7. 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 succesful predictions of the existence and approximate orbits of new ones. Irregularities in the orbit of Mercury (its 'perihelium 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

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 elliptic orbits with the sun in one focus

Kepler II: A line from the planet to the sun sweeps over the same are in the same time (meaning that they mover faster when closer to the sun)

Kepler III: The squares 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.

s01a

Asteroids and comets

In the solar system we also have

· asteroids, smaller planets and rocks mainly in the "asteriod 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. 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 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 from the sun; Pluto about 6 light hours.

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

s01b

There are various types of galaxies such as spiral, elliptical and irregular ones. Many galaxies near each other from 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 lightyears common in our parts of the galaxy. These are stellar clusters (Sw. stjärnhop, Fi. tähtisikermä).

A constellation (Sw. stjärnbild, Fi. tähtikuvio) 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 1960s 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 make them appear as regularly flashing beacons, pulsating stars or pulsars.

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

Binary stars

Many stars are not, like our sun, the only 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 could be difficult to have as 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 cathegorized 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)

s01c

· 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 Cepheids

Most stars, including our sun, have periodically varying brightness or intensity. For some stars (e.g. Cepheids) 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 an 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 around. 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 earths imagined axis is not at a perfect 90^o angle to the plane of revolution, but rather at one of ca 66.5^o (in other words - a plane through the equator makes a 23.5^o 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 360^o for north, 90 for east, 180 for south, 270 for west) and the altitude, Alt (angle up from the horizon, that is 0^o 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 SkyMap 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 arcsecond = 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 here 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 how small angles they can measure, 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 the bulb to cover all directions. This could be done with a spherical surface with the bulb in 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

s04a

The intensity which hit the surface of this imagined spherical surface can be measure 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 films 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 for 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 measurments 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 frequence, 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 prevoius tone with 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)

·        m_A = 1, m_B = 2, m_C = 3, m_D = 4, m_E = 5 and m_F = 6

we should have the corresponding apparent brightness values (in Wm^-2) b_A,b_B , b_C, b_D, b_E and b_F where we should have

·        b_A/ b_F = 100 and m_A - m_F = 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, b_B » 2.5b_A, b_C » 2.5 b_B » 2.5² b_A, ...., b_F » 2.5b_E » 2.5⁵ b_A » 100 b_A.

Now for the stars X and Y with the apparent magnitudes m_X and m_Y and apparent brightnesses (intensities) b_X and b_Y we have, using the exact value ⁵Ö100 = ⁵Ö10² = 10^2/5 instead of the approximative 2.5:

·        b_X = 10^(2/5)(mX-mY)b_Y giving

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

·        log(b_X/b_Y) = log10^(2/5)(mX-mY) and using the rule log x^a = a log x

·        log(b_X/b_Y) = (m_Y - m_X)log10^(2/5) which by definition is

·        log(b_X/b_Y) = (m_Y - m_X)(2/5) and then

·        (m_Y - m_X) = (5/2)log(b_X/b_Y), or

m_Y - m_X = 2.5log(b_X /b_Y)

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 ration between the brightnesses b_X and b_Y 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 b_d and its brightness at the distance 10 pc from us d₁₀. Since it is the same star, its absolute luminosity (power) is the same; L_d = L₁₀ = L. We will then have

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

· b_d / b₁₀ = 100/d²

Using the earlier equation

m_Y - m_X = 2.5log(b_X /b_Y)

and letting m_X = m, m_Y = M, b_X = b_d and b_Y = b₁₀ we will get

· M - m = 2.5log(b_d /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. Luminosity (power) and temperature

The apparent and absolute magnitude scales were a sidetrack which is, by tradition, a part of astronomy but which 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 to get a picture of the structure of the universe is 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 can 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 displacment law ("the colour 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 raditation 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, l_max , 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"

l_max = 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 (kelvinmeters). 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 colour: 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 vaporised before that).

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

The remaining problem: size

In order to find L = sAT⁴ (and with the also measured b-value 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 l_max is primarily therefore 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 noone 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:

· horisontal 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. L_sun = 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 temporart 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 thos near us, we can measure the l_max which with Wiens 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 horisontal 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 lightyears. 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 varible 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 Cepheids are most known this variation is significant and has a regular period. By studying Cepheids 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 Cepheids 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 Cepheids 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 from 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 will it change from a nebula to a star (ca 10 000 years for very heavy stars, 10 million years for smaller) and the faster will it 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 more fresh 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" te inner planets when this happens) but the surface temperature drops and the colour following Wien's displacement law changes. The star now becomes a red giant or supergiant 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 its, m_sun :

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

·        if 1.4m_sun (the Chandrasekhar limit) < m < 8m_sun , then the star will first become a red supergiant, 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 supergiant and then a supernova as above, but eventually collapse to a black hole, see later section.

s12a

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:

·        E_k + E_p = 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 Schwardschild 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. 

R_Sch = 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 universe. 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 enourmous 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 maths again.

· The instensity 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 = L_star and that they are uniformly distributed in the universe, with a constant number N_average of stars per volume unit (this is not true in detail, stars are sticking togeter as galaxies, but on a large enough scale this would on average be true). Then we study a spherical shell which is thin (compared to the size of the universe):

s14a

·        the volume of the shell is the difference between that of an "outer" sphere with the radius r + Dr and an "inner" one with the radius r: V_shell = V_outer - V_inner.

·        for the inner sphere, V_inner = (4/3)pr³

·        for the outer sphere V_outer = (4/3)p(r + Dr)³

·        we now have (r + Dr)³ = r³ + 3r²Dr + 3rDr² + Dr³

·        since Dr is very "small" Dr² and Dr³ are extremely "small" and the terms including these can approximately be ignored:

·        V_outer - V_inner = (4/3)pr³ - (4/3)p(r³ + 3r²Dr) which gives

· V_shell = (4/3)p( r³ - r³ + 3r²Dr) = (4/3)p(3r²Dr) = 4r²Dr

Let us now assume that all stars in one shell are approximately at the distance d = r from us

·        the number of stars in the shell is N_shell = N_averageV_shell where N_average was the average number of star per volume unit

·        the total apparent brightness (intensity) of all the stars in a shell is b_shell = N_shellb_star, where bstar is the brightness of an individual star (recall that we assumed all stars were identical, with the luminosity L_star).

·        this gives b_shell = N_averageV_shellb_star so

· b_shell = N_average4r²Dr(L / 4pr²) = NLDr

Since r is cancelled and Dr is constant (we look at equally thin shells further and further away) we have

found that the total brightness (intensity) of every shell is constant, regardless of how far away from us it is. If the universe is infinite, then the total intensity of the starlight should be infinite, and we should have been barbecued by the starlight long ago!

There are several possible ways to resolve Olber's paradox. The universe may not be infinitely large or old, only very large and old (but size of the already observed universe is so large that the total starlight intensity should be larger than it is), or it may not be static - if the stars are moving further away from each other while the starlight is on its way, the intensity may decrease enough. One possible solution to Olber's paradox is the Big Bang model, which is also supported by other evidence outlined in the following sections.

7.15. Galactic redshift and Hubble's law

Doppler redshift

In the Waves sections we learned that the frequency of a sound we observed can be distorted from that sent out by the source if the source moves in relation us or we in relation to the source (or both). A similar phenomenon occurs for light (and other electromagnetic waves), although in a somewhat different way. As is presented in the Relativity section, the velocity of light relative to any observer is constant, and therefore we do not get two different formulas for cases where the source and the observer are moving, but one formula which can be shown to be (the proof not needed in the IB)

l' = lÖ[(1+v/c)/(1-v/c)]

where c = the speed of light in vacuum, v = the relative velocity, the velocity at which something moves away from us or we from it (the recession velocity), l = the original wavelength emitted by the light source, l' = the distorted wavelength observed by us. [If the relative motion was such that the source was moving towards us or we towards it, just use a negative value for v]

If v << c it can be shown (some manipulation and binomial expansion) that we approximately have

Dl / l » v / c [DB p.12]

where the change in wavelength Dl = l' - l

If the source and observer are moving further away from each other, then the observed wavelength is longer than the emitted, (and the frequency lower - this is the same as in the Doppler effect for sound waves) which for visible light would mean a shift towards the red end of the spectrum or redshift. The opposite phenomenon would be called a blueshift.

Redshift of spectral lines

When observing the spectra of star in other galaxies, we find the expected set of spectral lines, but often redshifted indicating that the star and therefore the galaxy is moving away from us, or receding. For stars in other galaxies this redshift is rather constant over time, unlike the redshift or blueshift of the earlier mentioned spectroscopic double stars (see that section above).

s15a

The recession speed v can be found using

Dl / l » v / c => v = cDl / l

where the l is known if the pattern of spectral lines (whose internal differences remain the same!) is recognised as one familiar from some atom, ion or molecule studied in the lab!)

Hubble's law

We are now able to measure and conclude the following for the galaxies of the universe:

·        their distance d from us, for galaxies primarily using Cepheid standard candles

·        their recession speed v from the spectral redshift

It turns out that the relation between these is linear and follows Hubble's law:

v = Hd

where the Hubble constant H which due to large uncertainties in the distance measurements for galaxies very far away is somewhere between 40 and 100 kms^-1(Mpc)^-1.

Hubble's law means that other galaxies are moving away from us (and/or we from them) and this faster the further away they are. This supports the Big Bang model.

The Hubble constant and the age of the universe

Assuming a that the Hubble constant has been the same over time, one can find a value for the age of the universe from Hubble's law.

·        Let us take an arbitrary galaxy and find the time t it took for it to recede away from us to its present distance d

·        if v was constant, then v = d/t or t = d/v

·        but now v = Hd so we get t = d / Hd or

· t_universe = 1 / H

This time is the same for all galaxies and indicates how long ago they all - the whole visible universe - was packed closely in one place.

Ex. Let H = 80 kms^-1(Mpc)^-1. Then 1/H = 1 Mpc/ 80kms^-1 = (1000 000 x 3.26 ly) / 80 000 ms^-1 =

100 x 3.26 x 9.46 x 10¹⁵ / 8 ms^-1 = 3.85 x 10¹⁷ s; using that one year is 60x60x24x365 s = 31536000s which is about 3 x 10⁷ years we get about (3.85 x 10¹⁷/3 x 10⁷) years or ca 13 000 million years, 13 billion years. Different H-values give ages of the universe usually between 10 and 20 billion years.

7.16. The Big Bang model

The Big Bang

The BB model states that at a certian time, about 10-20 billion years ago, all matter and energy but also all space-time in the universe was concentrated in one point from which it since has expanded quickly.

Not a conventional explosion

Let us assume that a conventional explosion has occurred somewhere in space and we, the observers (O) are sitting on a piece of hypothetical debris moving away from the place (X) in space where the explosion took place. When something explodes on earth the pieces flying away will be slowed down by air resistance and bent downwards by gravity, but in space this is not a problem so everything keeps moving away from the place of the explosion with a constant velocity. If we in one way or the other are able to find the distance and relative velocity of other pieces of debris from the same explosion we will find that some pieces (A) are far from us and also moving away from very quickly. Others (B) are nearer and move away more slowly.

s16a

So far it seems to fit Hubble's law rather well - the further awaw, the faster it is moving away. But in a conventional explosion one would notice that the pieces which are furthers away and move the fastest relative to us are in a certain direction, indicating that the place of the explosion is somehwhere on a line from O to A. But astronomical observations show no such direction - in all directions there are galaxies as far away as we can see, and receding faster in proportion to their distance. This indicates that the Big Bang was something more than an explosion happening in some place.

The rubber band and balloon comparisons

Instead, what the BB model means is not only all material and energy came from the BB, but also all of space-time. It is not the galaxies that are moving away in an "existing" space, but space itself is expanding whether or not there is a galaxy in it.

· One-dimensional expansion: take a straight rubber band and make dots 1 cm apart with a pen. If the band is stretched, every dot will get further away from the nearest, and even further away from those not so near. If the band is circular, then we cannot say that any dot on the band is the center of the expansion - the center of the expansion is the center of the circle, which is not on the band - nowhere in the one-dimensional universe of the band.

· Two-dimensional expansion: assume we have a balloon with a dotted pattern on it. If we inflate the balloon, then the distance from one dot to another increases, and it increases more the further away from each the dots are (this distance is counted as the way a flea would have to crawl from one dot to the other along the surface of the balloon). Again - the center of expansion is no particular dot on the balloon surface, but the point inside the balloon which is its center in three dimensions.

· Three-dimensional expansion of the universe: in a similar way, it is not possible to find any point in the three-dimensional space where the Big Bang would have happened. It did not happen anywhere in the three dimensions, it happened in the fourth dimension, time. We can say when the BB happenede (ca 10-20 billion years ago) but not where - such a question is not only impossible to answer, but meaningless (like: "is a Friday green or blue?" or "What does the area of a triangle taste like?").

Details of the BB model

The Big Bang model is related to the four fundamental forces or types of interaction (see the Atomic, Nuclear and Quantum Physics), gravity, strong force, weak force and electromagnetic force. It is assumed that these were one unified force in the beginning and later separated. In large particle accelerators conditions of extreme heat and pressure like those just fractions of a second after the BB can be achieved.

·        before 10^-43 s : all four forces unified

·        from 10^-43 s to 10^-35 s : gravity separated, the other three forces still unified

·        from 10^-35s to 10^-12 s : the strong force separated from the electroweak which still is unified

·        from 10^-12 s to now : all four forces separated

In the period from 10-35 s to 10-24 s (when the electroweak force was still united) we had the Inflationary epoch, a rapid expansion of the universe (for not yet quite clear reasons) In this period also ordinary particles outnumbered antiparticles. Other key timepoints:

·        10^-6 s : protons and neutrons formed, no more free quarks

·        300 000 years: the universe becomes transparent to photons emitted by the then still extremely hot materia. In the further expansion these turn into the microwave background radiation (see below). About at this time hot and dense gas clouds which become todays galactic clustes were formed.

7.17. Cosmic microwave background radiation

The photons (electromagnetic radiation) mentioned above increased their peak wavelength as the universe cooled and expanded. Recall wavelength = the distance between two wavecrests; if space itself between the wavecrests expands, then the wavelength increases. It should by now (as was calculated already in the 1940s) have increased so much that the radiation is in the microwave section of the EM-spectrum. It should also come from all directions, since in the early universe at 300 000 years after the BB it was emitted by material in the entire universe in all directions.

This cosmic microwave background (CMB) radiation was discovered by Penzias and Wilson in the 1960s and provides additional support for the BB model. If the peak wavelength of the CMB is inserted in Wien's law,

l_max = 2.90 x 10^-3 / T

it will give a value of T = ca 3 K for the temperature of a blackbody which would emit such radiation.

Summary of support for the BB

· Olbers paradox

· Galactic redshift : same in all directions

· CMB at predicted wavelength from all directions

7.18. The future of the universe

There are three possible alternatives for the future of the universe:

· A : it will continue to expand forever (open universe)

· B : it will continue to expand but the rate of expansion will eventually slow down to zero (flat universe) and reach a "steady state".

· C : it will under the force of gravity eventually start to contract back (closed universe) into a point (Big Crunch); possibly to undergo a new Big Bang.

Which of these will happen depends on the average density of the universe: if this is below a critical density, we get alternative A, if it is equal to the critical density we get B, if it is less we get C. The result depends on the existence of dark matter (invisible matter which should exist to explain the movements of visible matter in the galaxies). This whereabouts of this dark matter which may make up as much as 90% of the mass in the universe is currently unknown. Various suggestions of particles not yet detected have been made, such as WIMPs (Weakly Interacting Massive Particles), MACHOs (massive compact halo objects, that is dim stars or black holes difficult to observe) and others. If such dark matter exists, the universe may be flat or closed, otherwise open.