I. The Sun (our star)

Structure. Photosphere to corona:

The sun is an incandescent sphere of gas with a diameter of about 865,000 miles or a little less than 1.5 million km. Its surface temperature is about 6000K (5770); that is the temperature of the thin layer we called the photosphere, the visible “surface” of the sun. The photosphere is a convective zone about 1000 km thick.

The next layer outward from the photosphere is the chomosphere, so-called because it is seen as a pinkish fringe around the sun during a total solar eclipse. It is a transition zone, about 10,000 km thick, with the temperature at its base only about 4500K and at the top it transitions into the corona. The pinkish/reddish color is due to the hydrogen-alpha emission, which occurs when electrons in H-atoms fall from energy level 3 to 2.

Outside the chromosphere is the corona, a very tenuous (thin) envelope with a temperature of about 2 x 10 ⁶ K. It extends out several million miles from the sun. Its spectrum shows the existence of ionized Calcium and other elements, due to the very high temperature, which means very high thermal motions. The corona can only be seen during total solar eclipses or from outside the atmosphere.

“Surface” Features (the sun has no surface in the usual sense):

Sunspots: sunspots are magnetic storms on /in the photosphere which are cooler than the surrounding photosphere because energy is stored in the magnetic fields instead of heat (motion). They may be 1500K cooler than t heir surroundings. The dark, central, umbra is coolest, the paler outer penumbra is not so cool. Sunspot numbers rise and fall with a period of 11 years, and sunspots typically occur in pairs with opposite magnetic (N-S) polarity. The sun’s magnetic field reverses after 11 years, so that the cycle is really 22 years. The last sunspot maximum was 2000/2001. All solar activity, including the current flare activity (fall 2003), is highest near sunspot maximum.

Granulation: mot tling of the photospheric surface due to small convection cells which carry hot gases up to the surface by convection and back down again. There dimensions are about 1000 km, or 1 arc second as seen f rom the earth.

Prominences: veils or loops of mostly hydrogen gas which seem to slowly rise and hang above the photosphere, into the chromosphere and corona, whose shapes are controlled by sunspot groups on the sun’s limb.

Flares: the most violent outburts on the sun, releasing charged particles (mostly electrons and protons) and electromagnetic energy (ultraviolet, x-rays) when energy is dumped from the intense magnetic field of a complex sunspot group into heat. The EM radiation arrives at the earth in 8 minutes, if it is in the path of the “coronal mass ejection” but the charged particles take hours to reach the earth. The UV and x-rays affect ionization of the earth’s atmosphere and can disrupt long-distance radio communication, etc. The charged particles are mostly trapped in the earth’s bi-polar magnetic field, forming radiation belts (the “Van Allen belts”), which where these charged particles, which are spiraling along the magnetic field lines, encounter the upper atmosphere, mostly near the poles, the aurora (northern, southern lights) result from light emitted by excited oxygen and nitrogen molecules. These charged particles represent currents, which set up currents in the earth, affecting the earth’s magnetic field and even causing ground transmission of power to be affected.

Energy generation in the Sun’s core:

Energy is generated through thermonuclear fusion reactions in the sun’s core, where the temperature is 15 x 10 ⁶ K. The sequence of reactions is known as the proton-proton (or p-p) chain, and goes like this:

H ¹ + H¹à H² + e⁺ + ν

H² + H¹ à He³ + γ

He³ + He³àHe⁴ + 2 H¹

The symbol ν denotes a neutrino, a “ghost-like” particle which has almost no mass, has no charge, interacts through the weak force, and can travel through light years of lead without being stopped. The symbol γ stands for gamma-ray. Note that we should, but won’t, write ⁴He, instead of He⁴. e+ is a positron, a positive electron.

Note that 6 protons go in, 2 come out, so the net result is that:

4 H¹ à He⁴

This is called “hydrogen burning,” even though it is nuclear fusion rather than combustion. If one adds up the masses, there is about 1% more on the left hand side than the right. That matter is converted into energy via the Einstein formula E = mc ².

Consider the mass of the sun, 2 x 10 ³³ g. If that were converted entirely into energy, the amount of energy generated would be:

(2x10³³ g)(3x10¹⁰ cm/sec) 2 = 18 x 10 ⁵³ ergs (unit of energy in the cgs form of the metric system).

Now the luminosity of the sun (power output) is 4 x 10 ³³ ergs/sec (4x10²⁶ watts). How long could the sun shine at that luminosity? Answer:

Time=total energy/rate at which it is used=18x10⁵³ ergs/4x10³³ ergs per second= 4.5 x 10 ²⁰ s. But 1) only 1% of the matter is converted to energy, and 2) energy is only generated in the central 10-10% of the sun where the temperatures are millions of degrees, so we can divide by 500 to 1000, obtaining about 10¹⁷-10¹⁸ seconds. There are 3x10⁷ seconds in a year, so we obtain about 10¹⁰ years for the sun’s hydrogen burning, compared to its present age of 4.5x10⁹ years. You will not have to do calculations of this kind.

Read about Ray Davis’s neutrino experiment in the Homestake gold mine in SD, using 100,000 gal of perchloroethylene, C₂Cl₄ (cleaning fluid).

II. Stars

The most important properties of a star are its mass and its luminosity (power output). We will talk later about how to determine mass. Luminosity depends on knowing its distance.

The most direct method of determining distance to stars is the method of parallax. The method of parallax uses the diameter or radius of the earth’s orbit as a base-line, and by watching the displacement of a nearby star against the distant background, due to the earth’s motion about the sun, geometry or trigonometry yields the distance directly (see Figures in the text or in your notes). The parallax is the angle through which a star moves, during half a year, from a center position to one extreme or the other (half the total displacement from one extreme to the other). It also is the angle subtended by the earth’s orbit as seen from the star. If we measure distance in parsecs, at a distance of one parsec, the parallax is 1” (1 arc second). In terms of parsecs, d = 1/p where the parallax is in arc seconds. Example: if p=0.5”=1/2”, then d=1/(1/2)=2 parsecs. Since a second of arc is represented by a triangle with one side equal to one unit and the others to 206,265 units, a parsec represents a triangle with one side equal to the earth-sun distance (1 AU) and the others 206,265 AU. If we compute 200,000x93,000,000, we get about 19 x 10 ¹² miles or about 3.2 light years.

Other methods: moving cluster method, Cepheid variable, spectroscopic parallax, et c. (See later).

Once we have the distance, we can correct for it to determine the true brightness (luminosity of a star). But first we need a scale of stellar brightness.

Stellar magnitude scale—the stellar magnitude scale is a logarithmic scale based on the star catalogue of Hipparchus (2^nd B.C.). The naken-eye stars fall into six categories or magnitudes, with 6 being faintest. The modern version of this takes account of the fact that between Hipparchos’ 1^st and 2^nd magnitudes there is, on average, a factor of about 100X in brightness (therefore, one magnitude represents the fifth root of 100 =2.512 times). Adopting that, and some standard reference star, we have the modern stellar magnitude scale (for example, we could call Polaris 2.0, or Vega, 0.0). In any case, 5 magnitudes is a factor of 100X in brightness, so 10mags=100 ² or 10,000, 15 mags=100 ³ or 10 ⁶ times, etc.; 100X for each 5 magnitudes.

When the planets and the sun and moon are included, the magnitudes become negative, with the apparent magnitude of the sun being m=-26, venus has m=-4, and the brightest star, Sirius is of magnitude m=-1.4. The faintest objects which can be imaged are at about m=+30. All told a d ifference of over 55 magnitudes:

5mà 100

10mà 10,000

15mà 10 ⁶

20mà 10 ⁸

25 mà 10 ¹⁰

30mà 10 ¹²

35mà 10¹⁴

40mà 10 ¹⁶

45mà 10 ¹⁸

50mà 10 ²⁰

55mà 10 ²²

Example problem:

1) if star A has apparent magnitude 13.9 and star B has magnitude –1.1, how much brighter (fainter) is B than A? Answer: the difference is 15 magnitudes, which from the table above, represents 1,000,000 times; B is brighter than A by a factor of 1 million.

Next, we take distance into account, in order to directly compare the luminosity of stars which are at different distances. This requires the “inverse-square law,” which tells us that the brightness of a star depends inversely on the square of the distance: L =k/r ². If we double the distance, the star is ¼ as bright; if the distance is multiplied by 10, t he star is 1% (1/100^th) as bright. See the text for the basis for the inverse-square law.

We define absolute magnitude to be the magnitude a star would have at a distance of 10 parsecs.

Example problem:

1) Suppose a star has an apparent magnitude of m=11.8, and is at a distance of 1000 parsecs. What is its absolute magnitude? To move the star from 1000 up to 10 parsecs, it will be 100X nearer, meaning 100 ² = 10,000 times brighter. A factor of 10,000 means 10 magnitudes brighter. The result is either 11.8+10=21.8 (fainter!) or 11.8-10=1.8, which is brighter, and hence correct.

2) The sun is at a distance of 1 AU and has an apparent magnitude of –26. What is its absolute magnitude? One parsec is 206,265 AU so 10 parsecs is 2,000,000. To move the sun from 1 to 2 million parsecs will make it (2 x 10 ⁶) ² = 4x10 ¹² times brighter. How many magnitudes is that. The table above says that 10 ¹² times represents 30 magnitudes, and the f actor of 4 is another 1+. So the sun would be about 31 magnitudes fainter. Hence –26 +31 = +5, the absolute magnitude of the Sun. On the exam, you will only have to deal with differences of 5 magnitudes or its multiples.

We now have a way of comparing the luminosity of stars, independent of their distance. Is there another fundamental property of a star we might use in classifying them? The answer is yes, and that property is temperature, or color, or spectral type. The latter, spectral type, contains the most information, though it all depends on t emperature. The book has a t able of spectral types in the so-called Harvard classification of stellar spectra: O,B,A,F,G,K,M, from hottest to coolest stars. You only need to have a general idea of the correlation between types and temperature. But the temperature correlates directly with (blackbody) temperature: spectral type O and B stars, for example, are hot, and therefore blue. Spectral type M stars are cool and red. The sun is a spectral type G2 stars, and is yellow-white. Consult the table. Note that hydrogen lines are important in hot stars, especially around spectral type B,A, and F stars, while molecular bands are found in cool stars, e.g., type M, which have surface temperatures of only 3000K more or less.

If we plot t he luminosity or absolute magnitude vertically, with brightness increasing upwad, and the spectral type or temperature horizontally, with temperature increasing to the left, we obtain the Hertzsprung-Russell or H-R diagram, of which t here are many examples in your text. You should be able to identify 1) the main sequence, 2) red giants, 3) red supergiants, 4) white dwarf stars, and 5-6) know the labels for both axes. You should probably also know where the red dwarfs are, at the lower end of the main sequence. See p. 533, I think.

Luminosity: L=k x T ⁴ x area

So a star, like Sirius B, about twice as hot as the sun, emits 2 ⁴ =16 times as much energy per second from each square cm of surface. But it has only about 1% of the sun’s radius or about 1/10,000 of its area. The two factors combined, 16/10,000 makes Sirius B about 1/600^th as bright as the sun in absolute terms.

Mass-luminosity relation:

The luminosity or brightness of a star depends on the 3^rd or 4^th power of its mass: L = k M ³ . So a small change in mass results in a large increase in brightness. Stellar masses range f rom about 0.1 to 50 solar masses, but luminosities range from about 1/10,000 to 10 ⁶ times the luminosity of the sun.

III. Contents of the galaxy

We will discuss:

1) binary and multiple star systems

2) regular, pulsating variable stars

3) clusters of stars

4) interstellar matter

1) binary and multiple star systems:

According to the way they are detected, we classify binary star systems as:

a) visual

b) eclipsing

c) spectroscopic

Visual binaries are detected visually, that is, the separation is sufficiently wide (>.1”) to be observed that way. How do we know they are physically related? A simple statistical argument shows that if among the 6000+ naked eye stars, perhaps several hundred are pairs as close as one or a few seconds of arc, that cannot be line-of-sight accidents. Kepler’s first law, generalized, says that the orbits of any two bodies interacting through gravitational forces are conic sections (ellipse, parabola, hyperbola). If they are bound together, as in a binary star system, the orbits of both stars are ellipses. That is, the two stars have a center of mass (barycenter, center of gravity) somewhere between them (such that the lower mass star times its larger distance from the cg is balanced by the more massive star times its smaller distance: Mx=my, where (M,m) are the large/small masses, and (x,y) are the small/large distances. y/x=M/m, so if star A is 10 times the mass of star B, start B is 10 times as far from the center of mass; its orbit will be 10 times larger. In practice we see a relative orbit, of one star relative to the other, and a projected orbit, depending on how it is oriented in space.

An ellipse is characterized by its semi-major axis a, which is half the long axis, and e its eccentricity, which determines its shape. An ellilpse has the property that the sum of the distances from a point on the ellipse to the two foci is always a constant (=a). Kepler’s first law applied to the solar system says that “the orbits of the planets are ellipses with the sun at one focus”. In the case of binary star orbits, both orbits are ellipses with the cg at the focus of each of them.

Kepler’s third law allows us to determine the masses of the two stars through the relation:

P²=4π² a³/[G(m₁ + m₂)]

This provides the only direct method to measure the masses of stars. Or rather, binary stars, more generally, provide the only direct method. It gives the sum of the masses, and some other means must be used to determine each mass. For example, if the sum of the masses is 10 solar masses, but one star is 10 times as bright as the other, we might guess that the masses are 7 and 3. Note that a factor of 2X in mass should make a difference of 8X in brightness (mass-luminosity relation). This, by the way, is how we know the mass of the earth and of the planets that have moons.

In the solar system we can just write P²=a³ if we measure the period in years and the distance (semi-major axis) in AU.

Eclipsing binaries are close pairs whose orbital planes are tilted nearly in the line of sight from us to the stars, so that the stars alternately eclipse one another. See the figure I drew or the text. They have a characteristic light curve, usually with primary and secondary maxima, and from t he details of the light curve one can learn not only about masses, but sizes, shapes, etc. Best know eclipsing binary is β Persei, Algol. Note that a star which is an eclipsing binary as seen from the earth, might not be seen eclipsing from some other solar system.

Spectroscopic binaries are detected through their spectra alone. The stars are too close together or too distant to be separated in the largest telescope, and their orbits are not oriented such as to produced eclipses here at the earth. In some cases we see composite spectra of two different stars (say, a hot and a cool one), or we see lines due to one or both stars Doppler-shifted by their orbital motion. The Doppler effect is the shift in frequency of a source of sound or light depending on whether t he source is approaching or receeding. If a star is approaching us, its light is shifted to higher frequency, or toward the blue, if receeding, the light is shifted toward the red.

2) Variable stars. Regular pulsating variables (vs. cataclysmic, explosive stars)

These stars go through periods of instability in which the pulsate, and as they pulsate, they vary in brightness. Period of pulsation range from hours to many hundreds of days. Typically, the longer the period of pulsation, the greater the range of brightness change, sometimes exceeding 7 magnitudes or 600X.

I emphasized three of the many types of variable stars: 1) long-period variables, 2) Cepheid variables, and 3) RR Lyrae variable stars.,

Long period variables: these are giants relatively late in their evolution which pulsate with periods of 100-300 days or more, and brightness ranges of several magnitudes. The best known is Mira, ο Ceti , which has a period of about 330 days and ranges from mag. 2 or 3 to 9, that is, 250-600X in brightness.

Cepheid variables: these stars played a crucially important historical role in the determination of the distance scale of t he universe in the “teens” and 20s (1920s) when the distance to the fuzzy “spiral nebulae” was unknown. That is, they represent another cosmic yardstick, or rung on the “distance ladder.” It turns out t hat the period of pulsating of these variables, named after the prototype, δ Cephei, is directly correlated with their absolute magnitude, with the stars with the longest periods being the brightest (50d corresponding to something like M=-5, a period of 2d corresponding to about M=-1, for the so-called classical cepheids. Henrietta Leavitt of Harvard College Observatory calibrated this “period-luminosity relation” for cepheids. No Cepheid is near enough to allow direct determination of its parallax, so some other yardstick had to be used to determine the distances of enough to calibrate this relationship. Then, if both t he apparent magnitude m and the absolution magnitude M are known, the distance can be found. Example (which you will not have to reproduce): if a Cepheid variable is found in the Andromeda galaxy with apparent magnitude m=20, and with a period of a bout 50d, is found to have absolute mag. M=-5, what is the distance. The star is 25 magnitudes fainter than it would be at the standard distance of 10 parsecs, which is 10 ¹⁰ X. To be 10 ¹⁰ times fainter, it must be 10 ⁵ times farther away. But 10⁵ times 10 parsecs is 10⁶, or 1 million parsecs to the Andromeda galaxy. Cepheids have been identified using the Hubble Space Telescope in the Virgo cluster of galaxies, 20 million parsecs distant.

RR Lyrae variables: these are common in globular clusters and hence helped determine the distance to them. They all have about the same absolute magnitude (near 0), so that if one is identified, we have both m and M and can get distance. The designation RR Lyrae means a variable in Lyra, and variables are labeled in the sequence R,S, T….,Z, RR,RS,….,ZZ, etc.

3) Clusters

There are two main types of stellar clusters, a) globular, and b) open or galactic:

Globular clusters are dense aggregations of 100,000 or more old stars, located in the galactic halo and nucleus, containing Population II stars (see below). Example: the Hercules cluster, M13.

Open or galactic clusters are loose aggregations of 100 or more young stars, located in the galactic plane or disk, containing mostly Population I stars. Example: the Pleiades, M45

Population I stars are young stars, relatively rich in heavy elements (few percent), having formed in an interstellar medium enriched by the death of earlier stars which cast their heavy elements, formed in their cores, into space.

Population II stars are old stars, devoid or nearly devoid of heavy elements, having formed in a medium consisting mainly of H and He.

Stars form in clusters, that is, out of clouds of dust and gas (see below) which collapse to form stars, either a few hundred in the case of open clusters or a few hundred thousand in the case of globular clusters. The stars in clusters represent homogeneous samples of stars, because they all have t he same age, same d istance, and same composition. Their H-R diagrams are thus particularly “clean.” The H-R diagrams reveal their age, since in a young cluster, the main sequence may be continuous nearly all the way up to the hottest, most massive stars, while in an old cluster, it might be continuous only up to stars with the mass of the sun.

4) Interstellar matter:

Interstellar matter consists of a) dilute hydrogen gas, b) dense molecular clouds, c) organic molecules, and d) dust

Dilute H gas fills the space between the stars, perhaps only at the level of one H atom per cubic cm.

Molecular clouds are relatively dense clouds of mostly H gas (with He and trade elements) and dust which can collapse under gravity and fragment due to gravitational instability into smaller clouds which may form into stars. So these clouds are stellar nurseries, and a typical one will consist of an open cluster embedded in a cloud of dust and gas out of which stars continue to form. Examples: the Orion Nebular, M42, the Lagoon Nebula, M20, the Trifid Nebula, M8.

Organic molecules also fill the space between the stars, as indicated by the absorption of light that reaches us from distant stars. These include some fairly heavy (high molecular weight) molecules like methyl cyanide and chlorform (see text…?).

Interstellar dust is found in the plane of our galaxy and others, and helps obscure the center of the galaxy from examination using visible light. It consists of particles, perhaps of hydrogen or carbon, with dimensions of the order of a micrometer (10-6m). Such a particle will still contain perhaps a trillion atoms. Stars form out of clouds of dust and gas, and dust grains are essential in the formation of planets.

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p.s. no proof-reading or spell-checking done