2 Why are distances important? Distances are necessary for estimating: Total energy released by an object (Luminosity) Masses of objects from orbital motions (Kepler s third law) Physical sizes of objects
3 The problem of measuring distances Q: What do you do when an object is out of reach of your measuring instruments? Examples: Surveying & mapping Architecture Any astronomical object A: You resort to using GEOMETRY.
4 Method of trigonometric parallaxes June p December Foreground Star Distant Stars
5 Parallax decreases with distance Closer stars have larger parallaxes: Distant stars have smaller parallaxes:
6 Stellar parallaxes All stellar parallaxes are less than 1 arcsecond Nearest Star, α Centauri, has p=0.76-arcsec Cannot measure parallaxes with naked eye. First parallax observed in 1837 (Bessel) for the star 61 Cygni. Use photography or digital imaging today.
7 Parallax formula p = parallax angle in arcseconds d = distance in Parsecs
8 Parallax Second = Parsec (pc) Fundamental unit of distance in Astronomy A star with a parallax of 1 arcsecond has a distance of 1 Parsec. Relation to other units: 1 parsec (pc) is equivalent to 206,265 AU 3.26 Light Years 3.085x10 13 km
9 Light year (ly) Alternative unit of distance 1 Light Year is the distance traveled by light in one year. Relation to other units: 1 light year (ly) is equivalent to 0.31 pc 63,270 AU Used mostly by Star Trek, etc.
10 Examples α Centauri has a parallax of p=0.76 arcsec: A distant star has a parallax of p=0.02 arcsec:
11 Limitations If stars are too far away, the parallax can be too small to measure accurately. The smallest parallax measurable from the ground is about 0.01-arcsec Measure distances out to ~100 pc But, only a few hundred stars this close
12 Hipparcos satellite European Space Agency Launched in 1989 Designed to measure precision parallaxes to about ±0.001 arcseconds! Gets distances good out to 1000 pc Measured parallaxes for ~100,000 stars!
13 Future missions Space Interferometry Mission (SIM) Launch in about 2013 Goal of 4 microarcseconds Direct parallax to any star in our Galaxy Pointed observations of specific targets GAIA European mission; launch in about 2012 Roughly microarcseconds precision Measure every star in the Galaxy over roughly 5 years
14 In-class assignment The smallest angle that can be reliably measured for parallax purposes is about 0.01 second of arc. Suppose there are 0.08 stars/pc 3 observable near the Sun. How many stars, in principle, exist that could have their distances measured by the parallax method? If accuracy improved to second of arc (with Hipparcos, for example), how many stars would have measurable parallaxes?
15 How bright is an object? We must define Brightness quantitatively. Two ways to quantify brightness: Intrinsic Luminosity: Total Energy Output. Apparent Brightness: How bright it looks from a distance.
16 Luminosity Luminosity is the total energy output from an object. Measured in Power Units: Energy/second emitted by the object (e.g., Watts) Independent of Distance Important for understanding the energy production of a star.
17 Apparent brightness Measures how bright an object appears to be to a distant observer. What we measure on earth ( observable ) Measured in Flux Units: Energy/second/area from the source. Depends on the Distance to the object.
18 Inverse Square Law of Brightness The apparent brightness of a source is inversely proportional to the square of its distance: 2-times Closer = 4-times Brighter 2-times Farther = 4-times Fainter
19 d=1 B=1 d=2 B=1/4 d=3 B=1/9
20 Apparent brightness of stars Apparent brightness is what we measure. How bright any given star will appear to us depends upon 2 things: How bright it really is (Luminosity) How far away it is (Distance).
21 Appearances can be deceiving... Does a star look bright because it is intrinsically very luminous? it is intrinsically faint but located nearby? To know for sure you must know: the distance to the star, or some other, distance-independent property of the star that clues you in.
22 Flux-luminosity relationship Relates apparent brightness (Flux) and intrinsic brightness (Luminosity) through the Inverse Square Law of Brightness:
23 Measuring apparent brightness The process of measuring the apparent brightnesses of objects is called Photometry. Two ways to express apparent brightness: as Stellar Magnitudes as Absolute Fluxes (energy per second per area)
24 Magnitude system Traditional system dating to classical times (Hipparchus of Rhodes, c. 300 BC) Rank stars into 1 st, 2 nd, 3 rd, etc. magnitude. 1 st magnitude are brightest stars 2 nd magnitude are the next brightest and so on... Faintest naked-eye stars are 6 th magnitude.
25 Modern system Modern version quantifies magnitudes as: 5 steps of magnitude = factor of 100 in Flux. 10 th mag star is 100 fainter than 5 th mag 20 th mag star is 10,000 fainter than 10 th mag Computationally convenient, but somewhat obtuse.
26 Flux photometry Measure the flux of photons from a star using a light-sensitive detector: Photographic Plate Photoelectric Photometer (photomultiplier tube) Solid State Detector (e.g., photodiode or CCD) Calibrate the detector by observing a set of Standard Stars of known brightness.
27 Measuring luminosity In principle you just combine the brightness (flux) measured via photometry the distance to the star using the inverse-square law. The biggest problem is finding the distance.
28 d=1 B=1 d=2 B=1/4 d=3 B=1/9
29 Summary Distance is important but hard to measure Trigonometric parallaxes direct geometric method only good for the nearest stars (~500pc) Units of distance in Astronomy: Parsec (Parallax second) Light Year
30 Summary Luminosity of a star: total energy output independent of distance Apparent brightness of a star: depends on the distance by the inversesquare law of brightness. measured quantity from photometry.
31 Binary stars Apparent Binaries Chance projection of two distinct stars along the line of sight. Often at very different distances. True Binary Stars: A pair of stars bound by gravity. Orbit each other about their center of mass. Between 20% and 80% of all stars are binaries.
32 Types of binaries Visual Binary: Can see both stars & follow their orbits over time. Spectroscopic Binary: Cannot separate the two stars, but see their orbit motions as Doppler shifts in their spectra. Eclipsing Binary: Cannot separate stars, but see the total brightness drop when they periodically eclipse each other.
33 Visual Binary
34 Center of Mass Two stars orbit about their center of mass: a 2 a 1 M 2 a M 1 Measure semi-major axis, a, from projected orbit and the distance. Relative positions give: M 1 / M 2 = a 2 / a 1
35 Measuring masses Newton s Form of Kepler s Third Law: Measure Period, P, by following the orbit. Measure semi-major axis, a, and mass Ratio (M 1 /M 2 ) from projected orbit.
36 Problems We need to follow the orbits long enough to trace them out in detail. This can take decades. Need to work out the projection on the sky. Everything depends critically on the distance: semi-major axis depends on d derived mass depends on d 3!!
37 Spectroscopic binaries Most binaries are too far away to see both stars separately. But, you can detect their orbital motions by the periodic Doppler shifts of their spectral lines. Determine the orbit period & size from velocities.
38 B A B A B A A B
39 Problems Cannot see the two stars separately: Semi-major axis must be guessed from orbit Can t tell how the orbit is tilted on the sky Everything depends critically on knowing the distance.
40 Eclipsing binaries Two stars orbiting nearly edge-on. See a periodic drop in brightness as one star eclipses the other. Combine with spectra which measure orbital speeds. With the best data, one can find the masses without having to know the distance!
41 Eclipsing binary Brightness 1 2 Time 3 4
42 Problems Eclipsing Binaries are very rare Orbital plane must line up just right Measurement of the eclipse light curves complicated by details: Partial eclipses yield less accurate numbers. Atmospheres of the stars soften edges. Close binaries can be tidally distorted.
43 Stellar masses Masses are known for only ~200 stars. Range: ~0.1 to 50 Solar Masses Stellar masses can only be measured for binary stars.
44 Stellar radii Very difficult to measure because stars are so far away. Methods: Eclipsing binaries (need distance) Interferometry (single stars) Lunar Occultation (single stars) Radii are only measured for about 500 stars
45 Summary Types of binary stars Visual Spectroscopic Eclipsing Only way to measure stellar masses: Only ~150 stars Radii are measured for very few stars.
46 Questions What makes it necessary to launch satellites into space to measure very precise parallax? Would it be easier to measure parallax from Jupiter? From Venus?
47 Questions How much does the apparent brightness of stars we see in the sky vary? Why? Stars have different colors? So is the amount of light at different wavelengths the same? Can we tell the difference between a very luminous star that is far away and an intrinsically low luminosity star that is nearby?
48 Questions What star do we know the mass of very precisely? Why is it so unlikely that binaries are in eclipsing systems? Most binaries are seen as spectroscopic. Why? How can we know the sizes of more stars than masses?
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