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The IB Physics Compendium 2005: Astrophysics
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.
ObjectDist. from sun/mill.kmm/1024kgDiam./103kmMoons
Mercury600.334.9-
Venus1084.912.0-
Earth1506.012.81
Mars 2280.646.82
Jupiter778190014315+
Saturn143057012117+
Uranus2900875115
Neptune450010508
Pluto59200.012.31
More information about the solar system is found at The Nine Planets website,
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.
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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 100 billion 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.
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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)
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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 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 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 SkyMap demo version (
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 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.
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From this we find that:
tan = r / d => d = r / tan but since tan for very small (in radians) we get
d = r /
If we here used conventional SI units we would insert r in meters, in radians and get d in meters. If instead we use AU for r (which gives r = 1 in this unit), arc-seconds for 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 = 4r2 and we can define intensity = power/area with the unit 1 Wm-2 so that
I = P/A = 4r2
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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 = 4r2for 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-2of the starlight which hits an observer on earth, at a distance now called d so:
b = L / 4d2[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, 122 ]
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 5100 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 brightnesses (intensities) bX and bY we have, using the exact value 5100 = 5102 = 102/5 instead of the approximative 2.5:
bX = 10(2/5)(mX-mY)bY giving
bX/bY = 10(2/5)(mX-mY) which if the 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 5100 2.5 but the exact 1/[log(5100)] = 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 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 / 4d2 and b10 = L / 4102 , 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 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 / 4d2
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 = AT4[DB p. 12]
where Stefan-Boltzmann constant = 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
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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, 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"
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 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 = AT4 (and with the also measured b-value then get the distance d from b = L / 4d2 ) 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)r3 we can get the radius of the star r and then its surface A = 4r2 (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 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.