Our Place in the Cosmos

Lecture 10Observed 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/dd (parsecs) = 1/p (arcsecs)

Parallaxes and Distances

• Since parallaxes are very small, they are 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 (parallax-arcsecond), 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 = 4d2 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 4d2

• 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 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 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

0K = absolute zero, 0C = 273K

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 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 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 = 4d2 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 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• 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?• 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?