An Introduction to Astronomy Part X: Properties of Stars Lambert E. Murray, Ph.D. Professor of Physics

Starlight: The Key to the Universe F Everything we can know about a distant star, we determine from the light that reaches us from that star. F By analyzing the light from distant stars, astronomers can determine: –Distances to Stars –Surface Temperatures and Luminosities –The Motions of Stars –Stellar Masses –Stellar Chemistries

Time and Distance F Light travels at a rate of 3 x 10 8 m/sec, so when we see a star, we are actually seeing the light that was emitted from that star some time in the past. F The nearest star, Proxima Centauri, is about 4.3 light years (or 4.08 x m) away from our Sun, which means it takes light 4.3 years to reach us from that star. Thus, if the star were to explode today, we would not be aware of it for 4.3 years! F Similarly, when we observe the light from a distant star we are looking at that stars characteristics some time in the distant past – that star may not ever exist today!

Stellar Spectra F By studying the spectrum of starlight we can determine several things about a star: –The stars surface temperature –The rate at which energy is being emitted from the star per surface area –The chemical make-up of the star (at least its outer atmosphere) –The velocity of the star toward or away from us based upon the Doppler shift.

Color and Temperature F We have already discussed the relationship between the color of a star and the stars temperature. F The blackbody spectrum of a star’s continuous spectrum can be used to determine surface temperature of the star. F In addition, once the surface temperature is known, the rate at which energy is emitted from the star can be determined, using the Stephan- Boltzmann equation, if we know the size of the star.

The energy radiated by a blackbody takes a special form when the relative brightness (or strength) of the radiation is graphed as a function of the wavelength. Hotter objects emit more light (the total area under the curve) and this light is peaked at shorter wavelengths.

From a Star’s spectrum, we can determine the star’s chemical make-up.

The Doppler Effect The Doppler Effect can be used to obtain a star’s radial velocity by measuring the shift in wavelengths of certain spectral lines. is the reference wavelength in the laboratory, c is the speed of light, and  is the measured shift in the spectral line from the star.

Doppler Shifts can also be used to determine the rate of rotation of a few nearby, large stars.

Stellar Parallax: A Direct Method to Determine the Distance to Stars F This method, also known as triangulation, uses simple mathematical ratios to determine the distances to stars (see next slide). F It is the only direct method of determining stellar distances and can be used only for stars within about light years of our Sun. F The ancients were not able to see any stellar parallax because the stars are so far away. F To determine the distance to more distant stars, indirect methods must be used.

Parallax Formula F Distance in Parsecs (pc) = 1/parallax angle measured in seconds of arc (arc seconds) F 1 Parsec (pc) = 3.3 light years (ly) F The parallax angle shift of the nearest star is the angular size of a dime measured at a distance of 2.4 km – or about 0.77 arc seconds. This corresponds to about 4.3 ly.

Limitation of the Parallax Method F From the Earth’s surface, parallax measurements can be made accurately only for stars within about 100 parsecs. This is because turbulence in the atmosphere interferes with the accuracy of small angle measurements. F From above the atmosphere (using satellites such as Hipparcos) we have been able to accurately determine the distance to stars out to about 250 parsecs. Distances to an accuracy of better than 10% have been obtained for stars out to 150 parsecs (approx. 500 ly).

Proper Motion of Stars F Some stars are observed to move relative to the background stars over a period of time (this is in addition to the apparent back-and-forth motion due to the Earth’s orbit about the Sun).  This motion is called proper motion, , and is measured in arcseconds/year. F Barnard’s Star exhibits the largest proper motion of any star – arcseconds/year.

Proper Motion for Barnard’s Star – The Largest Observed Proper Motion for any Star – is arcsec/yr

The actual motion of a star is a combination of the proper motion and the radial motion.

Stellar Magnitudes F In the second century B.C., Hipparchus created the first star catalog in which he tabulated the location and apparent magnitude of a large number of stars. F He designated the brightest stars (the first order stars) as having a magnitude of +1, while the next brightest stars (the second order stars) had a magnitude of +2, etc. F The dimmest stars visible to the naked eye are approximately +6 magnitude stars. F Modern telescopes can image stars as faint as +25 th magnitude. F Using this scale, brighter objects, like the Sun, have a negative magnitude.

The Luminosity of a Star F The luminosity is the total amount of radiation emitted from the star. F The luminosity depends upon the temperature and the size of the star according to the equation: L is the Luminosity (total energy radiated/sec) T is the temperature in Kelvin  is a constant (5.67 x watts/m 2 K 4 ) A is the surface area of the radiating body L =  T 4 A =4  R 2  T 4

Brightness of Stars F The apparent magnitude (or brightness) or a star depends upon two things: –The actual brightness of the star – its luminosity, and –The distance to the star F The Luminosity of a star depends upon two things: –The surface temperature of the star, and –The size of the star

Relationship between Luminosity and Apparent Magnitude F Since stars are at different distances from the Earth, the apparent magnitude depends upon that distance and the actual luminosity, L, of the star. F The apparent magnitude M (sometimes called the brightness) is given by: where R is the distance from the Sun to the Star.

Absolute Magnitude F The absolute magnitude M o of a star is defined as the apparent magnitude of a star if that star were a standard distance (10 parsecs) from our Sun. F If we can determine a stars distance (by parallax) and the stars apparent magnitude, we can then determine by using ratios the absolute magnitude of that star.

Spectral Classification of Stars F Another method of classifying stars is based upon the absorption lines observed in the spectra of the stars. A classification scheme was developed using the letters: O B A F G K M F It turns out that this spectral classification is essentially equivalent to the temperature or color classifications, the O stars being the hottest (bluest), and the M stars being the coldest (reddest). This classification scheme is further subdivided into subunits (e.g., B0 – B9, etc.)

Spectral Classification of Stars

Spectral Classification

Hertzsprung-Russel Diagrams F Around 1910, attempting to gain a better understanding of the different types of stars, Ejnar Hertzsprung, from Denmark, plotted the star’s temperature (or color) versus the star’s absolute magnitude. F At about the same time, Norris Russell (USA) made a similar plot of spectral types versus absolute magnitude. F These plots, which are essentially identical, showed a surprising relationship between the temperature (color or spectral type) of a star, and the star’s absolute magnitude (or luminosity). F Sample HR diagrams are shown on the next few slides.

Observations from the HR Diagram F The stars are not found to be randomly distributed across this diagram – on the contrary, they seem to fall into groups: –A majority of the stars (about 90%) fall along a diagonal; from hot, luminous stars to cool, dim stars. This diagonal is called the main sequence, and stars falling along this line are often referred to as dwarf stars. –Another group of very luminous, yet cool (red) stars occurs in the upper right corner – these stars must be very large, and are known as red giants. –Yet another grouping of stars occurs in the lower left corner – these are hot, dim stars, known as white dwarfs. F This observation turns out to be significant in understanding the life cycle of stars.

Stellar Radii F Since stars act like blackbodies, we can use our equation for the luminosity of a star to determine the radii of different Stars. F Using this equation, we can plot stellar radii on the HR diagram on the next slide. L =  T 4 A =4  R 2  T 4

Hertzsprung-Russell Diagram with Stellar Radii

Dwarfs, Giants, and SuperGiants F Notice that many of the main sequence stars have a radius about equivalent to the Sun’s radius. (The cooler main sequence stars are all somewhat smaller.) F White dwarfs are all about 100 time smaller that our Sun (i.e., about the size of the Earth). F The giants and super-giants are from 10 to 100 times larger that our Sun. Betelgeuse, for example, is about 370 times larger that our Sun, with a radius of about 2 A.U. (larger than the orbit of Mars).

Luminosity Classes F In addition to the spectral classification OBAFGKM, which is related closely to temperature, astronomers have subdivide stellar spectra into luminosity classes which are related more to the size (or surface density) of a star. F For example, a B8 supergiant spectra exhibits narrower spectral lines than a main sequence B8 star. F This is because the density at the surface of the supergiant star is less than the main sequence star, and the spectral line exhibits less pressure broadening. F Astonomers have developed 5 luminosity classes, in order of decreasing luminosity: I, II, III, IV, and V, with I signifying supergiants and V main sequence stars. F Thus, the complete spectrascopic designation of our Sun is that it is a G2V star, designating the Sun’s surface temperature and luminosity.

Luminosity Classes The luminosity classification is based upon the absorption line spectrum of the star.

Spectroscopic Parallax F The HR diagram can actually help us to determine the distance to stars whose actual parallax cannot be measured. F If the stars temperature (color) is known, and if its spectral class can be determined from its spectra, the stars luminosity (or absolute magnitude) can be determined. F Knowing the absolute magnitude (the maganitude at 10 parsecs) and its apparent magnitude (as seen from the Earth) one can calculate, based on the 1/R 2 law, the distance to the star. F This technique is known as spectroscopic parallax.

F By looking at the continuous spectrum, we can determine the “color” of the star. F By looking at the absorption line spectrum, we can determine the spectral class of the star. F From these two, we can determine the absolute magnitude of the star – provided it is properly placed on the HR diagram. Spectroscopic Parallax

Binary Stars and Stellar Masses F The masses of individual or bachelor stars cannot be measured by any current techniques. F However, more than 50% of all stars are part of multiple star systems (the majority are binaries). F This means we can make use of Newton’s law of motion and gravity to determine the masses of the stars based upon the period of revolution and the the distance between the stars.

Multiple Star Systems F Optical doubles are two stars that have small angular separation as seen from Earth but are not gravitationally linked. F A binary star system is a system of two stars that are gravitationally linked so that they orbit one another. F A visual binary is an orbiting pair of stars that can be resolved (normally with a telescope) as two stars. F If one uses large telescopes, about 10% of the stars in the sky are visual binaries.

Visual Binary

The Orbit of 70 Ophiuchi, a faint double star in constellation Ophiuchus. The orbit is plotted with the more massive star held fixed. Recall that both stars actually orbit the common center of mass.

Mass of Binary Stars F This last equation determines the total mass of the binary system in terms of the period of revolution and the distance between the stars (which can be determined only if we know the distance to the binary system). Once the total mass of the binary system is determined, the mass of each individual star can be determined by observing the motion of each star relative to the center of mass, since:

Spectroscopic Binaries F Many binary (or other multiple) star systems are too close together or too far from the earth for the individual stars to be resolved. F Fortunately, many binaries that are visually unresolved, can be detected from the Doppler shift of their spectra. Such binaries are called spectroscopic binaries.

A Double-Line Binary This is the spectra of the same star at two different times. Notice the simultaneous red and blue shifts! Non-Shifted Lines Blue Shifted LinesRed Shifted Lines

Careful measurements of the Doppler shifts gives a radial velocity curve from which one can determine the periods of the orbits.

Eclipsing Binaries F Algol, discovered by Goodricke in 1783, is an eclipsing binary, in which one star moves in front of the other as viewed from Earth. F Algol’s light curve—a graph of the numerical measure of the light received from a star versus time—shows peaks and dips that indicate an unseen companion.

Mass – Luminosity Curve Data from binary stars enables us to determine the mass of various stars. If we plot the mass of these stars against their luminosity we find another interesting pattern.

Mass - Luminosity Data Plotted on an HR Diagram This plot looks almost like a plot of main sequence stars. We will notice this again when we talk about the birth of stars.

End of Part X