ASTROPHYSICS

 

ASTRONOMY (7.1. - 7.2)

7.1. The "geography" of the universe

 

Sun, planets and moons

 

In the center we have the sun, our closest star. There are so far 9 known planets, of which the 5 inner have been known since ancient times, Uranus was discovered in the 18th and Neptune in the 19th century, Pluto as late as 1930. The gravitational disturbances on the orbits of thus far known planets lead to successful predictions of the existence and approximate orbits of new ones. Irregularities in the orbit of Mercury (its 'perihelion precession') lead in the late 19th century to a search for a planet even closer to the sun (and it was tentatively named Vulcan) but none was found and the irregularities ca 1915 shown to be a side-effect of the theory of relativity.

 

The average distance of Earth from the sun, ca 150 million km or 1.5 x 1011 m is called 1 astronomical unit, 1 AU. The mass of the earth is ca 6 x 1024 kg. The radius of earth is ca 6370 km.

 

Object           Dist. from sun/mill.km           m/1024kg        Diam./103km                        Moons

 

Mercury         60                                        0.33               4.9                                       -                   

Venus            108                                      4.9                 12.0                                     -

Earth              150                                      6.0                 12.8                                     1

Mars              228                                      0.64               6.8                                       2

Jupiter            778                                      1900              143                                      15+

Saturn            1430                                    570                121                                      17+

Uranus           2900                                    87                  51                                        15

Neptune         4500                                    10                  50                                        8

Pluto              5920                                    0.01               2.3                                       1

 

More information about the solar system is found at The Nine Planets website, http://www.seds.org/nineplanets/nineplanets/nineplanets.html (now debateable!)

 

The orbits of the planets are elliptic but nearly circular. That such orbits should be followed can (not required here) be shown to be a necessary mathematical consequence of the Universal Law of Gravity,

 

F = Gm1m2/r2

 

Recall the laws of Kepler from mechanics

 

 

 

 

  • Kepler I: The planets follow or move in elliptic orbits with the sun in one focus
  • Kepler II: A line from the planet to the sun sweeps over the same area in the same time (meaning that they move faster when closer to the sun)
  • Kepler III: The Square of the periods (time to complete one orbit that is the local “year”) is proportional to the cubes of the radiuses.

 

T2 = kr3

 

The moons of the planets are in similar orbits around their planets, and a Kepler III (with a different k-value) could be used for several moons of one planet.

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Asteroids and comets

 

In the solar system we also have

 

·      asteroids, smaller planets and rocks mainly in the "asteroid belt" between the orbits of Mars and Jupiter, but to some extent also in other parts of the system

·        Comets, that is smaller (ca a few km) pieces of ice and frozen gases which have extremely eccentric elliptic orbits, that is they are sometimes very near the sun and sometimes very far from it. They become visible when they approach the sun and a tail of boiled-off gases reflects sunlight. The tail is always in the direction away from the sun (and therefore precedes the comet when it moves away from the solar system).

 

Stars and galaxies and...

 

For distances within the solar system, the astronomical unit is suitable. Outside that distance the light year, 1 ly, is used. This is the distance travelled by light in one year (= 60 x 60 x 24 x 365 seconds = 31536000s) so

 

1 ly = 3.00 x 108 ms-1 x 31536000s = 9.46 x 1015 m

 

The nearest stars are ca 4 ly (Alpha Centauri, a triple star) and 6 ly (Barnard's star) from us. For comparison, Earth is about 8 light minutes 20 seconds from the sun; Pluto about 6 light hours.

 

The stars "near" us form the Milky Way, a galaxy containing ca 100 billion stars shaped like a disc with some spiral arms. The size of our galaxy is in the order of magnitude 100 000 ly and it rotates around its center in ca 200 - 300 million years. Except for the stars, there is mostly thin interstellar matter between the stars in our Galaxy.

s01b

 

There are various types of galaxies such as spiral, elliptical and irregular ones. Many galaxies near each other form galactic clusters which in turn form superclusters, which make up the known Universe.

 

Stellar clusters and constellations

 

In some parts of a galaxy a number (maybe 100 - 10000) stars can be considerably closer to each other than the several light-years common in our parts of the galaxy. These are called stellar clusters

 

A constellation is a pattern of stars which seem to be near each other in the night sky. In 3-dimensional reality they do not have to be near each other.

 

Pulsars and quasars

 

In the 1960’s objects which emit light or other radiation in regular pulses were discovered and first briefly considered possible signs of extra-terrestrial life. They are more likely to be stars which emit radiation dominantly in one direction, which because of the star's rotation it makes them appear as regularly flashing beacons, pulsating stars or pulsars.

 

Certain stars emit much more radiation than a star regularly does and are named quasi-stellar objects or quasars.

 

Binary stars

 

Many stars are not, like our sun, the only one in a solar system. It is quite common for a star to be a double (binary) or triple star, that is to have two or three stars rotating around each other or some point in space. In such a solar system it would be difficult to have a stable planetary orbit, and even more difficult to have one in which the planet remains at roughly the same distance from a star providing a stable climate. Binary stars can be categorized as:

 

·        Visual binaries: a double star where the two components can be distinguished with a strong telescope

·        Spectroscopic binaries: a double star which appears to be one star, but where the spectral lines emitted change wavelength because of the Doppler Effect (see diagram below)

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·        Eclipsing binaries: a double star detected as such by one star getting in the way of the other, thus decreasing its brightness temporarily (not to be confused with a variable star, see below)

 

In addition to these there can be false (visual) binaries which appear to be very close but may be at very different distances from us.

 

Variable stars and Cepheid’s

 

Most stars, including our Sun, have periodically varying brightness or intensity. For some stars (e.g. Cepheid’s) the periodic variations in intensity are clearer and related to the "power" with which it emits light and other types of radiation. This will prove useful in the later sections.

 

7.2. Astronomic observations

 

Apparent motion of stars

 

Daily motion: As the earth rotates at 24 hours, the stars seem to rotate while keeping their positions relative to each other. In a direction where an axis can be imagined to go from the South Pole to the North Pole and onwards, one will find the point in the sky which stars seem to rotate in circles. Very near this direction the star Polaris is found.

 

Annual motion: As the earth makes a revolution around the sun, the set of stars visible above the horizon changes somewhat during the year since the earth’s imagined axis is not at a perfect 90o angle to the plane of revolution, but rather at one of ca 66.5o (in other words - a plane through the equator makes a 23.5o angle with the plane of revolution).

 

[Describing astronomic observations

 

The easiest way to describe where a star has been observed is to use the azimuth, Az (0 or 360o for north, 90 for east, 180 for south, 270 for west) and the altitude, Alt (angle up from the horizon, that is 0o at the horizon and 90o for zenith = the direction vertically upwards). This system, however, depends on where on earth the observation was made, and when.

 

Another system which is independent of the time and place of observation is the right ascension (RA) and declination (Dec) system. It is more useful for communicating discoveries with others. Conversions between the systems are made conveniently with astronomic software, e.g. the freeware Sky Map demo version (www.skymap.com).]

 

ASTROPHYSICS (7.3 - 7.13)

 

7.3. Stellar parallax

 

When the earth makes a revolution around the sun in one year, other stars (rather near us) will appear to be in a slightly different direction (compared to a background of stars very far away). The angle q which (a distance equal to) the radius r of earth's orbit, that is 1 AU, from a star at the distance d from our solar system is the parallax angle. This angle is very small, and often measured in the unit 1 arc second = 1/3600 of a degree.

s03a

 

From this we find that:

·        Tan q = r / d => d = r / tan q but since tan q » q for very small q (in radians) we get

·        d = r / q

If we now used conventional SI units we would insert r in meters, q in radians and get d in meters. If instead we use AU for r (which gives r = 1 in this unit), arc-seconds for q which we now call p (for parallax angle) then the value obtained for d will by definition be in a unit called 1 parsec = 1 pc, where

 

1 parsec = 3.26 ly              [DB p.2]

 

And

 

d(parsec) = 1 / p(arc-second)                  [DB p. 12]

 

Since there is a limit to the "resolution" of telescopes, that is shows how small the angles they can measure are, this method is relevant only for stars rather near us, currently up to about 100 pc (ca 300 ly). Within distance there are, however, a number of stars which can be used to check the validity of other distance measurement methods (recall that the nearest star is ca 4 ly from us; the 20 nearest are within ca 12 ly).

 

7.4. Absolute luminosity (power) and and apparent brightness (intensity)

 

Luminosity (power) and apparent brigthness (intensity)

 

If a light bulb emits 60 W of light in all directions (since its efficiency is not 100% it would be less in reality) the watts of light energy hitting a surface at some distance r from the bulb would be the total 60W only of the surface embraced by the bulb to cover all directions. This could be done with a spherical surface with the bulb at its center and the radius r. The area of the surface would then be A = 4pr2 and we can define intensity = power/area with the unit 1 Wm-2 so that

 

I = P/A = 4pr2

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The intensity which hit the surface of this imagined spherical surface can be measured with for example a solar cell; if we know with what efficiency it converts the light energy which hits it into electrical energy. If we have measured the I and know the P then we can solve I = 4pr2 for r and find out that.

 

In an astronomic context, we would use the terms:

·        Absolute luminosity L for the power in W of (the light emitted by) a star

·        Apparent brightness b for the intensity in Wm-2 of the starlight which hits an observer on earth, at a distance now called d so:

 

b = L / 4pd2                        [DB p. 12]

 

Apparent brightness can be measured using electronic components similar to a solar cell or photographic film for which some relation between the amount of reaction in the chemicals on the film and the amount of light energy that it has been exposed to in a given time is known.

 

The logarithmic scale for apparent magnitudes (m)

 

The intensity values of starlight are extremely small and historically the intensity or brightness of stars was first described, based on mere visual observations, by dividing stars into a magnitude of class 1 (the brightest), magnitude 2 (not so bright) etc to magnitude 6 (just barely visible to the naked eye). To connect the intensity value in a more mathematically precise way, a logarithmic scale has been developed to fit the historical scale as closely as possible. In modern measurements it turned out that a (historically) magnitude 1 star had an apparent brightness (intensity) about 100 times greater than a magnitude 6 star.

 

[Compare this to the frequencies of sound on a piano. Every time you go up one octave, you should double the frequency, so that if the tone A of one octave is 440 Hz, then that of the following A is 880 Hz. To get up one octave, you have to take 12 steps, so the factor to multiply the frequency of the previous tone to get the next one is the twelfth root of 2, 12Ö2 ]

 

So here if the stars A, B, C, D, E and F have the apparent magnitudes (no unit used in a logarithmic scale)

 

·        mA = 1,  mB = 2, mC = 3, mD = 4, mE = 5 and mF = 6

 

we should have the corresponding apparent brightness values (in Wm-2) bA,bB , bC, bD, bE and bF where we should have

 

·        bA/ bF = 100 and mA - mF = 6-1 = 5 steps on the magnitude scale

 

we should get the following brightness in Wm-2 by multiplying with the factor 5Ö100 » 2.5112 » 2.5. That is, bB  » 2.5bA, bC  » 2.5 bB  » 2.52 bA,  ....,  bF  » 2.5bE  » 2.55 bA  » 100 bA.

 

Now for the stars X and Y with the apparent magnitudes mX and mY and apparent brightness (intensities) bX and bY we have, using the exact value 5Ö100 = 5Ö102 = 102/5  instead of the approximate 2.5:

 

·        bX  = 10(2/5)(mX-mY)bY giving

·        bX/bY = 10(2/5)(mX-mY)  which if we take the logarithm (base 10, sometimes denoted lg) of both sides gives

·        log(bX/bY) =   log10(2/5)(mX-mY)  and using the rule log xa = a log x

·        log(bX/bY) = (mY - mX)log10(2/5)  which by definition is

·        log(bX/bY) = (mY - mX)(2/5)  and then

·        (mY - mX) = (5/2)log(bX/bY), or

 

mY - mX = 2.5log(bX /bY)

 

Note again that the 2.5 in this formula is not the 5Ö100 » 2.5 but the exact 1/[log(5Ö100)] = 2.5

 

The logarithmic scale for apparent and absolute magnitudes (M)

 

The apparent magnitude scale only gives a measure of the ratio between the brightness bX and bY of two stars. In order to get a standardized way to describe the absolute luminosity of a star, it has been defined that

 

the absolute magnitude M is the apparent magnitude m a star would have, if it was at the distance 10 pc from us

 

Let us call the apparent brightness (intensity) of the star at its actual distance d (measured in pc) from us bd and its brightness at the distance 10 pc from us d10. Since it is the same star, its absolute luminosity (power) is the same; Ld = L10 = L. We will then have

 

·        bd = L / 4pd2 and b10 = L / 4p×102  , dividing the first equation with the second

·        bd / b10 = 100/d2

 

Using the earlier equation

mY - mX = 2.5log(bX /bY)

 

and letting mX = m, mY = M, bX = bd and bY = b10 we will get

 

·        M - m = 2.5log(bd /b10) and then

·        M - m = 2.5log(100/d2) or

 

M = m + 2.5log(100/d2)

 

In short, the apparent magnitude m represents the apparent brightness b and the absolute magnitude M the absolute luminosity L.

 

7.5. The Stefan-Boltzmann law

 

The apparent and absolute magnitude scales were a sidetrack which is, by tradition, a part of astronomy but has little relevance for the astrophysical problems before us. The quantities absolute luminosity (which could just as well be called what it is: power in W) and apparent brightness (or better: intensity in Wm-2).

 

What we are interested in now is to get a picture of the structure of the universe, the distance to a star, and especially to those too far from us for the parallax method to work. The formula

 

b = L / 4pd2

 

where b can be measured here on earth would give us the d-value if only we could find out L.

 

Stefan-Boltzmann's law ("the hotter, the more power is radiated")

 

By studying various objects in laboratories on earth, their temperature T and power of radiation P (or here luminosity L) can be measured it is found that the Stefan-Boltzmann law holds:

 

L = sAT4                            [DB p. 12]

 

where Stefan-Boltzmann constant s = 5.67 x 10-8 Wm-2K-4    [in DB] and A = the surface area of the object. (Strictly, this formula is valid for a "black body", one that emits and absorbs radiation perfectly. For a shiny object like a thermos, one would have to include another factor, the emissivity, which would be 1 for a "black body" and between 0 and 1 for others. It turns out that hot gases have emissivities close to 1).

 

We could then get a value for L if we

·        assume that the same physics is valid for a star far away from us as for the objects in our lab

·        find out the surface temperature T of the star (without actually travelling there and sticking a thermometer into it)

·        find a value for its surface area A

 

7.6. Wien's displacement law ("the color changes with temperature")

 

Black-body radiation

 

The study of black-body radiators (which also caused Planck ca 1900 to first suggest that the energy of a photon of light or other electromagnetic radiation to be E = hf, later confirmed by Einstein's analysis of the photoelectric effect) gave among other results a number of curves of how much radiation was emitted at different wavelengths for objects at various surface temperatures.

 

Wien's displacement law

 

Such a graph for two objects at the temperatures T1, T2 and T3 where T1 < T2 < T3 could be

s06a

 

It can be noted that the peak of the curve will shift (be "displaced", though this does not have anything directly to do with the quantity displacement known from Mechanics) along a graph indicated by the dotted line. If one was to make a graph of this peak wavelength, lmax , as a function of surface temperature T, one would find that it follows a hyperbolic graph (similar to y = 1/x or generally y = k/x) giving "Wien's displacement law"

 

lmax = 2.90 x 10-3 / T                               [DB p. 12]

 

The constant in Wien's displacement law is usually called "the constant in Wien's displacement law" or sometimes for short "Wien's constant" and should be assigned units: 2.90 x 103 Km (Kelvin-meters). It is rarely given any symbol, but one can be assigned to it at will.

 

This law means that the hotter something gets, the shorter the wavelength (or the higher the frequency) of the electromagnetic radiation it emits most of. We will notice this as a change in color: if you heat up a piece of iron it will first look like it did before heating (but emit invisible infrared radiation, observable in a "heat camera"), then become red-glowing (red has the longest wavelength of visible light), then white-glowing (indicating that also other, shorter, wavelengths are emitted) and eventually blue-glowing (but iron would have melted and been vaporized before that).

 

Applied to starlight this means that if we can find out the peak wavelength lmax of a star's light then we can say what its surface temperature T is. (One would fit the telescope with different color filters to find out what type of light is dominant).

 

The remaining problem: size

 

In order to find L = sAT4 (and with the measured b-value, we then get the distance d from b = L / 4pd2 ) we still need the surface area A. We assume that the star is shaped like a sphere so if we find its volume V = (4/3)pr3 we can get the radius of the star r and then its surface A = 4pr2 (Notice the conceptual difference between the surface area of a spherical radiation source and the imagined sphere at a distance d from the source - or strictly the center of the source - over which its inner imagined surface its radiation is spread) or vice versa. This method of relating distance d, apparent brightness b, absolute luminosity L, surface temperature T and peak wavelength lmax is primarily not used to find the distance of stars very far away, but to find out more (e.g. the size of) about those near enough for the parallax method for finding the distance to work.  A summary of other distance measuring methods will come later, first we will turn to what more one can learn about a star by observing the light from it.

 

7.7. Stellar spectra and chemical composition

 

Information from the spectra and spectral classes

 

Light is produced in nuclear fission reactions deep in the core of a star (see later) and is absorbed and re-emitted many times on its way out to the surface, and therefore has a rather continuous distribution of wavelengths. Chemical elements, ions and molecules near the surface will cause absorption lines in the spectrum (missing wavelengths) which provide information about the elements that exist in a star even if no-one goes there to collect a sample. Except the hydrogen and helium (input and output of the fission reaction) traces of several other elements are found, and these are typical for stars of different surface temperatures. (It may be noted that the spectral line of the element helium was found in sunlight before helium had been found on earth. The element was given its name for an ancient Greek sun "god", Helios, and was later also detected in small amounts in the atmosphere).

 

The types of stars have been divided into spectral classes (the Harvard system) which for some unknown reason have been assigned the letters O, B, A, F, G, K and M (which can be remembered with the phrase Oh, Be A Fine Girl, Kiss Me. Despite this it must be pointed out that some astrophysicists are not perverts and do have a life).

 

Spectral class Surface temperature             Colour           Typical spectral lines

                                           

O                                         20000-35000 K                  blue                He- and other ions

B                                         ca 15000 K                         blue-white      Neutral He

A                                         ca 9000 K                           white              Neutral H, metals

F                                         ca 7000 K                           white-yell.      metal ions

G                                         ca 5500 K                           yellow            K, CN-, Ca-ions

K                                         ca 4000 K                           orange            metals, TiO

M                                        ca 3000 K                           red                 TiO

 

The temperature intervals and typical spectral lines vary in the literature. The classes are further divided with numbers (G0, G1, ..., G9, K1, K2, ....) and there are some other classes for types of stars with other properties. Our sun is a yellow G-class star with a surface temperature of ca 5800 K.

 

The Hertzsprung-Russell diagram

 

The spectral classes (based on observations of maximum wavelength => surface temperature from Wien's displacement law) and absolute luminosities L (which for near stars are found getting the distance d from the parallax and measuring b, then using b = L / 4pd2) can be systematized into the Hertzsprung-Russell or H-R-diagram:

s07a

 

Note that we have:

·        horizontal axis: the spectral classes O,B,A,F,G,K and M so the temperature decreases from left to right

·        vertical axis: the luminosity (power) on a logarithmic scale using either the value in watts or as in how many times the power of the sun a stars luminosity is. Lsun = 3.9 x 1026 W.

 

In the graph we notice these features:

·        the main sequence, most stars are placed along a band roughly from the upper left to the lower right corner of the H-R-diagram

·        red giants, stars with a low temperature (would be class K or M) but a much higher luminosity than main sequence stars which means the size must be bigger (recall L = sAT4               , same T but bigger L requires bigger surface area A). They are in the upper right corner of the H-R.

·        white dwarfs, hot stars with a lower L -> smaller size than the main sequence; in the lower left corner of the H-R.         

 

The red giants and white dwarfs differ from the main sequence stars also in their chemical composition and are temporary phases near the end of a star's "life" (see later).

 

7.8. Spectroscopic parallax

 

The basic features of the H-R can be found using the population of stars near enough for the parallax method for distance measurement. Making the assumption that stars far away have the same properties as those near us, we can measure the lmax which with Wien’s law gives the temperature T and observe the chemical absorption lines of a more distant star, and place it in the appropriate spectral class, or on the horizontal axis of the H-R diagram. If its chemical composition fits the main sequence stars of this class we may read an approximate L-value from the vertical axis of the H-R and together with the measured apparent brightness b we then get the distance from

 

b = L / 4pd2 => d = Ö(4pb/L)

 

This method is called the spectroscopic parallax method, which is not very appropriate since it does not have anything to do with the parallax method other than that one uses it to find out the same quantity, namely the distance from us to a star. It works up to distances of about 10 Mpc = 10 million pc or ca 30 million light-years. Recalling that our galaxy is ca 100 000 ly in diameters and the nearest other galaxies a few million ly away, this expands the range of distance measurements a lot from the ca 100 pc or 300 ly available to the parallax method.

 

7.9. Luminosity and Cepheid variables ("standard candles")

 

Another method of finding the luminosity L needed for a distance measurement are various types of variable stars, whose luminosity and intensity fluctuate periodically. The luminosity of all stars do that to some extent (for our sun, there are slight variations connected to the 11-year solar spot cycle), but for some types of stars among which the Cepheid’s are most known this variation is significant and has a regular period. By studying Cepheid’s near enough for a distance measurement with the parallax method and/or the "spectroscopic parallax" method (giving d, and then with a b-measurement L from b = L / 4pd2 => L = 4pbd2 ) the relation between L and the time period T of the fluctuation can be studied. Assuming that Cepheid’s further away follow the same relation as those near us, one can then measure the T, read the L off a graph like the one below and find d from b = L / 4pd2 (b, as always, can easily be measured). With powerful telescopes, individual Cepheid’s have been observed in other galaxies and used to determine the distance to these.

s09a

 

7.10. Summary of distance measurement methods

 

·        parallax method (up to ca 100 pc)

·        spectroscopic parallax (up to 10 Mpc) 

·        Cepheid (standard candle) method (up to ca 60 pc)

·        other types of standard candles (up to ca 900 Mpc)

 

7.11. Energy production in a star

 

Fusion processes

 

The main process for energy production in a star is nuclear fusion of hydrogen to helium, but there are several other nuclear reactions also taking place, which are depending on each other. In some of these cycles or chains, beta decay takes place and the neutrinos emitted can give us some information about what happens there, since neutrinos rarely interact with matter and most of them pass undisturbed from the center or core of the sun out to the surface and away into space - or to a neutrino detector on Earth. Temperatures in the core are much higher than on the surface of a star.

 

Balancing radiation and gravity

 

The radiation from the reactions in the core have to pass through the star on its way out, being absorbed and re-emitted many times. In this it exerts an outwards "radiation pressure" which for a stable star in the main sequence is in equilibrium with the force of gravity trying to make the star collapse. (Recall from Relativity that photons have a momentum even if they do not have a mass).

 

7.12. The "life" of a star

 

"Birth"

 

Wherever there is a large cloud (nebula) of hydrogen in the universe, gravity will make it contract and get denser and hotter. If it is large enough, it will first form a protostar which is glowing (sometimes brighter than the later "real" star it will become) because of the high temperature. If the temperature and pressure in the center of the protostar become high enough, fusion reactions ignite and the star enters the main sequence in the H-R-diagram. Where on the main sequence it appears (what spectral class it will have) depends primarily on its mass - the higher, the hotter.

 

"Life" in the main sequence

 

The more massive the star is, the faster it will change from a nebula to a star (ca 10 000 years for very heavy stars, 10 million years for smaller) and the faster it will burn up its hydrogen fuel and reach the end of its "life" (a few hundred million years for big stars, several billion for smaller. Our sun has been around for ca 5 billion years and is expected to last for several more).

 

The "death" of a star

 

When the fusion reactions in the core of a star run out of hydrogen the outward radiation pressure decreases and the equilibrium that was in place during its "life" in the main sequence is disturbed - the star collapses under the force of gravity. This will however bring fresher hydrogen fuel in towards the center and the fusion reactions will temporarily increase again. The radiation pressure pushes out the outer layers of the star so that its size increases dramatically (the sun is expected to "swallow" the inner planets when this happens) but the surface temperature drops and the color following Wien's displacement law changes. The star now becomes a red giant or super giant depending on its mass.

 

Nucleosynthesis

 

Already in the main sequence the fusion reactions involved more than just hydrogen and helium, but during the final stages of the star's "life" more nuclear reactions (e.g. He undergoing fusion to Si and onwards to Fe) take place when the pressure and temperature is higher, forming heavier elements. These are in subsequent phases spread out in the universe and believed to eventually end up (possibly via the life of another star on planets), including into us. The iron atoms in your blood cells were most likely produced inside a star far away from here billions of years ago.

 

The Chandrasekhar limit

 

What happens now depends on the mass of the star, expressed in how many times the mass of our sun it is, msun :

 

·        if m < 1.4msun , then the star becomes a red giant and then a white dwarf, a small but hot and short-lived star which when it runs out of all possible fusion fuel becomes a "brown" or "black" dwarf, a lump of material sitting in space and not doing anything special. See H-R diagram below.

·        if 1.4msun (the Chandrasekhar limit) < m < 8msun , then the star will first become a red super giant, then as this collapses and material falls quickly towards the center have a very violent explosion called a nova or supernova. Such an explosion lasts for only a few years or decades. The leftovers then contract so much that the quantum mechanical rules for how many electrons can be packed close together are overcome, e- and p+ form neutrons, and the star becomes a neutron star. The stars called pulsars may be a type of neutron stars.

·        if m > 8msun the star will become first a red super giant and then a supernova as above, but eventually collapse to a black hole, see later section.

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7.13. Black holes

 

One possible final fate of a star is to become a black hole from which nothing, including light, can escape (other than by a quantum-mechanical type of "evaporation", not needed to know here)

 

Recall the derivation of escape velocity in mechanics:

·        Ek + Ep = 0 giving ½mv2 + (- GMm/r) = 0 so

·        ½mv2 = GMm/r or  ½v2 = GM/r which gives v = (2GM/r)½.

 

Now let v = c so c = (2GM/r)½ so c2 = 2GM/r so r = 2GM/c2 , the Schwarzschild radius, which indicates the size to which an amount of matter must be compressed to become a black hole (The classical derivation here turns out to give the same result as a relativistic one). This is also part of the Relativity option. 

 

RSch = 2GM / c2                 [DB p.12]

 

COSMOLOGY (7.14 -7.18)

 

7.14. Olber's paradox: Why is it dark at night?

 

Newton assumed that the universe is infinite, uniform and static. This view was difficult to reconcile with a seemingly simple question:

 

Why it is dark at night? Because the sun is down, one would answer. But there are stars shining, should not they make the night anything but dark? They are too far away, one would reply; - yes, but there is an enormous number of them. To find out which of these counteracting facts - the one that there is a large number of stars and the one that the intensity of the light we receive from them decreases the further away they are, we need some math again.

 

·        The intensity I (or as we call it here, apparent brightness b, in Wm-2) of the light we get from a star with the power P (which we call the absolute luminosity in W) decreases with the square of its distance d from us:  b = L / 4pd2

 

Assume now that all stars in the universe have the same L = Lstar and that they are uniformly distributed in the un