optical / IR: photon counting flux density or magnitude corresponds to number of electrons per second (mean rate)
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- Erick Clarke
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1 optical / IR: photon counting flux density or magnitude corresponds to number of electrons per second (mean rate) N electrons/sec = ɛ F λ λa hc/λ 0 efficiency factor flux density x bandpass x collecting area mean energy per photon observe (integrate) for a time tint total number of electrons counted: N electrons = ɛ F λ λa hc/λ 0 t int integration time
2 optical / IR: photon counting N electrons = ɛ F λ λa hc/λ 0 t int this is the number of electrons expected for a given flux density passed through the telescope+detector system for a time tint expect that this count is uncertain by ± N electrons this is the standard deviation about the mean count Nelectrons (Nelectrons large-ish, >50 or so) expect actual count to lie within one standard deviation of the mean count ~67% of the time
3 optical / IR: photon counting if the star is the only source of photons (in general, it is not) count signal rises proportionally to N electrons = N electrons/sec t tint t int count noise rises proportionally to tint signal to noise ratio (S/N or SNR) rises proportionally to: t int tint = t int SNR given by: SNR = N electrons Nelectrons = N electrons
4 optical / IR: photon counting previous example: K = 22.0 mag star, (K filter, 8m telescope) calculated: we expect 116 photons/sec for efficiency factor of 0.3 (0.3 electrons per photon) expect 35 electrons/sec Question: how long to reach a SNR of 10? SNR of 10: need 100 electrons need integration time of 2.86 seconds Question: how long to reach a SNR of 100? SNR of 100: need electrons : 286 seconds
5 considering star(s) as only source of photons: observe (integrate) for a time tint total number of electrons counted: N electrons = ɛ F λ λa hc/λ 0 t int observe standard star(s) to establish calibration between electron ( counts per ) second ( and ) flux density Nelectrons hc/λ0 F λ = t int ɛ λa effectively calibrating efficiency factor can establish the 1 count per second zero point (flux density (or magnitude) of star yielding 1 electron per second) use calibration to convert observed electron counts per second into flux density for programme star(s)
6 easy to quote magnitudes: example: let a magnitude 0.0 calibrator star give 10 9 electrons/sec if we observe a star and find that it gives 10 electrons/sec: electron count rate is a factor of 10-8 lower flux density is a factor of 10-8 lower (same bandpass, collecting area, efficiency) magnitude given by flux density - magnitude relation: m m cal = 2.5 log 10 (F /F cal ) m cal = 0.0 log 10 (F /F cal ) = log 10 (10 8 ) = 8 m = ( 2.5) ( 8) = +20
7 F λ = ( Nelectrons quoting flux densities: ) ( ) hc/λ0 ɛ λa t int total electron count, with N electrons ± uncertainties N electrons electron rate, with uncertainties F λ N electrons t int electron rate N electrons ± N electrons t int N electrons/sec ± Nelectrons/sec t int t int uncertainty on electron rate falls 1/ t int uncertainty on flux density falls 1/ t int
8 quoting flux densities: example: if zero mag calibrator star has a flux density of 10-8 W m -2 micron -1 and corresponds to an electron count rate of 10 9 electrons/sec we measured a count rate of 10 electrons per second from a programme star say we observed for 1000 seconds to establish this rate (i.e. we counted electrons) count = rate = ( )/1000 = our example programme star therefore has a flux density ( )/(10 9 ) x 10-8 W m -2 micron -1 = ( ) x W m -2 micron -1 photometry here accurate to 1% how much longer for 0.1% photometry? (10x better = 100x longer integration time) need to integrate for 10 5 seconds useful: uncertainty on magnitude = x fractional uncertainty on flux density (approx, for small uncertainties) 1% approx mag uncertainty etc
9 optical / IR: photon counting the star is NOT the only source of photons! other sources: mainly sky: atmosphere space (cosmic background, zodiacal dust, interstellar medium, unresolved stars...) + telescope +detector dark current readout noise
10 measuring a star in practice 1. Measure total flux through aperture centred on star: includes star, background, dark current, read-noise 2. Measure average flux per pixel through larger annulus: measures mean background, dark current; usually annulus has larger area than aperture, so error small 3. Multiply average background / dark current flux per pixel by area of inner aperture Histogram of background flux: clearly a normal distribution 4. Subtract background / dark current flux from value measured through star aperture; leaves star flux alone, but with Poisson noise of star, background, and dark current, plus read-noise adapated from original slide by Mark McCaughrean
11 optical / IR: photon counting integrating stellar flux density over small 2D (angular) aperture need to determine flux density of background in this aperture use larger aperture, free of stellar light, to get a good measure of the background flux density IDIOTIC UNIT ALERT!!!! background flux density is usually quoted in equivalent magnitudes per square arcsecond since magnitudes are a logarithmic unit, this does not scale linearly with area e.g. 12 mag arcsec -2 corresponds to 12 mag in a 1 arcsec 2 aperture! (OK so far...) but: 12 mag arcsec -2 is not 24 mag in a 2 arcsec 2 aperture! (obviously, since 24 mag is smaller (in flux density terms) than 12 mag) 12 mag arcsec -2 is actually mag in a 2 arcsec 2 aperture! mag in aperture = mag per square arcsec 2.5 log 10 (aperture area/arcsec 2 )
12 background flux density take aperture used for star calculate flux density of background within this aperture convert this to number of electrons (per sec, or over tint) subtract background electrons from total electrons to find electrons generated by star s radiation OK so far, but the uncertainty on the number of electrons counted in the stellar aperture is given by sqrt(ne,tot) where Ne,tot is the total number of electrons counted (star + background) (should add to this the dark current electrons and contribution from readout but in most situations, this is a negligible amount) signal electron count is then N e, = N e,tot N e,bckg ( N e,dark ) noise on this electron count is then Ne,tot where N e,tot = N e, + N e,bckg (+N e,dark )
13 signal-to-noise with background signal electron count is N e, = N e,tot N e,bckg ( N e,dark ) noise on this electron count is Ne,tot where N e,tot = N e, + N e,bckg (+N e,dark ) have a measure of N e, /t int convert this to flux density or magnitude as before but signal-to-noise ratio given now by: N e, / N e,tot reduces to what we had previously if star is only source of photons ( N e, = N e,tot ) ( shot-noise-limited in this case) background-limited if N e,tot N e,bckg
14 signal-to-noise with background previous example: K = 22.0 mag star, yielding 35 electrons/sec assume that this count is achieved by integrating over 1 arcsec 2 aperture and assume no dark current or readout noise let (e.g.) background flux density be 12 mag arcsec -2 background provides electron count equivalent to 12 mag ( simple calculation for other aperture sizes...) 12 mag is brighter than 22 mag by a factor: 10 (12 22)/2.5 = 10 4 (i.e. flux density is larger by this factor; so too is electron count) electron count from background = 35 x 10 4 elec/sec = 3.5 x 10 5 elec/sec reference point: no background, SNR = 10 achieved in 2.86 seconds (time to accumulate 100 photons) with background, SNR calculation looks like: SNR = signal noise = N e, /sec t int Ne,tot/sec t int = 35 t int t int solve for tint when SNR = 10 : tint = 169 seconds (~60 x longer)
15 signal-to-noise with background with background, SNR calculation looks like: SNR = signal noise = N e, /sec t int Ne,tot/sec t int notice: again, as before, SNR increases proportionally to sqrt(tint) generally applies, since signal increases with tint, and noise increases as sqrt(tint), no matter whether the source of photons (electrons) is predominantly star or sky or dark (although there is an exception to this: READOUT noise, which does not depend on tint but will contribute a variable number of electrons per readout) straightforward to convert electrons/sec for star into flux density (as before) now use full noise budget to calculate SNR and therefore uncertainty on flux density e.g. flux density is F, SNR is 10, then uncertainty is F/10 again, obvious that uncertainty in F will fall 1/ t int
16 optical / IR: photon counting star+background SNR given by: SNR = signal noise = N e, /sec t int Ne,tot/sec t int N e, /sec F λ SNR F λ tint to achieve same SNR on two stars with different flux densities t int,1 t int,2 = ( ) 2 Fλ,2 F λ,1 integration times to specified SNR go as inverse flux density squared magnitude difference of 1 corresponds to flux ratio of ~6.31 times longer to observe (e.g.) K=23 than K=22 to same SNR
17 back to the Main Sequence (and beyond...) main sequence - range of masses: low mass: low temperature, low luminosity high mass: high temperature, high luminosity relations between M, L, R, T L,R,T we know: L R 2 T 4 need to relate mass to these... approximate proportionalities for Main Sequence R M 0.8 L M 3.5 can also find (by substitutions of above R-M and L-M relations) that: L R 2 T 4 M 1.6 T 4 L 1.6/3.5 T 4 L 1 1.6/3.5 T 4 L T 4/(1 1.6/3.5) L T 7.4
18 distribution of stars in H-R Diagram (or CMD) Wikipedia 643 Prof Mark McCaughrean PHY2019 Observing the Universe
19 Evolution of the Sun Main Sequence Longest phase, ~11 Gyr, ~ 90% of Sun s life Hydrogen fused into helium in core via p-p chain Helium ash accumulates in core Temperatures not high enough to ignite helium fusion Hydrogen burning core Non-burning envelope 1.4 x 10 6 km ~ 0.01 AU T ~ 6000 K L = 1 L 644 Prof Mark McCaughrean PHY2019 Observing the Universe
20 Evolution of the Sun Young red giant phase After ~ 11 Gyr, all H burned; core fusion stops Core still not hot enough to burn helium; core shrinks Pressure increases enabling H to burn in shell around core New energy causes star to bloat out to much larger radii Surface is cooler but overall luminosity much larger Non-burning helium core Hydrogen burning shell 1.5 x 10 8 km ~ 1 AU T ~ 3500 K L ~ 1000 L Non-burning envelope 645 Prof Mark McCaughrean PHY2019 Observing the Universe
21 Evolution of the Sun Helium flash After ~ 2 Gyr, core rises to ~100 MK: He burns to C and O Core degenerate: as temperature skyrockets, pressure cannot increase; He burns furiously for few hours When core reaches 3.5 x 10 8 K, He reverts to normal gas Normal pressure release operates: runway He burning stops Helium burning core Hydrogen burning shell 1.5 x 10 8 km ~ 1 AU T ~ 3500 K L ~ 1000 L Non-burning envelope 646 Prof Mark McCaughrean PHY2019 Observing the Universe
22 Evolution of the Sun Late stages of red giant phase Expanding core cools H burning shell; fusion there slows Reduced outward energy flux cools outer layers; they shrink Star decreases in radius; adopts new hydrostatic equilibrium Core burns He to C and O ash ( triple alpha process ) Helium burning core with C & O ash Non-burning He shell Hydrogen burning shell 7 x 10 7 km ~ 0.5 AU T ~ 3000 K L ~ 200 L Non-burning envelope 647 Prof Mark McCaughrean PHY2019 Observing the Universe
23 Evolution of the Sun End of red giant phase He runs out in core; core contracts But does not reach 600 MK required to burn C and O Outer layers cool and become neutral; effectively trap heat Layers expand and are blown off into space Non-burning C & O core He burning shell Non-burning He shell Planetary nebula Lifetime ~ yrs Diameter ~ pc 7 x 10 7 km ~ 0.5 AU Hydrogen burning shell Non-burning envelope 648 Prof Mark McCaughrean PHY2019 Observing the Universe
24 NGC 7923 Helix planetary nebula 649 NASA / ESA / O Dell
25 Evolution of the Sun White dwarf Roughly half of mass lost; remaining core contracts Solar-type star never achieves 600 MK for C & O burning C & O core is all that remains Contraction continues until electrons become degenerate Density is then ~ 10 6 higher than that of Sun: 1 tonne cm 3 Surface reaches K; then simply cools down forever Remnant core supported by electron degeneracy pressure ~ km ~ size of Earth Maximum white dwarf mass is 1.38 M (Chandrasekar limit) Above this, gravity wins over degeneracy pressure Collapses and explodes as Type Ia supernova: important! 650 Prof Mark McCaughrean PHY2019 Observing the Universe
26 post main sequence tracks for the Sun: image: ATNF/CSIRO
27 post main sequence tracks other stars: image: ATNF/CSIRO
28 post main sequence tracks other stars: stars evolve off the main sequence (initially) into red giant /supergiant region of H-R diagram image: ATNF/CSIRO
29 location/structure of main sequence in CMDs is useful stars are born in clusters if young enough stars have not had time to evolve off the main sequence
30 Schematic Main Sequence fitting When sequences matched, MV = 0.0 on Main Sequence lines up with V=5.5 in Pleiades (m M) = 5.5 = 5 log10(d) 5 d = 10 ((5.5+5)/5) d = 126 pc (Actual distance = 135 pc) Wikipedia 655 Prof Mark McCaughrean PHY2019 Observing the Universe
31 location/structure of main sequence in CMDs is useful as clusters get older, stars move off the MS highest mass stars first, lower mass stars later (remember MS lifetime falls as (M/Mo) -2.5 )
32 location/structure of main sequence in CMDs is useful as clusters get older, stars move off the MS highest mass stars first, lower mass stars later (remember MS lifetime falls as (M/Mo) -2.5 ) can get distance and age of cluster (models of MS evolved to range of ages)
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