Lecture 2

1. Lecture 2 The distance scale

2. 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)

3. 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 mref=0 in the equation above gives: • Colour is defined as the relative flux between two different wavebands, usually written as a difference in magnitudes

4. Apparent magnitudes 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.

5. 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?

6. What is the B-V colour of a source that has a flux proportional to l-4?

7. Absolute magnitudes • 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 D0=10pc: • The value of m-M is known as the distance modulus. (note the zeropoints have cancelled)

8. Example • Calculate the apparent magnitude of the Sun (absolute magnitude M=4.76) at a distance of 1 Mpc (106 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).

9. 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

10. Types of stars • Intrinsically faint stars are more common than luminous stars

11. Main sequence fitting NGC2437 • 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.

12. Main sequence fitting Nearby stars (parallax) distant cluster (apparent magnitudes) • 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

13. 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.

14. Break

15. 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?

16. 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?

17. 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

18. 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.

19. 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 W Virginis RR Lyrae Pulsating white dwarfs

20. 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 Period (days) Absolute Magnitude Schematic representation Log (Period)

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

22. Summary: the distance ladder • 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 • 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 • 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!

23. 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

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

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

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

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

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

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

30. 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 here • 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

31. Characteristics of spectral classes

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