Download Stars I - Astronomy Centre

Our Place in the
Cosmos
Lecture 10
Observed Properties of Stars
Distances to Stars
• Early astronomers considered the stars to be
located on the surface of a sphere, and hence
all at the same distance
• To understand most properties of stars we need
to know their distance
• For nearby stars distance can be measured via
parallax
• This works on the same principle as
stereoscopic vision - we are able to judge
distances to objects by the separation of our two
eyes
Parallax
• Stereoscopic vision only helps to judge
distances to a few hundred metres as our eyes
are only separated by about 6 cm
• We can tell a mountain is more than a few
hundred metres away, but not whether it is 2 or 5
km away
• Parallax is due to our changing viewpoint as
Earth orbits the Sun
• With 2 AU separating our two “eyes” we can
measure distances to nearby stars
Parallax is one-half of the angle through which a star
appears to move over the course of a year
A star with a parallax of 1
arcsecond is at a distance of 1
parsec
More distant stars have
smaller parallaxes p  1/d
d (parsecs) = 1/p (arcsecs)
Parallaxes and Distances
• Since parallaxes are very small, they are
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measured in arcseconds
One degree is divided into 60 arcminutes, one
arcminute is divided into 60 arcseconds
An object with a parallax of 1 arcsecond (the
diameter of a ping-pong ball at 5 miles) is
defined to be at a distance of 1 parsec (parallaxarcsecond), abbreviated pc
1 pc = 206,265 AU = 3 x 1016 m = 3.26 ly
Distance (pc) = 1/parallax (arcsec)
Parallaxes and Distances
• Closest star (apart from the Sun) is Proxima Centauri
• Its parallax is 0.75 arsec giving a distance of 1.3 pc
• First successful parallax measurement was made in
1838 by FW Bessel
• He measured a parallax to the star 61 Cygni of 0.314
arcsec giving a distance of 3.2 pc, or 600,000 times
further than the Sun
• This single measurement increased the known size of
the Universe by 10,000-fold!
• Today, only 54 stars in 37 systems (singles, binaries or
triples) are known within 15 light-years - stars are few
and far between
Limits of Parallax
• Accuracy of positional measurements limits
distance to which stars have a measured
parallax
• Hipparcos satellite launched in 1990s has
measured parallaxes for 120,000 stars accurate
to 0.002 arcseconds
• We can only measure parallaxes accurate to
10% for distances up to about 50 parsecs
• Beyond a few hundred parsecs other distance
estimators have to be used
Luminosity
• The apparent brightness of a star depends
strongly on its distance
• As with gravity, the intensity of light drops
inversely with the square of distance d from a
source as the light is spread out over the surface
A = 4d2 of a sphere of radius d
• The luminosity of a star (the total energy
radiated per second) is thus given by its
measured brightness multiplied by 4d2
• We thus need to know the distance d to a star to
calculate its luminosity
Inverse Square Law
Light spreads out
more to cover a
larger sphere and
so appears fainter
further from the
source
Brightness  1/d2
Luminosity Function
• We find that stars vary tremendously in luminosity, and
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that some apparently faint stars are in fact extremely
luminous
The most luminous stars exceed the Sun’s luminosity by
a million, we say they have a luminosity of 106 L
The least luminous stars have luminosities below 10-4 L
A plot of the number of stars as a function of their
luminosity is known as the luminosity function
This shows that the vast majority of stars are less
luminous than the Sun
Stellar Luminosity Function
Colour and Temperature
• A star radiates because it is hot
• The hotter an object the faster its constituent
particles jostle about
• Any charged particle (such as an electron) that
is accelerated will radiate energy known as
thermal radiation
• The energy of a photon of light is inversely
proportional to its wavelength : E  1/
• Hotter objects thus emit radiation that is both
more intense and of shorter wavelength, ie.
bluer
Blackbody Radiation
• An idealised object that emits exactly as much
radiation as it absorbs from its surroundings is
known as a blackbody
• In 1900 physicist Max Planck calculated how the
spectrum (intensity as a function of wavelength)
of such a blackbody should depend on its
temperature
• The resulting spectrum is known as a Planck
spectrum or blackbody spectrum
• As expected, hotter blackbodies emit more of
their radiation at shorter, bluer, wavelengths
Blackbody Spectrum
Intensity of Blackbody
Radiation
• The luminosity of a blackbody increases with the
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fourth power of temperature
L =  A T4
This is known as Stefan’s law after its empirical
discovery by Josef Stefan
L is the luminosity: energy radiated/second
 is known as the Stefan-Boltzmann constant
A is the surface area of the blackbody
T is the temperature in degrees kelvin
0K = absolute zero, 0C = 273K
Colour of Blackbody
Radiation
• The peak wavelength of the Planck or blackbody
spectrum is given by Wien’s law
peak = (2,900 m K)/T
• The wavelength at which a blackbody’s
spectrum peaks is inversely proportional to
temperature
• We can thus judge a star’s surface temperature
from its colour
• Spectrum of sunlight peaks around 0.5 m
giving a surface temperature of 5800 K
Colours and Surface
Temperatures of Stars
• Most stars radiate approximately as blackbodies
• We can thus immediately say that blue stars are hot, red
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stars are cool
By measuring the spectrum of a star, we can use Wien’s
law to find its surface temperature
Rather than measuring a spectrum, we can gauge a
star’s colour by measuring its brightness through two
different filters, say blue and yellow (or “visual”)
The brightness ratio between the two filters provides an
estimate of temperature
Cool stars are much more common than hot
Sizes of Stars
• Once we have determined the temperature of a
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star from its spectrum or colour, we can
determine its size via Stefan’s law L =  A T4
Luminosity L can be determined by measuring
apparent brightness l and distance d (via
parallax): L = 4d2 x l
Temperature T can be determined from
spectrum or from colour
 is a known constant and so surface area A
and hence radius r of star can be determined
Most stars are smaller than the Sun
Masses of Stars
• Luminosities and radii of stars are a poor
indicator of mass as they can vary considerably
in mass-to-light ratio and in density
• Gravity is the key to determining masses
• The Sun’s mass can be determined by studying
the orbits of its planets
• We cannot directly see planets orbiting other
stars, but we can observe binary stars - two
stars orbiting a common centre of mass
Binary Stars
• About half of all stars occur in binary systems
• Each star feels an equal force towards the other,
but the lower mass star will experience a greater
acceleration (Newton’s 2nd law)
• If the two stars were initially at rest, they would
meet at a point closer to the more massive star
called the centre of mass
• If star 1 is 3 times the mass of star 2, the centre
of mass will be 3 times further from star 2 than
star 1
Force on each star is the
same, but acceleration is
larger for the less massive
star and so it picks up
speed faster
As the stars fall toward
each other the centre of
mass remains stationary
If m1 = 3m2 star 2 will fall 3
times as far as star 1
They meet at the centre of
mass where the pivot of a
balance would be
Binary Stars
• In practice each star will have some velocity
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perpendicular to the line joining them
Instead of falling into each other, they will orbit
about each other with the centre of mass at one
focus of their elliptical orbits
The stars will always be on opposite sides of the
centre of mass which remains stationary
Orbit of more massive star will be smaller than
the orbit of the less massive star
Less massive star must move faster in order to
complete its longer orbit in the same time as the
more massive star
Less massive star
moves faster on a
larger orbit
Centre of mass
remains stationary
Equal time
steps
Binary Stars
• Semi-major axis and therefore length of orbit is
inversely proportional to the mass of the star
• Each star must complete an orbit in the same
time so that they remain on opposite sides of
centre of mass
• Therefore the orbital speed of each star is
inversely proportional to its mass
v1/v2 = m2/m1
• By observing relative velocities via Doppler shift
of binary stars we can infer their relative masses
Total Mass
• Kepler’s 3rd law gives the total mass of the
system
where P is the orbital period and A is the
average distance between the two masses
• If we measure P in years and A in AU then this
mass in Solar masses is simply
Stellar Masses
• By measuring the period of the binary and the
average separation, Kepler’s 3rd law gives us
the total mass of the binary system
• If we can also measure the sizes of the orbits, or
the star’s orbital speeds, we can determine the
relative masses m2/m1
• Knowing the sum of the masses and their ratio
allows us to determine the individual masses
• The range of stellar masses so determined is
around 0.08 to 100 M, much smaller than the
range of luminosities
Summary
• With a small number of straightforward
observations we can determine the
principle physical properties of stars
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Surface temperature
Radius
Luminosity
Mass (for binary stars)
• Most stars are cooler, smaller, less
luminous and less massive than the Sun
Other Solar Systems?
• Models of star formation generically predict the existence
of proto-planetary disks around protostars and so we
expect other planetary systems like the Solar System to
be quite common
• Planets around other stars (extra-solar planets) are
extremely hard to see due to glare from the host star
• However, since stars and massive planets are in orbit
about each other we can detect a “wobble” in the
position of stars with nearby massive planets
• The existence of many extra-solar planets is now
inferred from such observations
Seminar Quiz (Solar System)
• Nearly all extra-solar planets discovered to
date have Jupiter-like masses and are
located very close to their host stars.
Does this mean that the Solar System is
unusual?
Seminar Quiz (Solar System)
• A planet has two kinds of angular momentum orbital and spin - due to the orbit of the planet
around the Sun and rotation of the planet about
its own spin axis respectively
• Given that angular momentum depends on
mass, size of object/orbit and velocity of
rotation/revolution, which form contributes most
to a planet’s total angular momentum?
• More than 99% of Solar System’s mass resides
in the Sun, yet Jupiter, with 1/1000 of Sun’s
mass, possesses more angular momentum than
any other body including the Sun. Why?
Seminar Quiz (Stars I)
• Some properties of a star can only be
determined once the distance is known. Others
properties do not require us to know the
distance. In which of the above categories would
you place the following and why?
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Luminosity
Size
Mass
Temperature
Colour
Seminar Quiz (Stars II)
• Albiero is a binary system whose components
are easily separated in a small telescope.
Observers describe the brighter star as “golden”
and the fainter one as “sapphire blue”.
• What does this tell you about the relative
temperatures of the two stars?
• What does it tell you about their respective sizes?
Seminar Quiz (Stars III)
• Why, apart from the Sun, can we only measure
reliable masses for stars in binary systems?
• Sirius, the brightest star in the sky, has a
parallax of 0.379 arcseconds. What is its
distance in parsecs? In light years?
• Sirius is 22 times more luminous than the Sun;
Polaris is 2,350 times more luminous than the
Sun but appears 23 times fainter than Sirius.
What is the distance to Polaris?