3. Observed Fundamental Properties of Stars

Transcription

1 3. Observed Fundamental Properties of Stars Magnitude, colour, luminosity Spectra and spectral types Velocities and distance Radius Mass Rotation Magnetic field

2 3.1.1 Elements of astronomical photometry Solid angle (*) ω dθ Spherical coordinates: θ, φ, r Surface element da on the sphere θ da = r 2 sinθ dθ dφ da corresponds to dω = sinθ dθ dφ dω Unit of ω: sr (steradian) 1 sr = solid angle of a surface element of the size A = r 2 r Solid angle of complete sphere: π 2π 0 0 Ω [sr] = sinθ dθ dφ = 4π (*) dt: Raumwinkel dφ sin θ φ ω [sr] = A / r 2 da dφ

3 3.1.1 Elements of astronomical photometry Intensity I Consider a radiation flow passing a surface element da dω = sinθ dθ dφ Part of the flow propagates in direction θ. The energy flowing through da in direction θ - per time interval dt - per frenquency interval dν - per solid angle dω (therewith through da') is given by projection of da in direction θ: da' = da cosθ deν = I ν da' dt dν dω = I ν da cosθ dt dν dω where Iν specific intensity [J m -2 s -1Hz -1sr -1] I = I ν dν 0 intensity [J m-2 s -1 sr -1] da θ

4 3.1.1 Elements of astronomical photometry Flux density (flux) Φ = radiation power per surface element = energy per time, per surface area, integrated over all solid angles monochromatic Φ ν = Ω integral de da dt dν dω = Iν cosθ dω [J m -2 s -1 Hz-1 or W m -2 Hz -1 ] Ω Φ = I ν cosθ dω dν [J m-2 s -1 or W m -2 ] ν Ω Remark on units: Observed flux densities usually very small -> standard units not handy suitable unit (especially in radio astronomy): Jansky 1 Jy = W m-2 Hz -1

5 3.1.1 Elements of astronomical photometry Flux density (flux) Φ Special cases: (a) Isotropically radiating source (e.g. star) in the outer space If there are no other radiation sources or sinks, the energy E per solid angle ω is conserved E / ω = const i.e., E r 2 / A = const As Φ = E / A, we get Φ ~ r 2 In the outer field of an isotropic source, the flux density decreases with the the square of the distance. r A 2r 4A

6 3.1.1 Elements of astronomical photometry Flux density (flux) Φ Special cases: r (b) Within an isotropic radiation field (e.g. inside star) I = const (independent of direction) π Φ- θ 2π Φ = I cosθ dω = I sinθ cosθ dθ dφ 0 Ω 0 0 Φ+ In an isotropic radiation field, the net flux density is Φ 0. π/2 2π Outward flux: Φ+ = I sinθ cosθ dθ dφ = π I Inward flux: 0 π 0 2π π/2 0 Φ = I sinθ cosθ dθ dφ = π I Φ = - Φ +

7 3.1.2 Apparent magnitudes Apparent magnitude m Definition Or, alternatively m = log 10 Φ + const. m 1 - m 2 = log10 (Φ 1 / Φ 2) Φ measured from Earth, but corrected for atmospheric extinction Remarks: 1. The negative sign magnitude system dates back to Hipparchus of Nicaea (ca BC): - bightest stars are 1st class = magnitude 1 - stars barely visible to the (naked) eye are of 6th class = magnitude 6 i.e., the brighter the star, the smaller the value of the brightness (Note: historical misconception: brightness = bigness, size) 2. The logarithmic scale based on the fundamental law of psycho-physics (Weber-Fechner law, 1859): Relation between the differential change in perception (dt. Empfindung) dp and the differential increase ds in stimulus (dt: Reiz) dp ~ ds/s p ~ log S

8 3.1.2 Apparent magnitudes 3. The scaling factor 2.5 From experiments with artificial stars (Pogson 1856): m 6- m 1= 5 corresponds to Φ 1 / Φ 6 = 100 Φ m / Φ m+1 = 100 1/5 = 10 2/5 m (m+1) = a log Φm / Φm+1 = a log 10 2/5-1 = 25 a a = The normalisation constant - Vega system: Vega (α Lyr) has magnitude m=0 (in all filter bands) * - AB system: zero-point flux density Φ ν (m=0) = 3631 Jy mab = -2.5 log (Φν /3631 Jy) (*) However, Vega is found to vary in brightness, and other standards are in use as well

9 3.1.2 Apparent magnitudes 5. Examples m Sun Moon (full) ISS (max) -6 Venus (max) -5 Sirius -1.5 Limit for naked eye ~6 Limit for binoculars ~10 Faintest observable objects (e.g. HST) ~30 Cosmic sources have an enormous range of brightness! Δm = 57 mag corresponds to Φ /Φ = 10 57/2.5 = Fluxes Φ cover a range of (at least) ~23 orders of magnitude.

10 3.1.2 Apparent magnitudes Surface brightness μ [mag arcsec -2 ] = apparent magnitude related to radiative power per solid angle μ ~ log (Φ A) (A: area corresponding to solid angle ω) μ does not depend on distance r since E /ω is independent of r Useful for extended objects (e.g. nebulae, star clusters, galaxies,...)

11 3.1.2 Apparent magnitudes Measuring magnitudes Relative or absolute photometry Relative photometry measures Φ* (or a property dependent on Φ ) relative to a calibration object C of known apparent magnitude m = m c on the same exposure m* = mc log Φ* / Φ c Absolute photometry measures the absolute value of the flux Φ (to be corrected for atmospheric extinction) m* = log Φ* + const.

12 3.1.2 Apparent magnitudes Measuring magnitudes 100 State of the art: CCD (=charge-coupled device) detector * Old standard detector (until 1980/1990s): photographic emulsion (usually on glas plates) Quantum efficiency [%] Types of detectors CCD (thinned) 10 photographic emulsion 1 eye 0.1 * Invented in 1969 by W. Boyle and G. Smith; Nobel Prize in Physics Wavelength of radiation [μm] 1.1 Quantum efficiency: percentage of photons incident on detector which produce measurable signals

13 3.1.2 Apparent magnitudes CCD detectors Principle: 2-dim. array of light-sensitive silicon diodes ( = pixel, ca μm) photon hitting the pixel can release an electron (inner light-electric effect) electron remains trapped by the (outer) potential (positive voltage) applied to the gate electrodes Properties: linear, stable, and efficient, yet expensive high quantum efficiency (>90%), particularly for thinned, backside-illuminated chips dark current caused by thermal noise has to be removed by cooling different sensitivity of infdividual pixels ( flatfield correction) Mosaic of 8 x (2k x 4k) CCDs of the 8m Subaru telescope

14 3.1.2 Apparent magnitudes CCD detectors Reading out the CCD detector after exposure, the accumulated charges are shifted parallel, row by row, to the serial shift register varying potential differences are used to move the charges from the serial shift register the charges are moved pixel by pixel to the analog-digital converter sensitivity is strongly affected by readout noise of the electronics ( long readout times) A/D

15 3.1.2 Apparent magnitudes CCD detectors Reading out the CCD detector Array of buckets Co n (se veyo ria r b l s elt hif t) "Imagine an array of buckets covering a field. After a rainstorm, the buckets are sent by conveyor belts to a metering station where the amount of water in each bucket is measured. Then a computer would take these data and display a picture of how much rain fell on each part of the field. In a CCD the raindrops are photons, the buckets are pixels." Arnold, H.J.P. Astrophotography: An Introduction to Film and Digital Imaging. Firefly Books, 2002 Conveyor belt (parallel shift) metering station

16 3.1.2 Apparent magnitudes CCD detectors Raw CCD frame (TLS 2m Alfred Jensch Telescope) Saturation and blooming CCD structure (hot pixels) Cosmic CCD structure (cold pixels) CCD structure (cold pixels) Background noise

17 3.1.2 Apparent magnitudes CCD detectors Corrected CCD frame (TLS 2m Alfred Jensch Telescope)

18 3.1.2 Apparent magnitudes Relative photometry on CCD frames Detection (automated) of objects on the reduced CCD frame Integration of the pixel values within properly defined apertures around the detected objects (see Figure) Transforming the integral pixel values into internal magnitudes Determining the calibration curve (internal mags standard mags) by means of calibration stars from existing catalogues (see Figure) Using the calibration curve to calibrate all detected objects calibrated magnitude internal magnitude

19 3.1.3 Colours Both the Earth atmosphere and the observational instruments possess spectral characteristics ( response functions ) atmosphere A(λ) Measured flux F(λ) is not identical with the incident flux Φ(λ) at the top of the atmosphere λ2 F (λ1,λ 2) = Φ(λ) A(λ) T(λ) D(λ) dλ λ 1 λ2 = Φ(λ) G(λ) dλ λ telescope optics T(λ) detector D(λ) 1 G(λ) = A(λ) T(λ) D(λ) photometric weight (response) function Any observation has a colour characteristic that is determined by G(λ) and the λ range. Any magnitude measurement is related to a certain colour system.

20 3.1.3 Colours Broad-band multi-colour systems (a) UBV system (Johnson system) Remarks: Originally U, B, and V only (Johnson & Morgan, ~1950) B ~ photographic plate V ~ human eye later continued into the red and IR: filter bands R, I, J,H,K,L,M U,B,V,... denote both the λ band and the apparent magnitude in that band, i.e. U = m U Remember: in the Vega magnitude system, U = B = V = = 0 for Vega Band U B V R I width (nm) λ eff (nm) U B V R I

21 3.1.3 Colours Broad-band multi-colour systems (b) Sloan ugriz system (SDSS system *) Remarks: Dense wavelength coverage without much overlap As in the UBV system, u, g, r,... denote both the λ band and the apparent magnitude in that band, i.e. g = mg Fig. (right): total response function G(λ) for the Sloan bands. (Dashed: without atmospheric extinction, solid: with atmospheric extinction at l=30 ) * SDSS = Sloan Digital Sky Survey Band u g r i z λ (nm) u g r i z

22 Excurs: colour imaging with CCD observations Filterband 1 Filterband 2 Filterband 3 Usually CCDs provide grey scale images only To produce a colour image, several images are needed taken through different filters.

23 Excurs: colour imaging with CCD observations Filterband 1 Filterband 2 Filterband 3 Usually CCDs provide grey scale images only To produce a colour image, several images are needed taken through different filters. These images must be colour-coded... and then combined to e.g. a RGB image.

24 Excurs: colour imaging with CCD observations Combined RGB image Colour imaging makes sense because colours transport relevant physical information, namely on the spectral energy distribution.

25 Excurs: colour imaging with CCD observations Practical realization: CCD detector combined with spectral filter (colour filter) u g r i z Focal plane of the 2.5 m telescope of the Apache Point Observatory (new Mexico, USA) with an array of 5 x 6 CCDs with 5 different colour filters for the Sloan Digital Sky Survey. (Credits: SDSS)

26 3.1.3 Colours Colour index difference of the magnitudes (of the same object) in two different bands, m1 - m 2, where λ 1 < λ 2, e.g. (U-B), (B-V) Two colour index diagram (colour-colour diagram) (U-B) vs. (B-V) interstellar reddening blackbody U-B well defined relation between the colour indexes most stars along s-shaped curve roughly described by blackbody curve some stars are away from the mean relation due to interstellar reddening U-B B-V blackbody B-V

27 3.1.3 Colours Interstellar extinction A interstellar reddening U-B Scattering by interstellar dust grains is λ-dependent ( reddening ) changes the colour indexes (CIs) Colour excess = observed CI original (unreddened) CI e.g. E(B-V) = (B-V) (B-V) 0 = (B-B 0) - (V-V0) = A B- A V blackbody where A B= B-B 0, A V = V-V0 is the amount of extinction in B, V We define a normalized extinction at wavelength λ: E (λ-v) Aλ - A V F = = E (B-V) A B- A V F -3 for λ (i.e. for 1/λ 0) On the other hand, we expect A λ 0 for λ -AV Fλ = A - A -3 B V AV 3 E(B-V) and and therewith A λ [3+F(λ)] E(B-V) 4 F(λ) Observed extinction curve for Galactic dust (see Fig.): B-V / λ [μm] 4 5 Observed normalized exctinction curce (dots) and model curve for water ice particles (following Wickramasinghe)

28 3.1.3 Colours Example for very strong interstellar extinction: Bok Globule 68 Visual NIR Dense cloud of cold gas and dust in the constellation Ophiuchus. In the visible light (left), the light extinction is such strong that there seems to be a void in the heavens. Infrared light (right) penetrates the cloud (right), but the light is strongly reddened (i.e. light of short wave-lengths is much stronger suppressed than long wavelengths). Credits: FORS Team/VLT Antu/ESO; ESO] The visual extinction A v through the cloud as a function of the centrum distance r. ir/ir_results.html

29 3.1.3 Colours Blackbody radiation Blackbody = object that completely absorbs (and subsequently re-emits) all infalling radiation (i.e. no reflection or scattering). Illustrated by a (nearly) closed cavity where the walls and the radiation are in thermal equilibrium Corresponds to an ideal thermodynamic equilibrium. The re-emitted radiation depends only on the temperature T (thermal radiation). Spectral intensity distribution (Planck law) B: Intensity h: Planck const c: speed of light k: Boltzmann const. or, with Bν dν = Bλ dλ (energy conservation) and ν = c/λ 2hν 3 Bν (T) = c2 B λ(t) = 2hc 2 λ5 1 exp (hν/kt)-1 Intensity B λ 1 exp (hc/λkt)-1 Wavelength λ [nm]

30 3.1.3 Colours Blackbody radiation Approximations of the Planck formula B λ(t) = (a) short wavelengths hc/λkt 1 2hc 2 λ5 1 exp (hc/λkt)-1 (λ λ max) 2hc 2 B λ(t) = λ5 e -hc/λkt (Wien's approximation)

31 3.1.3 Colours Blackbody radiation Approximations of the Planck formula B λ(t) = (b) long wavelengths hc/λkt 1 1 exp (hc/λkt)-1 (λ λ max) e hc/λkt 1 + hc/λkt B λ(t) = 2hc 2 λ5 2kcT λ4 (Rayleigh-Jeans approximation) Remark: classical theory provides only the Rayleigh-Jeans approximation where the intensity grows however for short λ to infinity ( ultraviolet catastrophe ) Rayleigh-Jeans approximation for T = 5000 K

32 3.1.3 Colours Blackbody radiation Consequences of Planck's law (a) Flux density of blackbody radiation total intensity: B(T) = Bν dν = B λ dλ setting x = hν /kt yields 4 x dx B(T) = 2hk T c h ex - 1 = const. = at 4 = const. For an isotropic radiation field, the intensity I is related to the flux density Φ via Φ = π I = π B (see Sect ) Hence, Φ = σ T 4 with σ = const (Stefan-Boltzmann law) As the stellar radiation can be approximated by the radiation of a blackbody, the Stefan-Boltzmann law provides an important relation between the flux density and the temperature.

33 3.1.3 Colours Blackbody radiation Consequences of Planck's law (b) Wavelength λ max of the maximum intensity set db λ / dλ = 0 and solve for λ λ max T = b with b = K m (Wien's dispacement law) Flux density Procedure: Blue stars are hot, red stars are cool. hot star B-V < 0 hot Flux density Flux density solar type star B-V 0 cool star B-V > 0 cool Wavelength λ [nm]

34 3.1.4 Absolute magnitudes and luminosity The apparent magnitude depends on the distance r of the star. Now we define a magnitude that is independent of the distance and reflects thus the intrinsic radiation power of the star. Absolute magnitude M = apparent magnitude at a distance of 10 pc from the source (*) m corresponds to Φ 1 = Φ(r) M corresponds to Φ2 = Φ(10pc) As Φ~ r -2 (Sect ), we have Φ1 10 pc 2 =( ) Φ2 r and with the magnitude definition (Sect ) we get m - M = -2.5 log (10/r[pc]) 2 = 5 log r[pc] -5. Finally, taking the interstellar extinction A (Sect ) into account: m - M = 5 log r[pc] -5 + A (*) 1 pc (parsec) = m (see Sect. 3.5) distance modulus

35 3.1.4 Absolute magnitudes and luminosity Luminosity L = total flux of a source, i.e. flux density integrated over a concentric sphere around the source (i.e. the total power output) L = Φ(r) 4π r 2 (Note: L is independent of r!) The luminosity is related to the absolute magnitude: Assume that Φ(10) Φ (10) is the flux of a star at a distance of 10 pc from the star is the flux of the Sun at a distance of 10 pc from the Sun Φ(10) L / 4π (10pc) = Φ (10) L / 4π (10pc) With the definition of magnitudes (Sect we can wright): Φ(10) M - M = -2.5 log Φ (10) Thus we get M - M = -2.5 log L L

36 3.1.4 Absolute magnitudes and luminosity Bolometric magnitude, bolometric correction Usually, luminosity means the total flux integrated over all wavelengths) flux. The magnitude that corresponds to the integrated flux is called bolometric magnitude m bol, M bol Therefore M bol - M bol, = -2.5 log L L The difference between M bol and M V is the bolometric correction BC = M bol - M V (i.e. BC < 0, as M bol < M V ) BC depends on the spectral energy distribution and is tabulated as a function of the colour index. The luminosity can be computed when the apparent magnitudes in two bands and the distance are known. (Check it!) Figure: Relative BC for blackbodies (Weidemann & Bues 1967; ZfA 67, 415)

37 3.1.4 Absolute magnitudes and luminosity Colour - (absolute) Magnitude Diagram Not all possible combinations of M and (B-V) realized Most stars along diagonal sequence = Main Sequence (MS) Other concentrations above and below the MS MV Mai ns equ enc e B-V