Luminosity, Flux and Magnitudes

Transcription

1 Luminosity, Flux and Magnitudes Relativity and Astrophysics Lecture 13 Terry Herter Outline Blackbody Radiation Flux and Luminosity Inverse square law Magnitudes A90-13 Flux and Magnitudes A

2 Radiation rom objects All objects have internal energy which is maniested by the microscopic motions o particles. There is a continuum o energy levels associated with this motion. I the object is in thermal equilibrium then it can be characterized by a single quantity, it s temperature. An object in thermal equilibrium emits energy at all wavelengths. resulting in a continuous spectrum We call this thermal radiation. A90-13 Flux and Magnitudes 3 Blackbody Radiation A black object or blackbody absorbs all light which hits it. This blackbody also emits thermal radiation, e.g. photons! Like a glowing poker just out o the ire. The amount o energy emitted (per unit area) depends only on the temperature o the blackbody. In 1900 Max Planck characterized the light coming rom a blackbody. The equation that predicts the radiation o a blackbody at dierent temperatures is known as Planck s Law. B 3 h c 1 exph kt 1 W/m /Hz/sr h = Planck s constant, k = Boltzmann s constant, c = speed o light This is the power radiated per unit area in the requency range to + d into a unit solid angle (d = d sin d in spherical coordinates) A90-13 Flux and Magnitudes 4 A90-13

3 6 Blackbody Spectra 7000 K The peak shits with T (Wien s law). peak 1/T Intensity K 5000 K The area under the curve increase rapidly with T (Stean-Boltzmann law). 4 F T UV VISIBLE IR = 5.7x10-8 Wm - K -4 Object Person Sun Neutron Star T (K) peak 9.4 m 0.5 m 9 A F (W/m ) L (W) ~ Notes L is net radiation R sun = m R = 1,000 m A90-13 Flux and Magnitudes 5 Properties o Blackbodies The peak emission rom the blackbody moves to shorter wavelengths as the temperature increases (Wien s law). peak 900 /T in m and T in K Hot objects look blue, cool ones look red The hotter the blackbody the more energy emitted per unit area at all wavelengths (Stean-Boltzmann law). 4 F T = 5.7x10-8 Wm - K -4 Note - bigger objects emit more radiation Except or their suraces, stars behaves as a blackbodies A90-13 Flux and Magnitudes 6 A

4 Energy Flux and Luminosity The Energy Flux, F, is the power per unit area radiated rom an object. 4 F T W/m (at all ) The units are energy, area and time. Luminosity is the total energy radiated rom star o radius R is given by: So the luminosity, L, is: L 4 4 R T Watts I stars behave like blackbodies, stars with large luminosities must be very hot and/or very big. A90-13 Flux and Magnitudes 7 Luminosity and Flux Luminosity, L The total energy radiated rom an object per second. Measured in Watts Emitted Flux, F The low o energy out o a surace. Measured in Watts/m Observed lux, The power per unit area we receive rom an object Depends on the distance to the object. Measured in W/m e.g. sun = 1 kw/m Also called lux or apparent brightness Meaning o Observed Flux Make a sphere o radius, r, around an object (such as the Sun or a light bulb) which is radiating power. All energy radiated rom the object must pass through this sphere The size o the sphere does not matter! A90-13 Flux and Magnitudes 8 A

5 Flux alls o with distance r r r All energy radiated rom the object must pass through each sphere -- The size o the sphere does not matter! The low o energy per square meter passing through a give sphere decreases as the size o the sphere increases. 1/r. A90-13 Flux and Magnitudes 9 Flux: Sprinkler and Bucket Analogy A I bucket B is twice as ar away as bucket A, it collects 1/4 as much water. B The sprinkler is the star, one o the buckets is the telescope (or your eye), and the water jets are the photons. A90-13 Flux and Magnitudes 10 A

6 Inverse square law The lux,, o energy through a sphere o radius, r, is given by L 4 r ( W/m ) where L is the luminosity o the object Inverse square law Why do we care about lux? The lux is what we measure. We use a telescope (or our eye) and measure a small raction o the light passing through this sphere. A90-13 Flux and Magnitudes 11 An illuminating example? A 100 W light bulb about 1/5 o power goes into light It s total power output is always 100 W. It s apparent brightness to us depends upon how ar away it is. For instance at 1 m the lux is: Flux = 0.08 W/m [ = 100 W/(41m) ) W/m ] I we double the distance away rom the light bulb, the lux drops by a actor o 4. At m, the lux is 0.0 W/m A90-13 Flux and Magnitudes 1 A

7 Observed Flux Distance Example: A star like the sun has an observed lux o.4x10-10 W/m. I the lux o the sun at the Earth is 1 kw/m, how ar away is the star? 11 4 d m 1000 W/m L sun Now d Lsun / 4 L sun L sun d 310 W/ W/m d W m 10pc We could also have used ratios rather than compute L sun irst. d d star sun L sun sun L star star sun star d d star sun sun star Where we know everything but d star. A90-13 Flux and Magnitudes 13 Distances to stars: Stellar Parallax 1 AU Earth Now s d s d tan p d p p d s p Sun 6 Months Later p = parallax (angle) d = distance As stars get urther away, their parallax becomes smaller. Parallax can not be measured to better than ~0.0 rom the ground (d < 50 pc). Intererometry is improving on this or selected applications Parallax is measured in arcseconds. Equations are or distances in AU and parsecs (pc), respectively d(au) 0665 p(arcsec) or 1 d( pc) 1.0 arcsec => 1 pc, p 0.5 arcsec => pc A90-13 Flux and Magnitudes 14 A

8 The closest stars Star Parallax Distance Luminosity (arcsec) (pc) (L sun =1) Proxima Centauri x10-5 Centauri A Centauri B Barnard s Star x10-4 Wol x10-5 Lalande x10-3 Sirius A Sirius B x10-3 Parallax is motion o a star on the sky due to the Earth orbiting around the Sun. 1 parallax corresponds to 1 pc. The more distant a star the smaller the parallax. (1 pc = 3.6 lyr) 1 d(pc) p(arcsec) now 6 months later A90-13 Flux and Magnitudes 15 Current status The Hipparcos Satellite ( ) Astrometry mission, produced two catalogs Hipparcos catalog: ~10,000 stars Measured parallaxes to better than 0.00 => d < 500 pc Tycho catalog: ~ 1,100,000 stars Measured parallaxes and proper motions to ~ 0.05 (40 pc) Tycho catalog:,500,000 stars Update version o Tycho catalog Reprocessed raw Tycho data & used 144 other catalogs to obtain proper motions Proper motions to /yr Parallaxes are the key to knowing distances in the universe. Nearby stars are the stepping stone to measuring distance to everything else in the universe. We can now compute the luminosity o stars! A90-13 Flux and Magnitudes 16 A

9 Magnitudes We would like a way o speciying the relative brightness o stars Hipparchus Devised a the magnitude system 100 years ago to classiy stars according to their apparent brightness. He labeled 1080 stars as class 0, 1, was the brightest, 1 the next brightest, etc. The magnitude scale is logarithmic. An increase in magnitude by.5 means an object is a actor o 10 dimmer, e.g. a 0 mag star is 10 times brighter than a.5 mag star. a 0 mag star is 100 times brighter than a 5 mag star. A90-13 Flux and Magnitudes 17 Example magnitudes Star Sun Sirius Canopus Arcturus Vega Betegeuse Altair Deneb m v Designations CMa Car Boo Lyr Ori Aqu Cyg A dark adapted person with good eyesight can see to ~ 6 th magnitude. Hubble Space Telescope can observed objects ainter than 30 mag. 4x10 9 times ainter than the eye! A90-13 Flux and Magnitudes 18 A

10 Fluxes and Magnitudes Flux is the power per unit area received rom an object, e.g. sun = 1 kw/m I two stars, A and B, have luxes, A and B, their magnitudes are related by m A m B.5log( A / B ) Thus i B / A = 10, then m A - m B =.5 We can also write the inverse relation B A 10 m A m B.5 So that i m A = 5 and m B = 0, B / A = 100. A90-13 Flux and Magnitudes 19 Absolute & Bolometric Magnitudes m v apparent magnitude How bright a star appears in the sky. M v absolute magnitude Brightness i the star were at 10 pc This is an intrinsic property o the star! M absolute bolometric magnitude Brightness at ALL wavelengths (and 10 pc). To get M v or M we must know the distance to the star. Example: Suppose a star has m v = 7.0 and is located 100 pc away. It is 10 times the standard distance, thus, it would be 100 times brighter to us at the standard distance. Or 5 magnitudes brighter => M v =.0 A90-13 Flux and Magnitudes 0 A

11 Example Absolute Magnitudes Object m V M V Sun: Full Moon: -1.6 (3) Sirius: Canopus: Arcturus: Deneb: A90-13 Flux and Magnitudes 1 The Distance Modulus Equation The relation between m v and M v is written in equation orm as: m v -M v = log 10 ( d ) (d in pc) m v -M v is called the distance modulus. Examples: Deneb: m v = 1.6 and is 490 pc away. m v - M v = log 10 ( d ) M v = log 10 ( 490 ) = -8.5 => M v = -7. Sun: m v = -6.8, d = 1 AU M v = log 10 ( 1/0665 ) => M v = 4.8 A90-13 Flux and Magnitudes A

12 Bonus: Deriving the S-B Law We get the Stean-Boltzmann law by integrating the Planck unction over all requencies (area under the curve) Let h x kt 3 h d B B d c exph kt h kt B c h 4 kt h 0 x 3 x dx exp 1 x & kt d dx h 4 B T W/m /sr The total power emitted rom a surace is proportional to the temperature to the ourth power just as the S-B law The constant o proportionality is not quite the S-B constant because we need to integrate over all solid angles to get it (which gives an additional actor o ). The integral is related to the Riemann zeta unction giving k c h W/m /K 4 A90-13 Flux and Magnitudes 3 Another orm o Planck unction Let s rewrite the Planck unction in terms o power per unit wavelength rather than requency interval Note that c c B d Bd & d d where B is over the interval to + rather than to +. This relationship must be true since the integral over requencies and over wavelengths much be the same. Substituting gives 3 d h B B d c exp 1 c h kt 1 Which yields or B hc B 5 1 W/m /m/sr or W/m /m/sr exphc kt 1 Note that the units are now per m rather than per Hz. A90-13 Flux and Magnitudes 4 A

13 Bonus: Deriving Wien s Law To ind the peak we dierentiate with respect to and setting this equal to zero to get the peak db 1 5 exp hc hc kt hc 0 5 exp 1 exp 1 d hc kt hc kt kt Letting x = hc/kt and solving iteratively solving or x, x Putting back into the equation deining x T hc xk x 5 1 exp J -s310 m/s 10 μm J/K m 1 hc exp hc kt kt x T 900 μm K which is Wien s law. Note that I we had solved or the peak in B (rather than B ) this would yield, x =.8 and the orm o Wien s law is T 5100 μm K B space Both orms is correct since the peak is dierent between B and B space. A90-13 Flux and Magnitudes 5 A