# Chapter 10 Measuring the Stars

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1 Chapter 10 Measuring the Stars

2 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

3 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

4 As the distance increases to a star, the parallax decreases. Examples: If the measured parallax is 1 arcsec, then the distance of the star is 1 pc. If the measured parallax is 0.5 arcsec, then the distance of the star is 2 pc. Note: 1 parsec = 3.26 light-years.

5 Let s get to know our neighborhood: A plot of the 30 closest stars within 4 parsecs from the Sun. The gridlines are distances in the galactic plane (the plane of the disc of the Milky Way) The Solar Neighborhood

6 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)

7 A plot of the 30 closest stars to the Sun 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. Using the stellar parallax, the distance to these stars can be determined directly

8 A plot of the 30 closest stars to 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.

9 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

10 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 formula of Doppler shift to determine the radial velocity Use the proper motion 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 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]

11 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

12 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

13 Apparent Brightness and the Inverse Square Law: Proportional to 1/d 2 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 = 4d 2 ) The apparent brightness or Flux decreases with distance, it is inversely proportional to the square of the distance. It can be determined by: Flux = Luminosity 4d 2 To know a star s luminosity we must measure its apparent brightness (or flux) and know its distance. Then, Luminosity = Flux *4d 2

14 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

15 The Magnitude Scale 2 nd 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 in flux or brightness. The magnitude scale is logarithmic. Each magnitude corresponds to a factor of 100 1/ magnitudes = factor 100 in brightness. The magnitude scale was later extended to negative values for brighter objects and to larger positive values for fainter objects Brightest

16 Equivalence between magnitude and brightness Magnitude Brightness 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 = ^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 = 2.512^2 = x = 6.31

17 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.

18 Enhanced color picture of the sky Notice the color differences among the stars

19 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

20 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 The stars were classified by the Hydrogen line strength, and started as A, B, C, D, Annie Jump Cannon 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 cassification system officially adopted in 1910.

21 Spectral Classification A mnemonic to remember the correct order: Oh Be A Fine Girl/Guy Kiss Me An alternative mnemonic: Oh Brother, Astronomers Frequently Give Killer Midterms 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

22 Strengths of Lines at Each Spectral Type

23 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

24 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 arc seconds

25 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. But we can use the radiation laws to make an indirect determination of their size. According to Stefan law, the luminosity of a star is proportional to the fourth power of the surface temperature (T 4 ) The luminosity also depend on its surface area. Larger bodies radiate more energy. Luminosity Surface area * T 4

26 Stellar Radii: An indirect way to measure the radius (Read 10-2 More Precisely, Estimating Stellar Radii ) Stefan s Law F = T 4 Flux (F) is the energy radiated per unit area by a black body at the temperature T Luminosity (L) is the Flux (F) multiplied by the entire spherical surface (A) L = A * F Area of sphere A = 4R 2 (R is the radius of the star) Substituting A in the equation of L L = 4R 2 F Substituting F in the equation of L L = 4R 2 T 4 Expressing in solar units (dividing by the solar L, R and T), the constants disappear: L star = (R star /R sun ) 2 * (T star /T sun ) 4 * L sun

27 The relationship between Luminosity, Radius, and Temperature provides a means to evaluate these properties relative to the Solar values. L L L L sun sun 4 4 R R R R 2 For example, a star has 10 times the Sun s radius but is half as hot. (Since this is relative to the Sun, we will consider that the radius of the Sun is 1 and the temperature of the Sun is 1) How much is the luminosity respect to the Sun? sun 2 sun 2 T T T 4 4 sun T sun 4 L Lsun

28 Determining radii using radiation laws The equation L = 4R 2 T 4 can be expressed in solar units as: L(in solar luminosities) = R 2 (in solar radius) * T 4 (in solar surface temperature) If we need to calculate the radius, we can rearrange the equation : R 2 (in solar radius) = L(in solar luminosities) / T 4 (in solar surface temperature)

29 Understanding Stefan s Law: Radius dependence L star = (R star /R sun ) 2 * (T star /T sun ) 4 * L sun Let s consider a star that has a radius twice the radius of the Sun. What will be the luminosity of that start? (assume that the two stars have the same temperature) If we receive 100 photons from the Sun, we should receive 400 photons from a star twice the diameter of the Sun. The star will look four times brighter than the Sun

30 Understanding Stefan s Law: Temperature dependence L star = (R star /R sun ) 2 * (T star /T sun ) 4 * L sun Let s consider a star with a temperature twice that of the Sun and another star with a temperature one third of the Sun The luminosity of a star that has a temperature twice that of the Sun, must be 16 times larger. The luminosity of a star with a temperature 1/3 of the Sun, must be 1/81 that of the Sun The assumption here is that these stars have the same radius

31 Hertzsprung-Russell (HR) Diagram The HR diagram is a plot of star Luminosity versus Temperature (or spectral class) It also give information about: Radius Mass Lifetime Stage of Evolution The Main Sequence is the diagonal band of stars in the HR diagram Stars reside in the main sequence during the period in which the core burns H Most stars (like the Sun) lie on the main sequence. The Sun will spend most of its life in the main sequence (about 5 billion years) Main sequence

32 From Stefan s law... L = 4R 2 T 4 The HR diagram to the right has L and T on the axes. But we can plot R (The other parameter in the equation) also which will appear as straight lines crossing the diagram Let s use the equation and the HR diagram to learn more about L, R and T More luminous stars at the same T must be bigger! Cooler stars at the same L must be bigger!

33 The HR Diagram: 100 Brightest Stars Most luminous stars, because they are so rare, lie beyond 5 pc. If we know the luminosity, we can determine distance from their Flux (brightness). Flux = Luminosity 4d 2 The technique to determine distances to stars using the radiation laws and HR diagram is called: Spectroscopic Parallax

34 The HR Diagram: Spectroscopic Parallax An example to illustrate how this works: Main Sequence 1) We measure the Flux or apparent brightness of a star Apparent brightness is the rate at which energy from the star reaches a detector 2) From the spectrum of a star, we can determine its temperature or the spectral type. 3) Then using the HR diagram we can determine its luminosity assuming it is located in the Main Sequence 4) Use inverse square law to determine distance. Flux = Luminosity 4d 2

35 The HR Diagram: Luminosity & Spectroscopic Parallax What if the star doesn t happen to lie on the Main Sequence - maybe it is a red giant or white dwarf??? We determine the star s Luminosity Class based on its spectral line widths: Spectral lines get broader when the stellar gas is at higher densities - indicates smaller star. Wavelength A Supergiant star A Giant star A Dwarf star (Main Sequence)

36 The HR Diagram: Luminosity Class Bright Supergiants Supergiants Bright Giants Giants Sub-giants Main-Sequence (Dwarfs)

37 Example of absorption lines for different spectral classes The lines are wider for dwarf (denser) stars of spectral class V and narrower for giant stars of spectral class I. Isn t this getting a little circular? First we said that we derive Luminosities from measured Fluxes and Distances? Now we re saying we know the Luminosities and we use them together with Temperatures to derive Distances.. Let s clarify this!

38 More on Spectroscopic Parallax The answer: Now we made use of additional information obtained from the spectral analysis. The spectral analysis provide information to determine the temperature of the star or the spectral classification (Using the spectrum of the star). To do this, we didn t know or need the distance Next we also made use of the HR diagram. If we know the temperature (or the luminosity class), then we can deduce the luminosity

39 We get distances to nearby planets from radar ranging If we know the distance (and we can measure the orbital period), we apply Kepler s 3 rd law to obtain the distance Earth-Sun (AU) That sets the scale for the whole solar system (1 AU). It allows us to get a value for the AU in km (1AU = 150,000,000 km) Knowing the value of the AU in km, we use the stellar parallax, to find distances to nearby stars. The Distance Ladder Use these nearby stars with known distances, then we measure the Fluxes and determine the Luminosities, to calibrate Luminosity classes in HR diagram. Then knowing spectral class (or T) we determine Luminosity. Next we measure the Flux and get Distance for farther stars (Spectroscopic Parallax).

40 Stellar Masses: Visual Binary Stars Binary star are classified as visual, spectroscopic and eclipsing The example shows Sirius (visual binary), the brightest star in the sky. Sirius A has a companion Sirius B, a very dense object called white dwarf With Newton s modifications to Kepler s laws, the period and size of the orbits yield the sum of the masses. P² = a³ /(m 1 + m 2 ) The relative distance of each star from the center of mass yields the ratio of the masses. m 1 d 1 = m 2 d 2 The ratio and the sum provide each mass individually (Two equations and two unknowns). P, a, d1 and d2 are known (they are measured) Note: For Sirius, the plane of the orbit is not face on, it is inclined 46 degrees from the line of sight. A correction needs to be done first before using the values of size of orbit and distance to the center of mass

41 Stellar Masses: Spectroscopic Binary Stars Many binaries are too far away to be resolved or they orbit around the other star at a short distance, but they can be discovered from periodic spectral line shifts. The shift of the spectral lines is causes by the Doppler effect In this example, only the yellow (brighter) star is visible

42 Stellar Masses: Eclipsing Binary Stars How do we identify eclipsing binaries? We can identify an eclipsing binary by observing the light curve of the star, a plot of the apparent brightness of the star as function of time The occultation of the star in the system must be observed only if we can see the orbital plane edge on. This method also tells us something about the stellar radii (Through the deep of the eclipse).

43 The HR Diagram: Stellar Masses Why is the mass of a star so important? Together with the initial composition, mass defines the entire life cycle and all other properties of the star! The composition of the first stars was H and He. After the interstellar gas was contaminated with heavier elements produced in the interior of the starts, the composition of the mass of those starts incorporated those heavier elements The mass determine: Luminosity Radius Surface Temperature Lifetime Evolutionary phases And how the star will end its life. All is determined by the mass of the star.

44 Example: For stars on the Main Sequence, if we plot the luminosity as function of mass, we find that the luminosity depends of the mass: Luminosity Mass 4 Why the luminosity increases at such high rate? A star with more mass means more gravity more pressure in the core higher core temperatures faster nuclear reaction rates fast production of energy ( Mass 4) higher luminosities! shorter lifetime

45 Lifetime Fuel available / How fast fuel is burned So for a star Stellar lifetime Mass / Luminosity Or, since Luminosity Mass 4 (For main sequence stars) Stellar lifetime Mass / Mass 4 = 1 / Mass 3 How long a star lives is directly related to the mass! Example: The Sun lifetime is estimated to be about 10 billion years. A star with 10 times the mass of the Sun has an estimated lifetime of 10 million years! Do the calculation! Big (Massive) stars live shorter lives, burn their fuel faster.

46 H-R diagram Location of stars of different masses

47 The turn off point of two open star clusters in the H-R diagram showing their different ages The turn off points Stars of higher mass leave the Main Sequence earlier What can we deduce from the HR diagram and the turn off points about the relative age of these two clusters?

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