Chapter 10 Measuring the Stars Read material in Chapter 10 Some of the topics included in this chapter • Stellar parallax • Distance to the stars • Stellar motion • Luminosity and apparent brightness of stars • The magnitude scale • Stellar temperatures • Stellar spectra • Spectral classification • Stellar sizes • The Herzsprung-Russell (HR)diagram • The main sequence • Spectroscopic parallax • Extending the cosmic distance • Luminosity class • Stellar masses Parallax is the apparent shift of an object relative to some distant background as the observer’s point of view changes It is the only direct way to measure distances to stars . It makes use Earth’s orbit as baseline . Parallactic angle = 1/2 angular shift .A new unit of distance: Parsec . By definition, parsec (pc) is the distance from the Sun to a star that has a parallax of 1” (1 arc second) Parallax Formula: . Distance (in pc) = 1/parallax (in arcsec) . One parsec = 206,265 AU or ~3.3 light-years As the distance increases to a star, the parallax decreases…. Examples using the parallax formula: If the measured parallax is 1 arcsec, then the distance of the star is 1 pc. (Distance 1 pc = 1/1 arcsec) If the measured parallax is 0.5 arcsec, then the distance of the star is 2 pc. (Distance 2 pc = 1/0.5 arcsec) Note: 1 parsec = 3.26 light-years. The Solar Neighborhood Let’s get to know our neighborhood: A plot of the 30 closest stars within 4 parsecs (~ 13 ly) from the Sun. The gridlines are distances in the galactic plane (the plane of the disc of the Milky Way) The Nearest Stars More examples using the parallax formula The nearest star Proxima Centauri (the faintest star of the triple star system Alpha Centauri) has a parallax of 0.76 arcsec. • Therefore, distance = 1 / 0.76 = 1.32 pc (4.29 ly) • The next nearest star is Barnard’s star, with a parallax of 0.55” • Therefore, d = 1 / 0.55 = 1.82 pc (5.93 ly) • From the ground, we can measure parallactic angles of ~1/30 (0.03”) arcsec, corresponding to distances out to ~30 pc (96 ly). • There are several thousand stars within that distance from the Sun. • From space (Hipparcos satellite), parallax’s can be measured down to about 5/1000 arcsec, which corresponds to 200 pc (~660 ly). • There are several million stars within that distance. Using the stellar parallax, the distance to these stars can be determined directly Stellar Proper Motion • Parallax is an apparent motion of stars due to Earth orbiting the Sun. • But stars do have real space motions. • Space motion has two components: 1) “line-of-sight” or radial motion (measured through Doppler shift of emission/absorption lines) 2) “transverse” motion (perpendicular to the line of sight) observer star radial motion transverse space motion How to determine the two component of the space motion of a star? • Use the Doppler shift to determine the radial component. Observe the shift in wavelengths of the emission or absorption lines. Then apply the formula of Doppler shift to determine the radial velocity • Use the proper motion and the distance to determine the transverse component. First we need to measure the proper motion. Proper motion is measured in arc seconds/year. Then we need to know the distance to the star using parallax so we can determine the transverse component • Finally use trigonometry to calculate the transverse velocity • This method works for stars that are nearby so we can measure the proper motion. • The total velocity can be calculated using the Pythagorean theorem: ____________________________________ Total velocity = √ [(Radial velocity)² + (Transverse velocity)²] Stellar Proper Motion: Barnard’s Star • Two pictures, taken 22 years apart ( Taken at the same time of the year so it doesn’t show parallax!). Barnard’s star is a red dwarf of magnitude +9.5, invisible to the naked eye (limit of naked eye is +6) • Barnard’s star has a proper motion of 10.3 arcsec/year (it is the star with the largest proper motion) • Given d = 1.8 pc, this proper motion corresponds to a “transverse” velocity of ~90 km/s ! Question: What does the proper motion depend on? Answer 1: Space velocity Answer 2: Distance Some important definitions and concepts • Luminosity is the amount of radiation leaving a star per unit time. • Luminosity is an intrinsic property of a star. • It is also referred as the star absolute brightness. It doesn’t depend on the distance or motion of the observer respect to the star. • Apparent brightness or Flux. When we observe a star we see its apparent brightness, not its luminosity. The apparent brightness (or flux) is the amount of light striking the unit area of some light sensitive device such as the human eye or a CCD. It depends on the distance to the star. Apparent Brightness and the Inverse Square Law: Proportional to 1/d2 • Light “spreads out” like the distance squared. • Through a sphere twice as large, the light energy is spread out over four times the area. (area of sphere = 4d2) The apparent brightness or Flux decreases with distance, it is inversely proportional to the square of the distance. It can be determined by: Luminosity Flux = 4d2 To know a star’s luminosity we must measure its apparent brightness (or flux) and know its distance. Then, Luminosity = Flux *4d2 Luminosity and Apparent Brightness Two stars A and B of different luminosity can appear equality bright to an observer if the brightest star B is more distant than the fainter star A The Magnitude Scale 2nd century BC, Hipparchus ranked all visible stars Faintest He assigned to the brightest star a magnitude 1, and to the faintest a magnitude 6. Later, astronomer found out that a difference of 5 magnitudes from 1 (brightest) to 6 (faintest) correspond to a change in brightness of 100 To our eyes, a change of one magnitude = a factor of 2.512 in flux or brightness. The magnitude scale is logarithmic. Each magnitude corresponds to a factor of 1001/5 2.5 5 magnitudes = factor 100 in brightness. The apparent magnitude scale was later Brightest extended to negative values for brighter objects and to larger positive values for fainter objects Equivalence between magnitude and brightness Magnitude Brightness -1 2.512 0 2.512 1 2.512 2 2.512 3 2.512 4 2.512 5 2.512 6 The change of brightness between magnitude 1 and 6 is 2.512^5 = 100 In general, the difference in brightness between two magnitudes is: Difference in brightness = 2.512 ^n, where n is the difference in magnitude Example: What is the difference in brightness between magnitude -1 and +1? Answer: n=2, difference in brightness = 2.512² = 2.512 x 2.512 = 6.31 Absolute Magnitude is the apparent magnitude of a star as measured from a distance of 10 pc (33 ly). Sun’s absolute magnitude = +4.8 It is the magnitude of the Sun if it is placed at a distance of 10 pc. Just slightly brighter than the faintest stars visible to the naked eye (magnitude = +6) in the sky. Enhanced color picture of the sky Notice the color differences among the stars Stellar Temperature: Spectra • The spectra shows 7 stars with same chemical composition but different temperatures. • Different spectra result from different temperatures. Example: Hydrogen absorption lines are relatively weak in the hottest star because it is mostly ionized. Conversely, hotter temperatures are needed to excite and ionize Helium so these lines are strongest in the hottest star. Molecular absorption lines (TiO) are present in low temperature stars. The low temperatures allow formation of molecules Ti Titanium, TiO titanium oxide Spectral Classification: A classification of stars was started by the “Pickering’s women”, a group of women hired by the director of the Harvard College observatory, including Annie Cannon Annie Jump Cannon The stars were classified by the Hydrogen line strength, and started as A, B, C, D, … But after a while they realized that there is a sequence in temperature so they rearranged the letters (some letters were drop from the classification) so that it reflect a sequence in temperature. It became: O, B, A, F, G, K, M, (L) A temperature sequence! Cannon’s spectral classification system was officially adopted in 1910. Spectral Classification A mnemonic to remember the correct order: “Oh Be A Fine Girl/Guy Kiss Me” Each letter is divided in 10 smaller subdivisions from 0 to 9. The lower the number, the hotter the star. Example, G0 (hotter) to G9 (cooler). The Sun is classified as a G2 star, the surface temperature is 5800 K Strengths of Lines at Each Spectral Type Stellar Radii • Almost all stars are so distant that the image of their discs look so small. Their images appear only as an unresolved point of light even in the largest telescopes. Actually the image shows the Airy disk produced by the star. • A small number of stars are big, bright and close enough to determine their sizes directly through geometry. •Knowing the angular diameter and the distance to the star, it is possible to use geometry to calculate its size. Diameter/2π x distance = Angular diameter/360 Stellar Radii • One example in which it is possible to use geometry to determine the radius is the star Betelgeuse in the Orion constellation • The star is a red giant located about 640 ly from Earth • Betelgeuse size is about 600 time larger than the Sun • Its photosphere exceed the size of the orbit of Mars • Using the Hubble telescope it is possible to resolve its atmosphere and measure its diameter directly • The measured angular size is about 0.043-0.056 arc seconds An indirect way to determine the stars radii • Most of the stars are too distant or too small to allow the direct determination of their size.