Chapter 17: Properties of Stars

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Distance to Stars:

In order to learn anything about the composition, formation and evolution of stars, one must first determine their distances. For example, to determine the absolute luminosity of a star (the amount of energy emitted per second) you measure its apparent brightness (its luminosity from the Earth), then multiply by the distance from the Earth squared.

parallax

The most direct method to measure the distance to nearby stars is through the use of parallax. The Earth’s motion around the Sun every year produces a very small shift in nearby star’s position in the sky compared to distant, background stars. This shift is always less than one arcsecond for any star, which is very small (where a circle is 360 degrees, one degree is 60 arcminutes, one arcminute is 60 arcseconds).

prob_12

Proper Motion:

Although the stars appear fixed in the sky, they are actually moving through space at very high velocities. Their extremely large distances make this motion almost undetectable. This motion is called proper motion, and can also be used to judge the distance to stars.

proper_motion

The nearest stars should display the greatest proper motion, and the star with the greatest proper motion is Barnard’s star with a change of 10 arcsecs per year. Since only a few stars have proper motions greater than 1 arcsec per year, thus, the shape of the constellations have remained mostly unchanged since the dawn of man. The diagram below shows how the Big Dipper will look after 10,000 years.

big_dipper

Notice also that the Sun is moving through the Galaxy, so solar motion will give the appearance of stars in front of our motion moving towards an apex (like a car driving through snowflakes). Those behind us will appear to move towards an antapex.

barnard

Stellar Magnitudes:

The measure of the brightness of a star is, for historical and physiological reasons, called its apparent magnitude (see end of chapter). The human eye detects light in a logarithmic fashion, meaning that changes occur in powers of 10 rather than in a linear manner. So ancient astronomers divided stars into six classes or magnitudes where the brightest are first magnitude, the faintest are sixth magnitude. Later measurements showed that a change in 5 magnitudes is equal to a 100 increase in brightness.

Here is a list of common objects in the sky and their magnitudes. Note that the greater the magnitude (the more positive the number), the fainter the star). Negative numbers are bright stars.

Object Apparent Mag
—————————–
Sun -26.5
Full Moon -12.5
Venus -4.0
Jupiter -3.0
Sirius -1.4
Polaris 2.0
eye limit 6.0
Pluto 15.0
limit for telescope 25.0
—————————-

From the table it is clear that the Sun is the brightest object in the sky, but it is not the brightest star in the Galaxy. It’s apparent brightness is high since it is relatively nearby.

We also distinguish between apparent magnitude and absolute magnitude. Apparent magnitude is what we measure in the sky, absolute magnitude is the real luminosity of the star, corrected for distance.

prob_13

Solar Neighborhood:

Stars have different absolute luminosities. So the brightest stars in the sky are not necessarily the closest stars. Here is a list of the twenty brightest stars in the sky. And here is a list of the twenty nearest stars.

Twenty Brightest Stars:

star apparent distance
mag (parsecs)
——————————–
Sirius -1.50 2.6
Canopus -0.73 30.1
Alpha Centauri +0.10 1.3
Vega +0.04 8.0
Arcturus 0.00 11.0
Capella +0.05 13.8
Rigel +0.08 184.0
Procyon +0.34 3.5
Betelgeuse +0.41 184.0
Achernar +0.47 19.9
Beta Centauri +0.61 92.0
Altair +0.77 5.1
Alpha Crucis +1.58 119.6
Aldebaran +0.86 16.0
Spica +1.12 84.0
Antares +0.90 128.8
Pollux +1.15 11.3
Fomalhaut +1.18 6.9
Deneb +1.26 429.4
Beta Crucis +1.24 153.4
——————————–

Twenty Brightest Stars:

star apparent stellar distance
mag type (parsecs)
———————————————-
Proxima Centauri 11.5 M5 V 1.3
Alpha Centauri 0.1 G2 V 1.3
Barnard’s star 9.5 M5 V 1.8
Wolf 359 13.5 M6e 2.3
Lalande 21185 7.5 M2 V 2.5
Sirius -1.5 A1 V 2.6
Luyten 726-8 12.5 M6e V 2.7
Ross 154 10.6 M5e V 2.9
Ross 248 12.2 M6e V 3.2
Epsilon Eridani 3.7 K2 V 3.3
Luyten 789-6 12.2 M6 V 3.3
Ross 128 11.1 M5 V 3.3
61 Cygni 5.2 K5 V 3.4
Epsilon Indi 4.7 K5 V 3.4
Procyon 0.3 F5I V 3.5
Sigma 2398 8.9 M3 V 3.5
Groombridge 34 8.1 M1 V 3.6
Lacaille 9352 7.4 M2 V 3.6
Tau Ceti 3.5 G8 V 3.7
Lacaille 8760 6.7 M1 V 3.8
———————————————-

The nearest stars make-up what is called the solar neighborhood, shown below. Note that the nearest stars are mostly small dim stars. These types of stars are hard to see at great distances. The twenty brightest stars are mostly supergiant stars; which are rare, but very bright.

solar_neighbourhood

Stellar Masses:

Since stars are so far away, it is impossible to measure their masses directly. Instead, we look for binary star systems and use Newton’s law of gravity to measure their masses.

Two stars in a binary system are bound by gravity and revolve around a common center of mass (see end of chapter).  Kepler’s 3rd law of planetary motion can be used to determine the sum of the mass of the binary stars if the distance between each other and their orbital period is known.

prob_14

When you plot the mass of a star versus its absolute luminosity, one finds a correlation between the two quantities shown below.

mass_luminosity

This relationship is called the mass-luminosity relation for stars, and it indicates that the mass of a star controls the rate of energy production, which is thermonuclear fusion in the star’s core. The rate of energy generation, in turn, uniquely determines the stars total luminosity. Note that this relation only applies to stars before they evolve into giant stars (those stars which burn hydrogen in their core).

Notice that stars range in mass from about 0.08 to 100 times the mass of the Sun. The lower mass limit is set by the internal pressures and temperatures needed to start thermonuclear fusion (protostars too low in mass never beginning fusion and do not become stars). The upper limit is set by the fact that stars of mass higher than 100 solar masses become unstable and explode. Notice also that these range of masses corresponds to a luminosity range from 0.0001 to 105 solar luminosities.

Stellar Color:

Stars have a range of colors which represent their surface temperatures due to Wien’s law (which states that the peak emission of light from an object goes as the inverse of temperature). The color of a star is determined by that part of the visible spectrum where the peak amount of radiation is emitted.

Blue stars are extremely hot, red stars are relatively cool. Temperature here is a relative thing; cool means temperatures near 2,000 to 3,000K, about 15 times hotter than your oven. Blue stars have temperatures near 20,000K. The Sun is an intermediate yellow star with a surface temperature of 6,000K. The color of a star is determined by measuring its color index.

color_index

It is important to remember temperature and luminosity for a star are not strictly related. Stefan-Boltzmann’s law states that the amount of energy emitted goes as the temperature to the 4th power; but, this relation is only strictly true for an object that is a point source (i.e. it has no size). The temperature of a normal object is proportional to its surface area (for example, things cool faster if you spread them out = increase their surface area).

sphere_geo

So, it is possible for a star to be very bright (emit alot of energy) yet, be cool and red. We will see below that this means the star must be very large to be both bright and cool.

Stellar Spectral Type:

Stars are divided into a series of spectral types (see end of chapter) based on the appearance of their absorption spectra. Some stars have a strong signature of hydrogen (O and B stars), others have weak hydrogen lines, but strong lines of calcium and magnesium (G and K stars). After years of cataloging stars, they were divided into 7 basic classes: O, B, A, F, G, K and M. Note that the spectra classes are also divisions of temperature such that O stars are hot, M stars are cool.

Between the classes there were 10 subdivisions numbered 0 to 9. For example, our Sun is a G2 star. Sirius, a hot blue star, is type B3.

Why do some stars have strong lines of hydrogen, others strong lines of calcium? The answer was not composition (all stars are 95% hydrogen) but rather surface temperature.

As temperature increases, electrons are kicked up to higher levels (remember the Bohr model) by collisions with other atoms. Large atoms have more kinetic energy, and their electrons areexcited first, followed by lower mass atoms.

collisional_excitation

If the collision is strong enough (high temperatures) then the electron is knocked off the atom and we say the atom is ionized (see end of chapter). So as we go from low temperatures in stars (couple 1,000K) we see heavy atoms, like calcium and magnesium, in the stars spectrum. As the temperature increases, we see lighter atoms, such as hydrogen (the heavier atoms are all ionized by this point and have no electrons to produce absorption lines).

stellar_lines

As we will see later, hotter stars are also more massive stars (more energy burned in the core). So the spectral classes of stars is actually a range of masses, temperatures, sizes and luminosity. For normal stars (called main sequence stars) the following table gives their properties:
Type Mass Temp Radius Lum (Sun=1)
——————————————-
O 60.0 50,000 15.0 1,400,000
B 18.0 28,000 7.0 20,000
A 3.2 10,000 2.5 80
F 1.7 7,400 1.3 6
G 1.1 6,000 1.1 1.2
K 0.8 4,900 0.9 0.4
M 0.3 3,000 0.4 0.04
——————————————-

So our Sun is a fairly middle-of-the-road G2 star:

sun

A B star is much larger, brighter and hotter. An example is HD93129A shown below:

hd93129a

Luminosity Classes:

Closer examination of the spectra of stars shows that there are small changes in the patterns of the atoms that indicate that stars can be separated by size called luminosity classes.

The strength of a spectra line is determined by what percentage of that element is ionized. An atom that is ionized has had all its electrons stripped off and can produce no absorption of photons. At low densities, collisions between atoms are rare and they are not ionized. At higher densities, more and more of the atoms of a particular element become ionized, and the spectral lines become weak.

One way to increase density at the surface of a star is by increasing surface gravity. The strength of gravity at the surface of a star is determined by its mass and its radius (remember escape velocity). For two stars of the same mass, but different sizes, the larger star has a lower surface gravity = lower density = less ionization = stronger spectral lines.

sur_g

This was applied to all stars and it was found that stars divide into five luminosity classes: I, II, III, IV and V. Stars of type I and II are called supergiants, being very large (low surface gravity), stars of type III and IV are called giant stars. Stars of type V are called dwarfs. The Sun is a G2 V type stars.

So now we have a range of stellar colors and sizes. For example, Aldebaran is a red supergiant star:

Aldebaran

Arcturus is an orange giant star:

arcturus

HST imaging found that Betelgeuse is one of the largest stars, almost the size of our whole solar system.

betelgeuse

The other extreme was also found, that there exist a class of very small stars called white and brown dwarfs, with sizes close to the size of the Earth:

40EridanusB gl229b

Red and blue supergiant stars, as well as giant stars exist. The following is a comparison of these types.

compare_star_sizes

Luminosity Function:

Surveying the skies for stars is a very biased method of doing science since clearly the brightest stars are the easiest to observe. But are the brightest stars typical of the stellar population? To determine what a typical star is like we construct a luminosity function, the number of stars as a function of absolute magnitude in the form of a histogram.

A luminosity function is constructed by sampling a volume of space and counting all the stars in that volume. The resulting plot will look like the diagram below:

lum_function

Notice that the most common type of star is actually small, low luminosity stars. Bright stars are quite rare (although they can be seen from great distances). Since luminosity is correlated with mass, then this means that high mass stars are rare.

Russell-Vogt Theorem:

Despite the range of stellar luminosities, temperatures and luminosities, there is one unifying physical parameter. And that is the mass of the star. Hot, bright stars are typically high in mass. Faint, cool stars are typically low in mass. This sole dependence on mass is so strong that it is given a special name, the Russell-Vogt Theorem.

The Russell-Vogt Theorem states that all the parameters of a star (its spectral type, luminosity, size, radius and temperature) are determined primarily by its mass. The emphasis on `primarily’ is important since we will soon see that this only applies during the `normal’ or hydrogen burning phase of a star’s life. A star can evolve, and change its size and temperature. But, for most of the lifetime of a star, the Russell-Vogt Theorem is correct, mass determines everything.

Binary Stars:

Planet’s revolve around stars because of gravity. However, gravity is not restricted to between large and small bodies, stars can revolve around stars as well. In fact, 85% of the stars in the Milky Way galaxy are not single stars, like the Sun, but multiple star systems, binaries or triplets.

If two stars orbit each other at large separations, they evolve independently and are called a wide pair. If the two stars are close enough to transfer matter by tidal forces, then they are called a close or contact pair.

Binary stars obey Kepler’s Laws of Planetary Motion, of which there are three.

1st law (law of elliptic orbits): Each star or planet moves in an elliptical orbit with the center of mass at one focus.

ellipse

Ellipses that are highly flattened are called highly eccentric. Ellipses that are close to a circle have low eccentricity.

2nd law (law of equal areas): a line between one star and the other (called the radius vector) sweeps out equal areas in equal times.

equal_areas

This law means that objects travel fastest at the low point of their orbits, and travel slowest at the high point of their orbits.

3rd law (law of harmonics): The square of a star or planet’s orbital period is proportional to its mean distance from the center of mass cubed.

It is this last law that allows us to determine the mass of the binary star system (note only the sum of the two masses, see previous lecture).

binary_star_orbit

Visual Binaries:

Any two stars seen close to one another is a double star, the most famous being Mizar and Alcor in the Big Dipper. Odds are, though, that a double star is probably a foreground and background star pair that only looks near each other. With the invention of the telescope may such pairs were found. Herschel, in 1780, measured the separation and orientations of over 700 double stars and found that only about 50 pairs changed orientation over 2 decades of observation.

One such example is Sirius A and B shown below. Their motion through the sky is a complex, twisted path which takes decades to map and plot.

sirius_motion

The observations made relative to center of mass of the two stars shows their respective elliptical orbits.

Spectrum Binary:

Often a system of binary stars are too close (or too far away) to be resolved into an optical pair. However, a spectrum of such an object will display the spectral fingerprints of two different stellar types (if the stars are different in spectral type).

spectrum_binary

Of course, the problem with this method is that since faint, cool stars are more common than brighter stars, the odds are that the companion is too faint to be detected in a spectrum. Also, just detecting two spectrum will not determine their masses since relative velocities are needed.

Spectroscopic Binary:

Another avenue to determine the masses of stars is to measure their relative velocities via the Doppler shift of their spectral lines. This is used when the pair can not be resolved as an visual binary, but motion is seen in the spectra of one star.

spectroscopic_binary

Notice that you do not need to see two spectra, only the motion of one of the stars is needed to deduce the existence of the binary system (why would one star be moving on its own?). Most binary stars are too close to separate the components, yet their existence can be deduced from Doppler shifts.

Typical velocities between binaries are 3 to 5 km/sec, so very high resolution, Coude spectra must be taken to observe this phenomenon.

Eclipsing Binaries:

In the late 1600’s, Italian astronomers noticed that some stars occasionally drop in their brightness up to 1/3 their peak luminosity. Later measurements showed that these declines were periodic, ranging from hours to days. It is now recognized that these brightness changes are due to the eclipsing of one star by another (as they pass in front of each other).

Eclipsing binaries are studied by monitoring their light curves (shown below), the changes in brightness with time. When the smaller, dimmer star passes in front of the brighter star, there is a deep minimum. When the dimmer star passes behind the bright star there is a second, less deep, minimum. Notice the transition zone at the start and end of each eclipse.

eclipsing_binary

Eclipsing binaries are very rare since the orbits of the stars must be edge-on to our solar system. Notice that an eclipsing binary is the only direct method to measure the radius of a star, both the primary and the secondary from the time for the light curve to reach and rise from minimum.

Contact Binaries:

When two stars are close in separation it is possible for tidal forces to come into play. Since stars are not solid bodies, rather made of gases, then gravity can strip material and transfer it from one star to the other. Thus we say the binaries are in contact, even if their surfaces are not touching directly.

How stars exchange material is similar to the way a ball rounds over and down a hill. The ball must have enough kinetic energy to exceed the potential energy of the hill. Around two stars there are lines of equipotential. Imagine two nearby lakes. If the water rises it takes on the shape of the contours of the land, the equipotential contours. If the water level rises too high, the lakes merge.

equipotential

In the same way, there exist lines of equipotential around stars, where the gravitational pull from one star exceeds that of another. This line where the forces or energies balance is called the Roche lobe. When the star’s radii exceed the Roche lobe, the gases are free to transfer from one star to the other. Usually in the form of a tube or stream.

binary_roche

In some binary stars, such as Phi Persei (see end of chapter), one of the binary stars evolves and expands (see stellar evolution lecture). Its surface exceeds the Roche lobe and material is streamed from one star to the other.

Some contact systems, such as the Algol system require sophisticated super computer simulations to understand the mass exchange.

Algol System
Algol System

Hertzsprung-Russell Diagram:

In 1905, Danish astronomer Einar Hertzsprung, and independently American astronomer Henry Norris Russell, noticed that the luminosity of stars decreased from spectral type O to M. They developed the technique of plotting absolute magnitude for a star versus its spectral type to look for families of stellar type.

These diagrams, called the Hertzsprung-Russell or HR diagrams, plot luminosity in solar units on the Y axis and stellar temperature on the X axis, as shown below.

hr_diagram_1

Notice that the scales are not linear. Hot stars inhabit the left hand side of the diagram, cool stars the right hand side. Bright stars at the top, faint stars at the bottom. Our Sun is a fairly average star and sits near the middle.

A plot of the nearest stars on the HR diagram is shown below:

hr_diagram_2

Most stars in the solar neighborhood are fainter and cooler than the Sun. There are also a handful of stars which are red and very bright (called red supergiants) and a few stars that are hot, but very faint (called white dwarfs). We will see in a later lecture that stars begin their life on the main sequence then evolve to different parts of the HR diagram.

Several regions of the HR diagram have been given names, although stars can occupy any portion. The brightest stars are called supergiants. Star clusters are rich in stars just off the main sequence called red giants. Main sequence stars are called dwarfs. And the faint, hot stars are called white dwarfs.

hr_diagram_3

prob_15

On a log-log plot, the R squared term in the above equations is a straight line on an HR diagram. This means that on a HR diagram, a star’s size is easy to read off once its luminosity and color are known.

hr_diagram_4

The HR diagram is a key tool in tracing the evolution of stars. Stars begin their life on the main sequence, but then evolve off into red giant phase and supergiant phase before dying as white dwarfs or some more violent endpoint.

Star Clusters:

When stars are born they develop from large clouds of molecular gas. This means that they form in groups or clusters, since molecular clouds are composed of hundreds of solar masses of material. After the remnant gas is heated and blow away, the stars collect together by gravity. During the exchange of energy between the stars, some stars reach escape velocity from the protocluster and become runaway stars. The rest become gravitationally bound, meaning they will exist as collection orbiting each other forever.

star_cluster

When a cluster is young, the brightest members are O, B and A stars. Young clusters in our Galaxy are called open clusters due to their loose appearance. They usually contain between 100 and 1,000 members. One example is the binary cluster below:

jewel_box_cluster

And the Jewel Box cluster:

open_cluster

Early in the formation of our Galaxy, very large, globular clusters formed from giant molecular clouds. Each contain over 10,000 members, appear very compact and have the oldest stars in the Universe. One example is M13 (the 13th object in the Messier catalogs) shown below:

m13

Cluster HR Diagrams:

Since all the stars in a cluster formed at the same time, they are all the same age. A very young cluster will have a HR diagram with a cluster of T-Tauri stars evolving towards the main sequence. As time passes the most massive stars at the top of the main sequence evolve into red giants. Therefore, the older the cluster, the fewer stars to be found at the top of the the main sequence, and an obvious grouping of red giants will be seen at the top right of the HR diagram.

This effect, of an evolving HR diagram with age, becomes a powerful test of our stellar evolution models. A computer model can be built that follows the changes in stars of various masses with time. Then an theoretical HR diagram can be built at each timestep. Observations of star clusters are compared to these computer generated HR diagrams to test of our understanding of stellar physics.

Observations of star clusters consist of performing photometry on as many individual stars that can be measured in a cluster. Each star is plotted by its color and magnitude on the HR diagram. Shown below is one such diagram for the globular cluster M13.

m13_hr

Note that the main sequence only exists for low mass G, K and M stars. All stars bluer than the turn-off point have exhausted their hydrogen fuel and evolved into red giants millions and billions of years ago. Also visible is a clear red giant branch and a post-red giant phase region, the horizontal branch.

Plotting various star cluster HR diagrams together gives the following plot

HR_clusters

Understanding the changes in the lifetime of a main sequence star is a simple matter of nuclear physics, where we can calibrate the turn-off points for various clusters to give their ages. This, then, provides a tool to understand how our Galaxy formed, by mapping the positions and characteristics of star clusters with known ages. When this is done it is found that old clusters form a halo around our Galaxy, young clusters are found in the arms of our spiral galaxy near regions of gas and dust.

 

Variable Stars:

Astronomers noticed that there was a gap in the HR diagram of old clusters. Stars, that are rarely found in this gap, were found to vary in their brightness. The gap came to be known as the instability strip, since the stars inside that region appeared unstable. And the stars within the instability strip are known as variable stars.

The two most famous types of variable stars are Cepheid variables and RR Lyrae variables. Both these types of stars obey a period-luminosity relationship, meaning that their period of variability corresponds to a unique absolute luminosity. Thus, by knowing the period of star determines its absolute luminosity. The difference between the apparent luminosity and its absolute luminosity is the distance to the star. Measurements of the variable stars in nearby galaxies allows us to determine the distance to those galaxies.

instability_strip

Cepheid’s are the brightest variable stars, with luminosities from 1,000 to 100,000 times brighter than the Sun. Cepheids have periods around 1 day from maximum to maximum. RR Lyrae stars are fainter than Cepheids, although still 100 times brighter than the Sun. They have periods from hours to a half a day.

The mechanism behind variable stars is pulsation. In normal stars, the internal pressure balances the force of gravity. However, stars that are evolving undergo sharp changes in their energy outputs. Much like pushing a swing, these changes in energy production puts the structure of the star out of balance. If the pressure exceeds the surface gravity, then the star expands until the pressure decreases to equal the surface gravity (volume goes up, pressure goes down). But the inertia of the outward moving layers carries the surface of the star past the balance point. Now the weight of the layers exceeds the pressure, and the surface drops, again falling past the balance point and the cycle begins again.

The cycle of pulsations is show below along with the changes in size and temperature of the variable star and the resulting light curve. Notice that the luminosity of the variable star is at a maximum when the size and temperature of a star are at a minimum. This is because Stefan-Boltzmann’s law tells us that the luminosity is much more dependent on temperature than size.

variable_star

Of course, like a bouncing ball, a variable star should stop pulsating after some time as the energy is radiated away. However, a layer of ionized helium serves as a “valve” to store the energy. When the star contracts, the helium ionizes and stores the gravitational energy. Ionized helium increases the opacity of the layers, which traps heat and the star expands.

helium_layer


 

Apparent Magnitude

Stellar magnitude is measure of the brightness of a star or other celestial body. The brighter the object, the lower the number assigned as a magnitude. In ancient times, stars were ranked in six magnitude classes, the first magnitude class containing the brightest stars. In 1850 the English astronomer Norman Robert Pogson proposed the system presently in use. One magnitude is defined as a ratio of brightness of 2.512 times; e.g., a star of magnitude 5.0 is 2.512 times as bright as one of magnitude 6.0. Thus, a difference of five magnitudes corresponds to a brightness ratio of 100 to 1. After standardization and assignment of the zero point, the brightest class was found to contain too great a range of luminosities, and negative magnitudes were introduced to spread the range.

Apparent magnitude is the brightness of an object as it appears to an observer on Earth. The Sun’s apparent magnitude is -26.7, that of the full Moon is about -11, and that of the bright star Sirius, -1.5. The faintest stars visible through the largest telescopes are of (approximately) apparent magnitude 20. Absolute magnitude is the brightness an object would exhibit if viewed from a distance of 10 parsecs (32.6 light-years). The Sun’s absolute magnitude is 4.8.

Bolometric magnitude is that measured by including a star’s entire radiation, not just the portion visible as light. Monochromatic magnitude is that measured only in some very narrow segment of the spectrum. Narrow-band magnitudes are based on slightly wider segments of the spectrum and broad-band magnitudes on areas wider still. Because ordinary photographic plates are more sensitive to blue light than is the eye, photographic magnitude is sometimes called blue magnitude. Visual magnitude may be called yellow magnitude, because the eye is most sensitive to light of that color.


 

The word particle has been used in this article to signify an object whose entire mass is concentrated at a point in space. In the real world, however, there are no particles of this kind. All real bodies have sizes and shapes. Furthermore, as Newton believed and is now known, all bodies are in fact compounded of smaller bodies called atoms. Therefore, the science of mechanics must deal not only with particles but also with more complex bodies that may be thought of as collections of particles.

To take a specific example, the orbit of a planet around the Sun was discussed earlier as if the planet and the Sun were each concentrated at a point in space. In reality, of course, each is a substantial body. However, because each is nearly spherical in shape, it turns out to be permissible, for the purposes of this problem, to treat each body as if its mass were concentrated at its centre. This is an example of an idea that is often useful in discussing bodies of all kinds: the centre of mass. The centre of mass of a uniform sphere is located at the centre of the sphere. For many purposes (such as the one cited above) the sphere may be treated as if all its mass were concentrated at its centre of mass.

To extend the idea further, consider the Earth and the Sun not as two separate bodies but as a single system of two bodies interacting with one another by means of the force of gravity. In the previous discussion of circular orbits, the Sun was assumed to be at rest at the centre of the orbit, but, according to Newton’s third law, it must actually be accelerated by a force due to the Earth that is equal and opposite to the force that the Sun exerts on the Earth.

center_of_mass

This remarkable result means that, as the Earth orbits the Sun and the Sun moves in response to the Earth’s gravitational attraction, the entire two-body system has constant linear momentum, moving in a straight line at constant speed. Without any loss of generality, one can imagine observing the system from a frame of reference moving along with that same speed and direction. This is sometimes called the centre-of-mass frame. In this frame, the momentum of the two-body system is equal to zero.


 

Spectral Types

Most stars are grouped into a small number of spectral classes. The Henry Draper Catalogue lists spectral classes from the hottest to the coolest stars. These types are designated, in order of decreasing temperature, by the letters O, B, A, F, G, K, and M. This group is supplemented by R- and N-type stars (today often referred to as carbon, or C-type, stars) and S-type stars. The R-, N-, and S-type stars differ from the others in chemical composition; also, they are invariably giant or supergiant stars.

The spectral sequence O-M represents stars of essentially the same chemical composition but of different temperatures and atmospheric pressures. This simple interpretation, put forward in the 1920s by the Indian astrophysicist Meghnad N. Saha, has provided the physical basis for all subsequent interpretations of stellar spectra. The spectral sequence is also a colour sequence: the O- and B-type stars are intrinsically the bluest and hottest; the M-, R-, N-, and S-type stars are the reddest and coolest.

In the case of cool stars of type M, the spectra indicate the presence of familiar metals, including iron, calcium, magnesium, and also titanium oxide molecules (TiO), particularly in the red and green parts of the spectrum. In the somewhat hotter K-type stars, the TiO features disappear, and the spectrum exhibits a wealth of metallic lines. A few especially stable fragments of molecules such as cyanogen (CN) and the hydroxyl radical (OH) persist in these stars and even in G-type stars such as the Sun. The spectra of G-type stars are dominated by the characteristic lines of metals, particularly those of iron, calcium, sodium, magnesium, and titanium.

The behaviour of calcium illustrates the phenomenon of thermal ionization. At low temperatures a calcium atom retains all of its electrons and radiates a spectrum characteristic of the neutral, or normal, atom; at higher temperatures collisions between atoms and electrons and the absorption of radiation both tend to detach electrons and to produce singly ionized calcium atoms. At the same time, these ions can recombine with electrons to produce neutral calcium atoms. At high temperatures or low electron pressures, or both, most of the atoms are ionized. At low temperatures and high densities the equilibrium favours the neutral state. The concentrations of ions and neutral atoms can be computed from the temperature, the density, and the ionization potential (namely, the energy required to detach an electron from the atom).

The absorption line of neutral calcium at 4227 Angstroms is thus strong in cool M-type dwarf stars, in which the pressure is high and the temperature is low. In the hotter G-type stars, however, the lines of ionized calcium at 3968 and 3933 Angstroms (the “H” and “K” lines) become much stronger than any other feature in the spectrum.

In stars of spectral type F, the lines of neutral atoms are weak relative to those of ionized atoms. The hydrogen lines are stronger, attaining their maximum intensities in A-type stars, in which the surface temperature is about 9,000 K. Thereafter, these absorption lines gradually fade as the hydrogen becomes ionized.

The hot B-type stars, such as Epsilon Orionis, are characterized by lines of helium and of singly ionized oxygen, nitrogen, and neon. In very hot O-type stars, lines of ionized helium appear. Other prominent features include lines of doubly ionized nitrogen, oxygen, and carbon and of trebly ionized silicon, all of which require more energy to produce.

In the more modern system of spectral classification, called the MK system (after the American astronomers William W. Morgan and Philip C. Keenan who introduced it), luminosity class is assigned to the star along with the Draper spectral class. For example, Alpha Persei is classed as F5 Ia, which means that it falls about halfway between the beginning of type F (i.e., F0) and of type G (i.e., G0). The Ia suffix means that it is a particularly luminous supergiant. The star Pi Cephei, classed as G2 III, is a giant falling between G0 and K0 but much closer to G0. The Sun, a dwarf star of type G2, is classified as G2 V. A star of luminosity class II falls between giants and supergiants; one of class IV is called a subgiant.


 

Excitation

Excitation is the addition of a discrete amount of energy (called excitation energy) to a system–such as an atomic nucleus, an atom, or a molecule–that results in its alteration, ordinarily from the condition of lowest energy (ground state) to one of higher energy (excited state).

collisional_excitation

In nuclear, atomic, and molecular systems, the excited states are not continuously distributed but have only certain discrete energy values. Thus, external energy (excitation energy) can be absorbed only in correspondingly discrete amounts.

Thus, in a hydrogen atom (composed of an orbiting electron bound to a nucleus of one proton), an excitation energy of 10.2 electron volts is required to promote the electron from its ground state to the first excited state. A different excitation energy (12.1 electron volts) is needed to raise the electron from its ground state to the second excited state.

Similarly, the protons and neutrons in atomic nuclei constitute a system that can be raised to discrete higher energy levels by supplying appropriate excitation energies. Nuclear excitation energies are roughly 1,000,000 times greater than atomic excitation energies. For the nucleus of lead-206, as an example, the excitation energy of the first excited state is 0.80 million electron volts and of the second excited state 1.18 million electron volts.

The excitation energy stored in excited atoms and nuclei is radiated usually as ultraviolet light from atoms and as gamma radiation from nuclei as they return to their ground states. This energy can also be lost by collision.

The process of excitation is one of the major means by which matter absorbs pulses of electromagnetic energy (photons), such as light, and by which it is heated or ionized by the impact of charged particles, such as electrons and alpha particles. In atoms, the excitation energy is absorbed by the orbiting electrons that are raised to higher distinct energy levels. In atomic nuclei, the energy is absorbed by protons and neutrons that are transferred to excited states. In a molecule, the energy is absorbed not only by the electrons, which are excited to higher energy levels, but also by the whole molecule, which is excited to discrete modes of vibration and rotation.


 

Ionization

Ionization is any process by which electrically neutral atoms or molecules are converted to electrically charged atoms or molecules (ions). Ionization is one of the principal ways that radiation, such as charged particles and X rays, transfers its energy to matter.

Ionization by collision occurs in gases at low pressures when an electric current is passed through them. If the electrons constituting the current have sufficient energy (the ionization energy is different for each substance), they force other electrons out of the neutral gas molecules, producing ion pairs that individually consist of the resultant positive ion and detached negative electron. Negative ions are also formed as some of the electrons attach themselves to neutral gas molecules. Gases may also be ionized by intermolecular collisions at high temperatures.

Ionization, in general, occurs whenever sufficiently energetic charged particles or radiant energy travel through gases, liquids, or solids. Charged particles, such as alpha particles and electrons from radioactive materials, cause extensive ionization along their paths. Energetic neutral particles, such as neutrons and neutrinos, are more penetrating and cause almost no ionization. Pulses of radiant energy, such as X-ray and gamma-ray photons, can eject electrons from atoms by the photoelectric effect to cause ionization. The energetic electrons resulting from the absorption of radiant energy and the passage of charged particles in turn may cause further ionization, called secondary ionization. A certain minimal level of ionization is present in the Earth’s atmosphere because of continuous absorption of cosmic rays from space and ultraviolet radiation from the Sun.


Phi Persei

4.1.1
4.1.1

Life near the double-star system of Phi Persei is never dull, as this illustration shows. Taken from the perspective of one of the Hubble Space Telescope observations of Phi Persei, this artist’s depiction provides a taste of the double- star system’s unstable existence. The bright “Be” star – a type of hot star with a broad, flattened disk – is the white, semicircular object looming in the upper right of the illustration. The red, pancake-shaped object surrounding the star is a gas disk. The gas is material the star is losing because of its rapid rotation. The small, hot subdwarf is in the lower left of the illustration. The blasts of white light represent particles of material – called a stellar wind – being released by the star. This powerful stellar wind is heating part of the “Be” star’s gas disk. The red ring of material surrounding the subdwarf was probably formed from the “Be” star’s outflow of gas. The subdwarf is moving toward the right in its 126-day orbit around the “Be” star.

phiperi