Distances to Stars • Early astronomers considered the stars to be located on the surface of a sphere, and hence all at the same distance Our Our Place Place in in the the Cosmos Cosmos • To understand most properties of stars we need to know their distance • For nearby stars distance can be measured via Lecture 10 parallax Observed Properties of Stars • 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 Parallaxes and Distances Parallax is one-half of the angle through which a star appears to move over the course of a year • 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 A star with a parallax of 1 More distant stars have • 1 pc = 206,265 AU = 3 x 10 16 m = 3.26 ly smaller parallaxes p � 1/ d arcsecond is at a distance of 1 parsec d (parsecs) = 1/ p (arcsecs) • Distance (pc) = 1/parallax (arcsec)
Parallaxes and Distances Limits of Parallax • Closest star (apart from the Sun) is Proxima Centauri • Accuracy of positional measurements limits • Its parallax is 0.75 arsec giving a distance of 1.3 pc distance to which stars have a measured parallax • First successful parallax measurement was made in 1838 by FW Bessel • Hipparcos satellite launched in 1990s has • He measured a parallax to the star 61 Cygni of 0.314 measured parallaxes for 120,000 stars arcsec giving a distance of 3.2 pc, or 600,000 times accurate to 0.002 arcseconds further than the Sun • We can only measure parallaxes accurate to • This single measurement increased the known size of 10% for distances up to about 50 parsecs the Universe by 10,000-fold! • Today, only 54 stars in 37 systems (singles, binaries • Beyond a few hundred parsecs other distance or triples) are known within 15 light-years - stars are estimators have to be used few and far between Luminosity Inverse Square Law • The apparent brightness of a star depends strongly on its distance Light spreads out • As with gravity, the intensity of light drops more to cover a inversely with the square of distance d from a larger sphere and source as the light is spread out over the so appears fainter surface A = 4 � d 2 of a sphere of radius d further from the • The luminosity of a star (the total energy source radiated per second) is thus given by its measured brightness multiplied by 4 � d 2 Brightness � 1/ d 2 • We thus need to know the distance d to a star to calculate its luminosity Luminosity Function Stellar 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 10 6 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
Colour and Temperature Blackbody Radiation • A star radiates because it is hot • An idealised object that emits exactly as much radiation as it absorbs from its • The hotter an object the faster its surroundings is known as a blackbody constituent particles jostle about • In 1900 physicist Max Planck calculated how • Any charged particle (such as an electron) the spectrum (intensity as a function of that is accelerated will radiate energy known wavelength) of such a blackbody should as thermal radiation depend on its temperature • The energy of a photon of light is inversely • The resulting spectrum is known as a Planck proportional to its wavelength � : E � 1/ � spectrum or blackbody spectrum • Hotter objects thus emit radiation that is • As expected, hotter blackbodies emit more of both more intense and of shorter wavelength, their radiation at shorter, bluer, wavelengths ie. bluer Intensity of Blackbody Blackbody Spectrum Radiation • The luminosity of a blackbody increases with the fourth power of temperature L = � A T 4 • 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 Colours and Surface Radiation Temperatures of Stars • The peak wavelength of the Planck or • Most stars radiate approximately as blackbodies blackbody spectrum is given by Wien’s law • We can thus immediately say that blue stars are hot, � peak = (2,900 µ m K)/ T red stars are cool • By measuring the spectrum of a star, we can use • The wavelength at which a blackbody’s Wien’s law to find its surface temperature spectrum peaks is inversely proportional to • Rather than measuring a spectrum, we can gauge a temperature star’s colour by measuring its brightness through two • We can thus judge a star’s surface different filters, say blue and yellow (or “visual”) temperature from its colour • The brightness ratio between the two filters provides an estimate of temperature • Spectrum of sunlight peaks around 0.5 µ m • Cool stars are much more common than hot giving a surface temperature of 5800 K
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 T 4 • Luminosity L can be determined by measuring apparent brightness l and distance d (via parallax): L = 4 � d 2 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 Binary Stars • Luminosities and radii of stars are a poor • About half of all stars occur in binary indicator of mass as they can vary systems considerably in mass-to-light ratio and in • Each star feels an equal force towards the density other, but the lower mass star will experience • Gravity is the key to determining masses a greater acceleration (Newton’s 2nd law) • The Sun’s mass can be determined by studying • If the two stars were initially at rest, they the orbits of its planets would meet at a point closer to the more massive star called the centre of mass • We cannot directly see planets orbiting other stars, but we can observe binary stars - two • If star 1 is 3 times the mass of star 2, the stars orbiting a common centre of mass centre of mass will be 3 times further from star 2 than star 1 Force on each star is the same, but acceleration is Binary Stars larger for the less massive star and so it • In practice each star will have some velocity picks up speed faster perpendicular to the line joining them • Instead of falling into each other, they will As the stars fall toward orbit about each other with the centre of each other the centre of mass at one focus of their elliptical orbits mass remains stationary • The stars will always be on opposite sides of the centre of mass which remains stationary If m 1 = 3 m 2 star 2 will fall 3 times as far as • Orbit of more massive star will be smaller star 1 than the orbit of the less massive star • Less massive star must move faster in order They meet at the centre to complete its longer orbit in the same time of mass where the pivot as the more massive star of a balance would be
Binary Stars Less massive star • Semi-major axis and therefore length of orbit moves faster on a is inversely proportional to the mass of the larger orbit star • Each star must complete an orbit in the same Centre of mass Equal time time so that they remain on opposite sides of remains steps centre of mass stationary • Therefore the orbital speed of each star is inversely proportional to its mass v 1 / v 2 = m 2 / m 1 • 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 Summary • By measuring the period of the binary and the • With a small number of straightforward average separation, Kepler’s 3rd law gives us observations we can determine the the total mass of the binary system principle physical properties of stars • If we can also measure the sizes of the orbits, • Surface temperature or the star’s orbital speeds, we can determine • Radius the relative masses m 2 / m 1 • Luminosity • Knowing the sum of the masses and their ratio • Mass (for binary stars) allows us to determine the individual masses • Most stars are cooler, smaller, less • The range of stellar masses so determined is around 0.08 to 100 M � , much smaller than the luminous and less massive than the Sun range of luminosities
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