Download Stefan-Boltzmann`s law Wien`s law

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Option E: Astrophysics
L = AT
4
-
𝜆max(meters) =
d(parsec) =
-
Stars’ an planets’ radiation spectrum is approximately the same as
black-body radiation/ Plank’s law.
Intensity as a function of wavelength depends upon its temperature
b =
m – M = 5 log (
Constellation is a group of stars that form a pattern as seen from the Earth, but not bound by gravitation.
Stellar cluster is a group of stars held together by gravitation in same region of space, created roughly at the
same time.
Galaxy is a huge group of stars, dust, and gas held together by gravity, often containing billions of stars,
measuring many light years across.
Star is a massive body of gas held together by gravity, with fusion going on at its center, giving off
electromagnetic radiation. There is an equilibrium between radiation pressure and gravitational pressure.
)
Wien’s law: Wavelength at which the intensity
2.9×10-3
of the radiation is a maximum λmax, is:
max (m) 
T(K)
Luminosity (of a star) is the total power (total energy per second) radiated by an object (star). If we regard stars as black body, then luminosity is:
4
2
L = A σT = 4πR σT
4
(Watts)
Stefan-Boltzmann’s law
A is surface area of the star, R is the radius of the star, T surface temperature (K), σ is Stefan-Boltzmann constant.
(Apparent) brightness (b) is the power from the star receive per square meter of the Earth’s surface
b=
2
(W/m )
L is luminosity of the star; d its distance from the Earth
Magnitude Scale • Magnitudes are a way of assigning a number to a star so we know how bright it is
Apparent magnitude (m) of a celestial body is a measure of its brightness as seen from Earth. The brighter the object appears the lower its
apparent magnitude. Greeks or ere the stars in the sky from brightest to faintest… Later, astronomers accepted and quantified this system.
3
• Every one step in magnitude corresponds to a factor of 2.51 change in brightness. Ex: m1 = 6 and m2 = 9, then b1 = (2.51) b2
Absolute magnitude (M) of a star is the apparent magnitude that a star would have if it were at distance of 10 pc from Earth.
It is the true measurement of a star’s brightness seen from a set istance.
m – M = 5 log (
)
m – apparent magnitude
M – absolute magnitude of the star
d – its distance from the Earth measured in parsecs.
• If two stars have the same absolute magnitu e but ifferent apparent magnitu e they woul have the same brightness if they were both
at distance of 10 pc from Earth, so we conclude they have the same luminosity, but are at different distances from Earth !!!!!!!!!!!!!!
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• Every one step in absolute magnitude corresponds to a factor of 2.51 change in luminosity. Ex: M1 = – 2 and M2 = 5, then L1 / L2 = (2.51)
Binary star is a stellar system consisting of two stars orbiting around their common center of mass.
The ONLY way to find mass of the stars is when they are the part of binary stars. Knowing the period of
the binary and the separation of the stars the total mass of the binary system can be calculated (not here).
Visual binary: a system of stars that can be
seen as two separate stars with a telescope
and sometimes with the unaided eye.
They are sufficiently close to Earth and the
stars are well enough separated.
Sirius A, brightest star in the night sky and
its companion first white dwarf star to be
discovered Sirius B.
Spectroscopic binary: A binary-star system
which from Earth appears as a single star,
but whose light spectrum (spectral lines)
shows periodic splitting and shifting of
spectral lines due to Doppler effect as two
stars orbit one another.
Eclipsing binary: (Rare) binary-star system
in which the two stars are too close to be
seen separately but is aligned in such a
way that from Earth we periodically
observe changes in brightness as each star
successively passes in front of the other,
that is, eclipses the other.
The Hertzsprung–Russell (H – R) diagram(family portrait) is a scatter graph of stars showing
the relationship between the stars' absolute magnitudes / luminosities versus their
spectral types(color) /classifications or surface temperature. It shows stars of different ages
and in different stages, all at the same time.
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L = sun luminosity = 3.839 × 10 W
main sequence stars: fusing hydrogen into helium, the difference between them is in mass:
left upper corner more massive than right lower corner.
white dwarf compared to a main sequence star: • has smaller radius • more dense
• higher surface temperature • energy not produced by nuclear fusion
Techniques for determining stellar distances: stellar parallax, spectroscopic parallax and Cepheid variables.
Stellar parallax • two apparent positions of a close star with respect to position of distant stars as seen by an
observer from two widely separated points are compared and recorded; • the maximum angular variation
from the mean, p, is recorded; • the distance (in parsecs) can be calculated using geometry
astronomical unit
Sun-earth istance
11
tan p =
=
for small angles: tan θ ≈ sin θ ≈ θ (in ra ians)
1 AU = 1.5 x 10 m
Sun-star istance
d=
16
if p = 1 sec of arc, d = 3.08x10 m defined as 1 pc
limit because of small parallaxes:
d (parsecs) =
p(arcsecon s)
≤ 00 pc
Spectroscopic parallax: no parallax at all!!!! (a lot of uncertainty in calculations)
• light from star analyze (relative amplitu es of the absorption spectrum lines) to give indication of stellar class/temperature
• HR iagram use to estimate the luminosity • istance away calculate from apparent brightness
limit: ≤ 0 Mpc
Cepheid variables are stars with regular variation in absolute magnitude (luminosity) (rapid brightening, gradual dimming) which is caused by
periodic expansion and contraction of outer surface (brighter as it expands). This is to do with the balance between the nuclear and gravitational
forces within the star. In most stars these forces are balanced over long periods but in Cepheid variables they seem to take turns, a bit like a mass
bouncing up and down on a spring.
Left: graph shows how the apparent
magnitude
(the brightness) changes,
.
getting brighter and dimmer again
with a fixed, measurable period for a
particular Cepheid variable.
There is a clear relationship between the
period of a Cepheid variable and its
absolute magnitude. The greater the period
then the greater the maximum luminosity
of the star. Cepheids typically vary in
brightness over a period of about 7 days.
Left is general luminosity – period graph.
So, to find out how far away Cepheid is:
• Measure brightness to get period • Use graph absolute magnitude M vs. period to find absolute magnitude M • Measure maximum
2
brightness • Calculate d from b = L/4π
• Distances to galaxies are then known if the Cepheid can be ascertained to be within a specific galaxy.
Newton assumptions about the nature of the universe:
• universe is infinite in extent • contains an infinite number of stars uniformly istribute • is static an exists forever
• these assumptions le to Olber’s para ox
Olber’s paradox
density of stars n = N/V = number of stars per unit volume • divide the whole universe into concentric shells around Earth
of constant thickness t • look at one shell of thickness t at distance d from the Earth • since stars are uniformly
2
2
distributed the number of stars seen from Earth increases as d : number of stars in shell = ensity x volume = n 4π t
2
2
• brightness of one star ecreases as / ; b = L/4 π
• brightness of shell is constant; assuming that luminosity L is
the same for all stars, the received energy per sec per unit area (brightness) from all stars in the thin shell is:
= Lnt = const.
• amount of light we receive from shell oes not epen upon how far away the shell is • a ing all shells to infinity; each contributing a
constant amount of energy • the total energy is infinite • sky would be uniformly bright • but it’s ark in night
Solutions to Olber’s paradox
• Perhaps the Universe is not infinite. But current model of the Universe is that it is infinite.
• Perhaps the light is absorbed before it gets to us. But then Universe would warm up and eventually reradiate energy.
Real help: the Big Bang model leads to the idea that the observable universe is not infinite and to the idea of the expansion of the universe;
• Universe is not static, it is expanding, hence the most distant stars/ galaxies are strongly red - shifted, out of the visible part of the spectrum.
• There is a finite time since the Big Bang. Some 12 to 15 billion years. That means we can only see the part of it that lies within 12 to 15 billion
light-years from us. And the observable part of the universe contains too few stars to fill up the sky with light.
Calculation shows that the helium produced by nuclear fusion within stars cannot account for the real amount of helium in Universe (24%). In 1960
it was proposed that sometime during the early history of the Universe, long before any star, Universe was at a sufficiently high temperature to
produce helium by fusion. In this process many high energy photons would be produced. The CMB (Cosmic Microwave Background Radiation)
radiation was emitted only a few hundred thousand years after the Big Bang, long before stars or galaxies ever existed. The photons would have a
black body spectrum corresponding to the then temperature of the Universe. As the Universe expanded and cooled the photon spectrum would also
cha g w th th r max mum wa
gth sh ft g accorda c w th W ’s aw It s st mat d that at th pr s t t m th photons should have a
maximum wavelength corresponding to a black body spectrum of an extremely cold object of temperature of 2.7 K.
Cosmological background radiation / Cosmic microwave background radiation (CMB) is microwave radiation - left over from the Big Bang that fills
the universe roughly uniformly in all directions. The Big Bang predicts an expanding universe that had a very high temperature at the beginning;
during the expansion the universe is cooling down and the temperature of the radiation should fall to its present low value of about 2.7 K.
That radiation corresponds to a black body spectrum of about 2.7 K.
The other way of explaining CMB is: Big Bang producing initially produced very short wavelength photons /EM radiation. As the universe
expands, the wavelengths become red shifted to reach current value.
█ Explain how knowledge of the spectrum of a black body and the existence of cosmological background radiation is consistent with the “Big Bang”
model of the universe.
► Big Bang predicts a low temperature radiation at 2.7 K (i.e. CMB radiation). The Big Bang theory also predicts an expanded universe which we
observe through red- shifting of the galaxies and the lowering of CMB radiation temperature. This expanding universe is the result of the initial
energy released in the Big Bang
█ State one piece of evidence that indicates that the Universe is expanding.
► light from distant galaxies/stars is red-shifted (which means they move away from us – as the red-shifting occurs in all direction, the universe
must be expanding)
► existence of CMB
► the helium abundance in the universe which is about 25 % and is consistent with a hot beginning of the universe;
The eventual fate of the Universe is determined by the amount of mass in the Universe.
Critical density is the density of the Universe which produces a flat universe, i.e. it would take an infinite amount of time to stop expansion of it.
Critical density is the density of the Universe that would be necessary to stop the expansion after an infinite amount of time.
• Closed Universe A model of the universe in which density of the Universe is such that gravity will stop the universe expanding and then cause
it to contract. Eventually the contraction will result in a ‘Big Crunch’ after which the whole creation process coul start again.
• Open Universe A model of the universe in which density is such that gravity is too weak to stop the Universe expanding forever.
• Flat Universe means that the density is at a critical value whereby the Universe will only start to contract after an infinite amount of time.
non-coincident starts (not at beginning);
correct shapes and correctly labelled;
coincident at appropriate place;
Dark matter is the matter that makes up for most of the mass in the universe, but cannot easily be detected because it does not emit radiation .
Examples of dark matter.
► two of Neutrinos / WIMPS / MACHOS / black holes / exotic super symmetric particles / grand unified predicted particles / magnetic monopoles
etc.; or maybe our current theory of gravity is again not correct