.) To see how all of this works in more detail, we need to consider a different kind of magnitude – the kind that relates directly to the luminosity of the star and is independent of its distance from us. –3– 3. Absolute Magnitudes The luminosity (L) of a star, which has the units of Watts (in mks), is a measure of the intrinsic brightness of a star – i.e. how bright it truly is, as opposed to how bright it appears from Earth. Astronomers use the quantity Absolute Magnitude (M) to describe this. They further have agreed on the convention that M is defined to be the apparent magnitude (m) that a star would have if it were at a distance of 10 pc. I am not really certain why they chose 10 pc as opposed to 1 pc or any other standard distance to define M, but that is what they did (in the 1920’s I believe) and that is what we have lived with ever since. So – get used to it! Absolute magnitudes of stars range from about -7 for the brightest supergiants out there, to about +20 for the very faintest little red dwarfs that we find. The absolute magnitude of the Sun is near +5. The concept can be extended to talk about the absolute magnitude of whole galaxies and of quasars and other astronomical stuff. 4. Distance Modulus Clearly there must be a relationship between m, M and d and this can be derived from the inverse square law if we neglect any dimming of a star that might occur due to interstellar absorption (i.e. “fog”). Assuming that the only factor causing a star to dim as it moves away from us is the geometric effect of spreading its light over an increasingly large sphere, we can write that L f= 4πd2 where L is the luminosity of the star, d is its distance and f is its flux measured at Earth. This is a statement of the so-called inverse square law for light, that the flux of an object changes as the inverse square of its distance. Using the definition of M as m (at 10 pc) and employing the magnitude equation, we may write: m − M = −2.5log10 f f (at 10 pc) Now, using the inverse square law, we have 102 f = 2 f (at 10 pc) d so, m − M = 5log10 d − 5 where we have employed some well-known properties of logs, as given in class. –4– The quantity m - M is known by astronomers as the distance modulus. Clearly, if we know it, we know the distance to the star. In practice, we always can measure m for a star, since this is just dependent on its flux at Earth (where we live). If we can measure d for at least one star of a particular spectral type (e.g. G5V) then we can use the above equation to get M for that star and, by extension, for all stars of its spectral type. Once we have calibrated the spectral type - M relation for stars of all spectral types, we can get the distance to any star using the same equation, just by measuring its apparent magnitude and its spectral type. 5. Correction for Interstellar Absorption (Extinction) In reality, space is not quite perfectly clear of any “fog”. There are tiny grains that are produced in stellar outflows from novae, and supernovae and some stars and these permeate space creating a kind of fog that is more intense in some directions than others. These grains are largely confined to the plane of disk galaxies and within our Milky Way galaxy tend to block our view along the plane of the galaxy, but not perpendicular to the plane of the galaxy. Hence, if we look outside the Milky Way we can see to great distances because there is very little obscuration due to these dust grains in those directions. When we look along the Milky Way our view is much more obstructed and limited. The absorption depends heavily on wavelength. At optical wavelengths we cannot see even to the center of our own galaxy (about 8 kpc away) because of this effect. In the infrared we can see the center because the dust obscuration is much less. Longer wavelengths are much less affected than shorter wavelengths, just like scattering in the Earth’s atmosphere (it is the bluer, shorter wavelength light that gets scattered most by our atmosphere and by the interstellar dust grains). Correcting for interstellar absorption (or, extinction, is the better term, because it encompasses both absorption of light and scattering of light, both of which diminish a star’s apparent brightness) is difficult. Formally, we add an extinction term to the distance modulus equation. This is usually represented as the letter “A” (for absorption) and defined to be the amount of extinction affecting the star’s light, in magnitudes. With this definition, the distance modulus formula becomes: m − M = 5log10 d − 5 + A In practice, determining what value A has for any particular star or galaxy can be challenging. We will discuss this more a little later. Here I note that we generally use the fact that extinction depends on wavelength to calibrate the total amount of extinction. A star suffering –5– from a great deal of extinction will look redder – its longer wavelength radiation penetrates to us more effectively than its shorter wavelength radiation. If we know the spectral type of a star then we know its surface temperature and we know what its “intrinsic color” should be. For example a star like the Sun should appear yellow, putting out most of its light near 6000 angstroms. If we see a star which appears redder than it should given its spectral type we can identify that as an effect of interstellar dust and use its degree of reddening to calibrate the amount of extinction the star suffers. In practice, as I say, this can be tricky and often knowledge of how much interstellar extinction is affecting any particular object’s light is hard to determine. 6. The Hertzsprung-Russell (H-R) Diagram This is a plot of absolute magnitude (or Luminosity) of a star versus its spectral class (or Temperature or Color) first employed by the astronomers whom it is named after. Examples were shown in class and can be seen on the Wikipedia links. Arguably, this is the most important diagram in astronomy, since it allows one to characterize every (normal) kind of star we find in the Universe. The main features are described in the Wikipedia reading but are quickly summarized here. There is a main sequence of stars that cuts diagonally across the diagram from upper left (hot and bright) to lower right (cool and dim). [Note that in typical astronomical fashion, the diagram does not follow usual conventions of science in having smaller values to the left and larger values to the right – the temperature sequence has hotter to the left and cooler to the right!]. The main sequence is where we find the vast majority of stars, including the Sun. The luminosity class of main sequence stars is class V, or “dwarfs”. We will see later that the reason this sequence is so well populated is that it represents the locus of stars that are burning Hydrogen in their cores. This is the longest-lived phase of stellar evolution. Once a star exhausts its Hydrogen it evolves through other phases quite rapidly, hence we do not find many stars in these other phases. The second-most populated region in the H-R diagram is the so-called red giants. These are class III (giant) stars. Most giants are of K or M spectral class, indicating relatively cool atmospheres. They are quite luminous, in spite of being cool, because their radii are truly enormous – a hundred times larger than the Sun’s, or even more. Again, we will discuss the nature of these stars more when we deal with stellar evolution theory, but we do know that they represent stars that have exhausted hydrogen in their cores but still shine by hydrogen burning in a shell around the (helium-rich) core. When a star leaves the main sequence this is where it heads on the H-R diagram and this is the next most long-lived phase of stellar evolution for a star, which is why it is fairly well populated. –6– Other regions where we find stars on the H-R diagram (although not that many) are among the supergiants, which are the more massive examples of stellar evolution, and among the so-called “white dwarfs”. Note that a white dwarf is not a class V star. These are much less luminous than the class V (dwarf) stars that populate the main sequence. The term “white dwarf” is always used with both words together, indicating that we are talking about something totally different than class V, dwarf, stars. White dwarfs are the burnt-out cores of previous main sequence stars after they have gone through the red giant phase. They are extremely hot only because they were once the core of the star and have not yet cooled off substantially. We will discuss them in greater detail later. They are very small, with radii approximately equal to the radius of the Earth! Masses of stars vary systematically across the H-R diagram. Along the main sequence (class V stars) the earlier the spectral class the more massive the star. [Note: early type refers to hotter stars like O or B-type, while later type refers to cooler stars like K or M. These terms were originally thought to relate to the ages of stars as well, or when they formed, but we now know this is not true. Nonetheless, the terms “early type” and “late type” continue to be widely used in astronomy today.] O-type stars have masses up to 50 solar masses or perhaps a little larger. M-type stars have masses down below 0.1 solar masses. The smallest mass object that can start its nuclear fusion of Hydrogen is thought to be about 0.07 solar masses, so this represents the smallest stars we see. These are of class M. We do find cooler objects in space that are classified as L or T type objects but these are not stars, per se, they are called “brown dwarfs”. They shine only by the heat generated through gravitational collapse. They are extremely low luminosity and hard to find! If we took a small volume of the galaxy around the Sun, say the nearest 20 pc or so, and plotted every star in that volume on the H-R diagram, we would find that almost all of them were low luminosity, low temperature stars of class K or M. M-type dwarf stars are the most abundant stars in the Universe by far. There are roughly 100 million M stars for every O star! Most of the mass in stars is in the form of these low luminosity, low mass stars, typically having masses of about 0.1 solar mass. On the other hand, if we plotted the brightest stars (by apparent magnitude) we would find that most of these stars are more massive than the Sun. This tells us that while the mass of our galaxy is primarily in the form of low mass stars, the light is primarily coming from high mass stars (O and B stars as well as K and M giants and supergiants). This turns out to be true of galaxies in general. The light of a galaxy is like the tip of an iceberg, showing us only the tiny fraction of stars that are exceedingly luminous because they are quite massive. The real mass of a galaxy resides in its M dwarfs, which even collectively do not produce that much light. –7– 7. Standard Magnitudes and Colors The definition of apparent magnitude (m) given so far has not included one important aspect – namely the wavelength range over which the flux (f) is measured. Since stars have different colors they can have different values of f depending on what range is chosen. Astronomers define a set of “standard” magnitudes that are measured over agreed upon, conventional, wavelength ranges using a set of “standard” stars as the flux standards. The most common standard system is known as the UBVRI system and has five separate wavelength regions over which flux from a star is measured. In practice this is done by using colored glass filters in front of the instrument measuring the flux. In the UBVRI system, the width of the band passes of those filters are relatively broad, with widths of about 1000 angstroms or more. The five bands are centered on five different wavelengths and named according to the spectral region that they sample: U is for Ultraviolet, and the bandpass is center is around 3600 angstroms B is for Blue, and the bandpass is center is around 4400 angstroms V is for Visual, and the bandpass is center is around 5500 angstroms R is for Red, and the bandpass is center is around 6700 angstroms I is for Infrared, and the bandpass is center is around 7900 angstroms We refer to the apparent magnitude of a star in the V bandpass as mV or, more commonly, just its V magnitude. Similarly we can speak of the B magnitude of a star as its apparent magnitude in the B bandpass. In practice, these standard magnitudes are determined for stars by observing them with telescopes, using the standard filter sets and comparing brightnesses of stars you wish to measure with standard stars whose magnitudes are agreed upon by convention. Astronomers have set up a whole lot of standard stars spread around the sky so that at least some of them can be observed at any time from any location. In making these observations, we must also correct for the effect of the Earth’s atmosphere. This is done by the astronomer before reporting the standard magnitude. Standard magnitudes, therefore, are really the brightness in the appropriate band pass that one would observe from just above the Earth’s atmosphere. I note in passing that it is sometimes desirable to talk about the flux from a star integrated over ALL wavelengths. This is called the “bolometric” magnitude of a star and often designated as mbol . It can be very difficult to determine mbol , in part because many wavelengths of light are totally blocked by the Earth’s atmosphere and can only be accessed –8– through satellite or balloon measurements. In practice, astronomers have calibrated a “bolometric correction” for stars of a given spectral type that indicates how much light is falling outside of the standard band (e.g. V) which can be measured more easily. We will not deal with bolometric magnitudes any further in this class, but I do note that they are important for determining the true luminosity of a star, since stars can emit much of their energy at wavelengths other than just UBVRI. Using the definition of the magnitude scale, we can write the color of a star as its difference in magnitudes through two of the standard bands. By convention, we usually take the bluer bandpass first and subtract a redder bandpass, producing colors such as B-V and V-R and R-I. The definition of B-V is just B − V = −2.5log10 fB fV where fB is the flux measured in the B bandpass and similarly for V and for the other colors. Hence a color is a measure of the ratio of the fluxes at different wavelengths. The B-V of bluer stars is more negative than the B-V of cooler stars. O stars have B-V = -0.3 or so, while the Sun, a G star, has B-V = +0.6 and M stars have B-V = +2.0 or so. If interstellar extinction effects are unimportant, then color simply depends on the surface temperature of the star and so it can be used in place of spectral class as the x-axis on the H-R diagram. When this is done, the resulting plot of, say, absolute magnitude in the V-band (designated MV ) versus B - V is referred to as a Color-Magnitude Diagram. It is equivalent to an H-R diagram. Finally, we note that standard colors quantify color measurement and allow us to correct for interstellar extinction in a fairly precise way. In particular, we can define the “color excess” (e.g. E(B-V)) of a star as E(B − V ) = (B − V ) − (B − V )0 where B-V is the measured color of a star and (B-V)0 is the color appropriate to the star’s spectral type – i.e. the color it would have if it were unaffected by interstellar absorption. We determine (B-V)0 for a star from its spectral type, having calibrated it with nearby stars that we believe are unaffected by absorption. Then we can compute E(B-V) for any star from its measured color. Studies throughout the galaxy have generally found that the grain properties are quite similar in most (but not all) directions and so we can calculate the amount of total extinction (AV ) in some standard band (usually V) as AV = 3.1E(B − V ) This is the value of A that then gets used in the distance modulus equation to correct for the “fog” of interstellar space.