Lecture 2 The distance scale

Apparent magnitudes The magnitude system expresses fluxes in a given waveband X, on a relative, logarithmic scale:  Note the negative sign means brighter objects have lower magnitudes  Scale is chosen so that a factor 100 in brightness corresponds to 5 magnitudes (historical)

The magnitude scale One common system is to measure relative to Vega  By definition, Vega has m=0 in all bands. Note this does not mean Vega is equally bright at all wavelengths!  Setting m ref =0 in the equation above gives: Colour is defined as the relative flux between two different wavebands, usually written as a difference in magnitudes

Apparent magnitudes ObjectApparent mag Sun-26.5 Full moon-12.5 Venus-4.0 Jupiter-3.0 Sirius-1.4 Polaris2.0 Eye limit6.0 Pluto15.0 Reasonable telescope limit (8-m telescope, 4 hour integration) 28 Deepest image ever taken (Hubble UDF) 29 The faintest (deepest) telescope image taken so far is the Hubble Ultra-Deep Field. At m=29, this reaches more than 1 billion times fainter than what we can see with the naked eye.

Imagine a hypothetical source which has a constant flux of 10 Jy at all frequencies. What is its magnitude in the U band? In the V and K bands? Band name Central Wavelength (  m) Bandwidth (  m) Flux of Vega (Jy) U B V R I J H K

What is the B-V colour of a source that has a flux proportional to -4 ? Band name Central Wavelength (  m) Bandwidth (  m) Flux of Vega (Jy) U B V R I J H K

It is also useful to have a measurement of intrinsic brightness that is independent of distance Absolute Magnitude (M) is therefore defined to be the magnitude a star would have if it were at an arbitrary distance D 0 =10pc: The value of m-M is known as the distance modulus. Absolute magnitudes (note the zeropoints have cancelled)

Example Calculate the apparent magnitude of the Sun (absolute magnitude M=4.76) at a distance of 1 Mpc (10 6 pc) Recall that the deepest exposures taken reach m=29 The nearest large galaxy to us is Andromeda (M31), at a distance of about 1 Mpc  Detecting stars like our Sun in other galaxies is therefore very difficult (generally impossible at the moment).

The colour-magnitude diagram Precise parallax measurements allow us to plot a colour-magnitude diagram for nearby stars.  The Hertzsprung- Russel (1914) diagram proved to be the key that unlocked the secrets of stellar evolution  Colour is independent of distance, since it is a ratio of fluxes:  Absolute magnitude (y-axis) requires measurement of flux and distance

Types of stars Intrinsically faint stars are more common than luminous stars

Main sequence fitting Stellar clusters:  Consist of many, densely packed stars  For distant clusters, it is a very good approximation that all the constituent stars are the same distance from us.  Typical clusters have sizes ~1 pc; so for clusters >10 pc away this assumption introduces a 10% error.  Therefore, we can plot a colour-magnitude diagram using only the apparent magnitude on the y-axis, and recognizable structure appears. NGC2437

Main sequence fitting We can take advantage of the structure in the HR diagram to determine distances to stellar clusters  Colour is independent of distance, so the vertical offset of the main sequence gives you the distance modulus m-M Nearby stars (parallax) distant cluster (apparent magnitudes)

Main sequence fitting Example: NGC2437: At a colour of B-V=1.0 mag, the main sequence absolute magnitude is 6.8. In NGC2437, at the same colour, V=17.5. Thus the distance modulus is: This gives a distance of 1.4 kpc to NGC2437, reasonably close to the accepted distance of 1.8 kpc.

Main sequence fitting Instead of fitting at one colour, you can fit the whole main sequence. Find the distance modulus that gives you the best overlap.

Break

Variable stars The images above show the same star field at two different times. One of the stars in the field has changed brightness relative to the other stars – can you see which one?

Variable stars The images above show the same star field at two different times. One of the stars in the field has changed brightness relative to the other stars – can you see which one?

Variable stars Many stars show fluctuations in their brightness with time. These variations can be characterized by their light curve – a plot of their magnitude as a function of time

Variable stars Certain intrinsically variable stars show a remarkably strong correlation between their pulsation period and average luminosity Modern calibration of the Cepheid P-L relation in the Magellanic clouds, yields: Where the period P is measured in days, and the magnitude is measured in the I band.

Instability strip Classical Cepheids are not the only type of pulsating variable star, however There is a narrow strip in the HR diagram where many variable stars lie Cepheids are the brightest variable stars; however they are also very rare Cepheids RR Lyrae Pulsating white dwarfs W Virginis

RR Lyrae Stars RR Lyrae stars (absolute magnitudes M=+0.6) are much fainter than Cepheids; but have the advantage that they almost all have the same luminosity and are more common. They are easily identified by their much shorter periods Absolute Magnitude Period (days) Log (Period) Schematic representation

RR Lyrae variables RR Lyrae stars have average absolute magnitudes M=+0.6. How bright are these stars in Andromeda?

Summary: the distance ladder 1.Find parallax distances to the nearest stars Dedicated satellites are now providing these precise measurements for thousands of stars Plot stellar absolute magnitudes as a function of colour 2.Measure fluxes and colours of stars in distant clusters Compare with colour-magnitude diagram of nearby stars (step 1) and use main-sequence fitting method to compute distances Identify any variable stars in these clusters. Calibrate a period- luminosity relation for these variables 3.Measure the periods of bright variable stars in remote parts of the Galaxy, and even in other galaxies Use the period-luminosity relation from step 2 to determine the distance Note how an error in step 1 follows through all subsequent steps!

Spectroscopy In 1814, Joseph Fraunhofer catalogued 475 sharp, dark lines in the solar spectrum. Discovered but misinterpreted in 1804 by William Wollaston Spectrum was obtained by passing sunlight through a prism

Example: the solar spectrum What elements are present in the Sun? Solar spectrum

Example: the solar spectrum What elements are present in the Sun? Balmer lines (Hydrogen)

Example: the solar spectrum What elements are present in the Sun? NaD

Example: the solar spectrum What elements are present in the Sun? Ca H+K

Example: the solar spectrum So: the Sun is mostly calcium, iron and sodium?? No! Not quite that simple… Solar spectrum

Stellar spectra Stellar spectra show interesting trends as a function of temperature: Increasing temperature

Spectral classification Stars can be classified according to the relative strength of their spectral features:  There are seven main classes, in order of decreasing temperature they are: O B A F G K M  For alternative mneumonics to the traditional ‘O be a fine girl kiss me’, see herehere  Each class is subdivided more finely from 0-9. So a B2 star is hotter than a B9 which is hotter than a A0  Additional classes are R, N, S which are red, cool supergiant stars with different chemical compositions

Characteristics of spectral classes Spectral Type ColourTemperature (K) Main characteristicsExample OBlue-white>25000Strong HeII absorption (sometimes emission); strong UV continuum 10 Lacertra BBlue-white HeI absorption, weak Balmer linesRigel AWhite Strongest Balmer lines (A0)Sirius FYellow-white CaII lines strengthenProcyon GYellow Solar-type spectraSun KOrange Strong metal linesArcturus MRed<3500Molecular lines (e.g. TiO)Betelgeuse

The HR diagram revisited Henry Norris’ original diagram, showing stellar luminosity as a function of spectral class. The main sequence is clearly visible Spectral Class O B A F G K M Luminosity The original HR diagramA modern colour-magnitude diagram