Astrophysics option E3: Stellar objects

1. Astrophysics option E3: Stellar objects Determining the Distance to the Stars

2. objectives • Describe the method of parallax, , the method of spectroscopic parallax and the Cepheids method for determining distances in astronomy; • Define the parsec; • State the definitions of apparent brightness, , and apparent and absolute magnitude. • Solve problems using apparent brightness and luminosity; • Use the magnitude-distance formula.

3. Angular sizes • 360 degrees (360o) in a circle • 60 arcminutes (60’) in a degree • 60 arcseconds (60”) in an arcminute

4. The parallax method • Takes advantage of the fact that, when an object is viewed from two different positions, it appears displaced relative to a fixed background. • Measure the angular position of a star and repeat the measurement some time later, the two positions are different, relative to a background of stars, because of the fact that in the intervening time, the Earth has moved in its orbit around the sun.

5. The parallax method Make 2 measurements of the same star 6 months apart. The distance between the two positions of the earth is D=2R, the diameter of Earth’s orbit around the sun. The distance to the star, d, is given by so ; since the parallax angle is very small, when measured in radians, so .

6. Earth in June Distant stars Near star 1 AU d θ Sun θ 1 AU Earth in January The parallax method d = 1 AU θ d (parsecs) = 1 p (arc-seconds)

7. Stellar parallax has its limits The farther away the star is, the smaller its parallax angle. Eventually, the movement of the star is too small to see.

8. The parallax method • The parallax angle is the angle at the position of the star that subtends a distance equal to the radius of the Earth’s orbit around the sun (1 AU = 1.5x1011 m). • Parallaxes are measured quite accurately provided they are not too small. For example, parallaxes down to 1 arcsecond, symbol 1” (or 1/3600 of a degree), are easily measured. • If the star is too far away, however, the parallax is too small to be measured and this method fails. Typically, measurements from observations on Earth allow distances up to 300 ly (roughly 100 parsec) to be determined with the parallax method; used mainly for nearby stars. • Satellites above Earth’s atmosphere can use this method for distances up to 500 pc.

9. The Parallax Method & the parsec • Parallax method used to define a common unit of distance – the parsec (parallax second) is the distance to a star whose parallax is 1 arcsecond. • The factor of converts degrees into radians. In terms of a light year (), 1 pc = 3.26 ly • 1AU = 1.5 x 10 11 m • 1 ly = 9.46 x 1015 m • 1 pc = 3.09 x 1016 m • If the parallax of a star is p arcseconds, the distance is 1/p parsecs or

10. Stellar disTANCES

11. Our 5 nearest stars

12. Absolute & apparent magnitudes • Ancient astronomers Hipparchos & Ptolemy devised a relative system of classifying stars according to how bright they appeared to an observer on Earth; apparent magnitude, m – the higher the apparent magnitude the dimmer the star. Six classes – magnitude 6 supposed to be 100 timeS dimmer than magnitude 1. magnitude 2 star must be 1001/5 ≈2.512 times dimmer than magnitude 1.

13. Modern Magnitude Scale More recently it was thought that the brightest stars visible from earth are about 100 times brighter that the dimmest. So, using the same classification of m = 1 to 6: Fractional change = 5√(1/100) = 1 / 2.512  Each apparent magnitude on the modern scale is 2.512 times dimmer than the previous

14. Over time, adjustments have had to made to account for dimmer and brighter stars. • e.g. • Vega has apparent magnitude 0, and Sirius (the brightest star) has a negative magnitude (-1.4). • So for the apparent magnitude scale... • the bigger the positive number the dimmer the star • the brightest stars have negative numbers

15. Modern magnitude scale Given a star of brightness b, we assign to that star an apparent magnitude m defined by Where is taken as the reference value for apparent brightness– a star w/ apparent magnitude = 0 Taking (base 10) logarithms gives Since 1001/5 = 2.512 the first equation above can be written as

16. Example Consider a star whose apparent brightness is . Then Similarly, a star of apparent magnitude m = 4.35 has an apparent brightness given by Note that a star of apparent brightness equal to the reference value b0=2.52x10-8 W m-2 is assigned an apparent magnitude m = 0.

17. The apparent magnitude scale • The larger the magnitude, the dimmer the star. Consider now 2 stars of apparent magnitude m1 and m2 and apparent brightness b1 and b2. • We have and We may also compare the brightness of Sirius to Bernard’s Star to find This means that Sirius is about 25,000 times brighter than Bernard’s Star. Proxima Centauri is about 5000 times brighter (check my work).

18. Apparent Magnitude • The human eye can detect a star of apparent magnitude not larger than about 6. Add simple binoculars – increases to about 9. largest telescope – up to 27

19. Asteroid ‘65 Cybele’ and 2 stars with their apparent magnitudes labelled

20. Apparent vs. absolute Magnitude • Two stars that have the same apparent magnitude are not necessarily equally bright intrinsically, since they may be at different distances. To establish a system of absolute magnitudes that will tell us if one star is intrinsically brighter than another, we imagine that all stars are positioned at the same distance from the Earth, 10 pc. • Absolute magnitude - the apparent magnitude a star would have if placed at a distance 10 pc from Earth

21. Apparent & absolute magnitudes Absolute & apparent magnitudes and distance are then related. Let’s compare the apparent brightness b of a star to the apparent brightness B that it would have if it were only 10 pc from the Earth. We have and and B And so Taking (base 10) logarithms gives Remember that d is in parsecs!

22. Apparent & absolute magnitudes of our 5 brightest stars

23. Example Calculate the absolute magnitude of a star whose distance is 25.0 ly and whose apparent magnitude is 3.45. F: M=? G: d=25.0 ly = 7.67 pc, m=3.45 E: W: A: M = 4.03

24. Example Calculate the distance to Sirius using the data in your apparent and absolute magnitude table. F: d=?G: m = -1.43, M = 1.4 E: W: A: 2.7 pc

25. Spectroscopic parallax • A method of finding the distance to a star given the star’s luminosity and apparent brightness. No use of parallax… • Assume we know the luminosity L and apparent brightness b of a star • Determine luminosity using spectrum to deduce temperature. Knowing temperature and using HR diagram (assuming main sequence) allows determination of luminosity. • Thus, the distance can be found (estimate to several thousand parsecs).

26. Example A main sequence star emits most of its energy at a wavelength of 2.4x10-7 m. Its apparent brightness is measured to be 4.3x10-9 W m-2. How far is the star? F: d = ? G: λ= 2.4x10-7m, b = 4.3x10-9 W m-2, HR diagram E: , W: ⇒ L = 100Lsun from HR diagram

27. The cepheids • Stars whose luminosity varies from a minimum to a maximum periodically (periods of a couple of days to a couple of months). The brightness of the star increases sharply, then fades off gradually.

28. The cepheids • First discovered by 19 year-old English astronomer John Goodricke in 1748, two years before his death. • Reason for periodic behavior has to do with interaction of radiation with matter in the atmosphere of the star. The interaction causes the outer layers of the star to undergo periodic expansions (brightest) and contractions (dimmest). • Beginning of 20th century – Henrietta Leavitt discovers that the longer the period, the larger the luminosity; makes Cepheid stars standard candles – finding the period of a Cepheid allows determination of its luminosity and distance.

29. Standard candles • A standard candleis a star of known luminosity. This means that the luminosity of all other stars in its galaxy can be estimated by comparing their apparent brightness with the standard candle. • Thus the existence of a Cepheid variable star in a distant galaxy enables the luminosity of all the stars in the galaxy to be determined.

31. Luminosity Cepheid variables The luminosity of a Cepheid varies periodically: Changes in the surface and atmosphere cause the star to increase in surface area and thus increase in luminosity periodically.

32. There is a direct relationship between the peak luminosity of Cepheids and their time periods (‘the luminosity-period relationship’):

33. The distance to a cepheidcan be found by • Measure the average period of luminosity • Use the relationship graph to find peak luminosity from period • Measure peak apparent brightness on earth • Find distance using apparent brightness formula

34. The cepheids For example, the Cepheid whose light curve is shown below has a period of about 22 days, this corresponds to a luminosity of about 7000 solar luminosities or 2.73 x 1030 W. The peak apparent magnitude is about m = 3.7. peak apparent brightness can be found from

35. The cepheids Using the relationship between apparent brightness, luminosity & distance Thus, one can determine the distance to the galaxy in which the Cepheid is assumed to be. This method is accurate up to a few Mpc.

36. Ways to measure stellar distances

37. Summary • Describe with the aid of a clear diagram what is meant by the parallax method in astronomy. Explain why the parallax method fails for stars that are very far away. • The distance to Epsilon Eridani is 10.8 ly. What is its parallax? • Give definitions of apparent magnitude of a star and absolute magnitude of a star. • The parallax of a star is 0.025” and its absolute magnitude is M=0.8. Is it’s apparent magnitude greater than or less than 0.8?