More general definitions on magnitudes and colors Monochromatic magnitude of a source:

1. ASTRONOMICAL PHOTOMETRYII - STELLAR APPLICATIONSG. COMTELaboratoire d’Astrophysique de MarseilleFebruary 2010

2. More general definitions on magnitudes and colors Monochromatic magnitude of a source: Let fl(l) be the flux density of a source in energy units per wavelength interval, (erg.cm-2s-1.A-1). The general expression of the monochromatic magnitude is : ml (l) = - 2.5 log10 fl(l) - 21.100 (this system is named « STMAG ») If we consider the energy density per unit frequency f n(n) , expressed in erg.cm-2s-1.Hz-1, the corresponding definition of the monochromatic magnitude is : mn (n) = - 2.5 log10 fn(n) - 48.598 (this system is named « AB ») (Oke, 1965) (Both systems are based on the absolute flux of Vega at 5550 A)

3. (from O’Connell)

4. Absolute flux calibration based on Vega (alpha Lyr) Vega has an apparent magnitude V = 0.03 Monochromatic flux densities for a V = 0 star of spectral type A0 V at 5450 A : 3.63 10-9 erg.cm-2.s-1.A-1 3.63 10-20 erg.cm-2.s-1.Hz-1 1005 photons.cm-2.s-1.A-1 (more on these in O’Connell Lecture 14 in ASTR 511 course on the web)

5. Spectral types of main sequence stars (dwarfs, luminosity class V) Ca+ Hd Hg Hb Mg0 Na0 Ha O5 B0 B5 A1 A5 F0 G0 K0 K5 M0 M5 u.v. violet blue green yellow red near i.r. T* / TSun He0 He0 He+ He+ He0 Ca0 CH TiO TiO TiO TiO TiO

6. « Blanketing » by absorptions in stellar atmospheres strongly affect the colors 4500 K BB

7. The « effective » temperature of a star Theeffective temperature of a star of radius R* is defined as the temperature of the blackbody of same radius that emits the same total flux. L * = 4 p R*2 Teff4 Sun spectrum in the visible 5790 K BB spectrum l

8. How different systems « see » the line blanketing ? 4500 K BB

9. Suppose we have two photometric systems : • Syst 1 (in bold line) : blue filter Dl = 1000 A , centered at 4500 A • red filter Dl = 1000 A , centered at 7900 A • -Syst 2 (dotted line) : blue filter Dl = 1000 A , centered at 4900 A • red filter identical to red filter #1 • We observe a typical K star through these filters. • The red filter in both systems will basically sample the continuum (cool BB with Teff ~ 4500 K ) (only weak TiO bands there) • In system #1, the blue filter is affected by « moderate » blanketing (basically CN molecular absorption at 4100 A and CH absorption at 4300 A) plus various narrow metal (FeI) lines. But the continuum at 4400 A is much lower than at 4800 A ! • In system #2, the blue filter is affected by the extremely severe blanketing of the MgI – MgH absorption that is in the center of the passband ( in real filters, the best transmission is there !) • If you look carefully, you will see that the color :blue – red will not be so different in the two systems : Mg blanketing « compensates » the continuum fading shortwards from 4300 !!  « degeneracy » between Teff and metal content !

10. The connection between broad-band colors and spectra of objects In the following 4 slides, various examples are given of how colors correlate with the content of the spectra of stars. For this, we use spectra extracted from the public data base of the Sloan Digital Sky Survey. These spectra are calibrated in energy units. On each spectrum, are drawn the bandpasses of three « ideal » filters (ideal rectangular spectral windows with 100 % transmission) of uniform width, 50 nm, centered at 465, 630and730 nm respectively. The integral fluxes F through the filters are approximated (within the multiplicative factor of the bandwidth, which is constant) by thecentral flux density value, f465, f630, f730. The color C465-730 = m465 – m730 is evaluated by : C465-730 = m465 – m730 ~ -2.5 [log10 ( f465 ) – log10 (f730)] Exercise : in the same way, compute the C465-630 and C630-730 colors

11. Example # 1 : a hot star F465nm = 20 F630nm = 8 F730nm = 5 «color» m465 - m730 = -1.5

12. Example # 2 : a star cooler than the preceding one F465nm = 85 F630nm = 70 F730nm = 55 « color » m465 - m730 = - 0.45

13. Example # 3 : a cool star (K type) F465nm = 9 F630nm = 14 F730nm = 15 « color » m465 - m730 = 0.55

14. Example # 4 : a very cold star F465nm = 2 F630nm = 3 F730nm = 7 « color » m465 - m730 = 1.38

15. Synthetic photometry : • the toolbox to understand how photometric systems work • Synthetic photometry computes the response of photometric systems : • or on real spectra of spectral standard stars whose atmospheric parameters are supposed well-known (derived by detailed fitting of atmosphere models) • either to model atmosphere spectra (where you have total control on atmospheric stellar parameters) • It is of common use to predict the behaviour of photometric systems on peculiar classes of stars, define selection criteria to mine data bases to search for peculiar objects, etc… • What you need is the table of the bandpass efficiencies of the photometric system and, either a library (or several !) of stellar model atmospheres (or the tools to compute it) or a library of real stellar spectra with well-determined atmospheric parameters. • You then compute the fluxes transmitted through the system bandpasses, convert them into colors and play with various color-color indexes to select what you want. • (n.b. : what has been done before on the SDSS spectra is an initiation to synthetic photometry)

16. To compute magnitudes and colors in a given photometric system from either model stellar atmospheres or existing flux-calibrated spectral energy distributions, you convolve these energy distributions with the system standard passbands, known as « response functions » RX (l) for filter X. Modern photometry (including CCD image analysis) is always done in photon-counting mode. The magnitude measured across bandpass X is : mX = -2.5 log10∫(fl(l)/hn).RX(l).dl + Cst mX =-2.5 log10∫l(fl(l)/hc).RX(l).dl + Cst That translates into : mX = -2.5 log10<Fl>X + Cst = -2.5 log10 (1/hc).∫fl(l).[lRX(l)].dl +Cst

17. Then the constant Cst is eliminated by raccording the computed integrals to the zero points of the magnitude scale. When Vega is the primary calibrator: mX = -2.5 log10. ∫fl(l).[lRX(l)].dl / ∫flVega(l).[lRX(l)].dl + 0.03 For colors, the definition is generalized as: CXY = mX – mY = -2.5 log10 (<Fl>X / <Fl>Y) + ConstXY where the <F> are the integrals. Note that there is generally a zero point for the colors because the zero points for the magnitudes differ from one filter to another! (See example of computation with the Strömgren photometry) V mag of Vega

18. Alpha Lyr (Vega) : spectral energy distribution 1000 A - 25000 A

19. The Johnson-Cousins UBVRcIc system in the visible U-B measures the Balmer jump in hot stars or the 4000 A spectral « break » in cooler stars, due to metal absorptions. Therefore, it is sensitive to metallicity. B-V measures the slope of the Paschen continuum between 4400 A and 5500 A. The U band is also sensitive to surface gravity (low pressure plasma  more UV) Basically, B-V, V-I, B-R, B-I are all sensitive to Teff. But the smoothest calibration for stars of moderate effective temperature is obtained with V-I, less sensitive to metallicity than B-V or B-R. (An even better calibration is obtained with the infrared K band using V-K) For hot stars, and only for these, U-B is a good Teff indicator.

21. Effective temperature vs V-K color

24. The Johnson infrared extended JHK system Atmosphere transmission 0.8 1.2 1.6 2.0 2.4 wavelength in microns

25. The Strömgren system (uvbyb) The « continuum » bands figured above are often completed by a set of two filters both centered on Hb (4861 A), one of narrow width (~30 A at half-max), the other broader (~150 A at half-max). The difference of fluxes through these filters is a measure of the strength (« equivalent width ») of the Hb absorption, and is very useful for luminosity classification of B to F stars. The following colors are used : b-y (sensitive to Teff , not very much to metallicity) m1 = v - 2b + y = (v - b) - (b - y) (metals ~ 4100 A) c1 = u - 2v + b = (u - v) + (b - v) (Balmer jump) (m1 and c1 are almost reddening independant)

27. Clem & VandenBerg (2004)

29. The David Dunlap Observatory system (DDO)

30. The DDO system is very good for investigating metal abundance, but passbands are narrow, and thus the faint stars need comfortable telescopes. The filters are a bit difficult to reproduce exactly as prescribed. The extinction coefficients are painful to reduce because of the 7 bands! The system has been more widely used to study composite stellar populations in their integrated light (globular clusters, galaxies)

31. Literature sources about calibrations of photometric systems in terms of stellar atmosphere parameters: UBVRcIcJHK: Bessell et al. 1998, Astron. & Astrophys. vol. 333, p. 231 Vandenberg & Clem 2003, Astron. J. vol. 126, p. 778 Stromgren uvby(Hb) :Balona 1994, Monthly. Notices RAS, vol.268, p. 119 Lester et al. 1986, Astrophys. J. Suppl. vol. 61, p. 509 Schuster & Nissen, 1989, Astron. & Astrophys. Vol. 221, p. 65 Nissen, 1994, Rev. Mexicana Astr. Astrofis. Vol. 29, p.129 Clem & VandenBerg, 2004, Astron. J. vol. 127, p.1227 DDO system : Piatti et al. 1993, J. Astrophys;& Astron. vol. 14, 145 Claria et al. 1994, Monthly Notices RAS, vol. 268, p. 733 Claria et al. 1994, Pub. Astr. Soc. Pacific, vol.106, 436 Sloan SDSS system: ( Fukugita et al. 1996, Astron. J. vol. 111, p. 1748 ) Lenz et al. 2008, preprint

32. II . 2 PHOTOMETRY OF VARIABLE SOURCES II . 2 . 1 : Light curve analysis: principles II . 2 . 2 : On some modern experiments in variable star search: * gravitational microlensing experiments, *CoRot satellite search for planetary transits

33. LIGHT CURVE ANALYSIS By repeating observations of the same object, one gets a time series of the value of the observed brightness and / or color of the object. The temporal scale unit is the Julian date of observation. For very rapidly variable objects (e.g. pulsating white dwarfs, where typical variation timescales are of the order of a few seconds, the second of time is sufficient). The time series is ~ never continuous : there are ~ almost always interruptions in the observations, due to night/day alternance, failure of equipment, meteo problems, etc… Thus, the light curve is generally an incomplete time series with unequally spaced data. The variation of the observable parameter of interest (brightness  magnitude or color) can be periodic or non-periodic. It is fundamental, before saying that it is non-periodic, to search for periodicity : if no reasonable period is discovered, the phenomenon will then be classified as non-periodic. However, a periodic character may still be present in apparently non-periodic data : if the total observation time span is smaller than the phase, no period will be found ! (e.g.: an eclipsing binary of period 2 yr observed during 6 months)

34. PERIOD DETERMINATION BY EMPIRICAL METHOD : This has only an historical interest (automated mathematical methods are more powerful and accurate) but can be done just for training. It will provide good results only for very clear-cut variations (Cepheid and RR Lyr stars; Algol-type eclipsing binaries, etc…) . The principle is simple : - compute the observed average value of the variable parameter using all the observed values. - subtract this average to all data : you get a « difference » light curve, easier to manipulate. - search for a clear maximum (or minimum) value that seems being repeated several times at regular intervals T. - « fold » the data to the phase assuming that T is the period : that is, if a data is taken at time ti and the first data of the series t1, the phase of each data point is : fi = (ti-t1)/ T - Integer [ (ti – t1)/ T] for i = 1 … N fi being in [0, 1]

35. MODERN AUTOMATED METHODS : • Non-parametric methods : • in fact they are a generalization of the empirical method, introducing objective mathematical statistics to decide the quality and robustness of the results. • The principle is to group the data into bins, and, using a « guess » period, to look at the variation of data taken at adjacent phases for this period. • If the guess period is the real one or close to it, the variation of data taken at adjacent phases should be minimal. • If not, another guess period must be tried. • The two best applications in use to-day are : • The phase dispersionminimization method (PDMM) (Stellingwerf 1978) • -The signal dispersion minimization method (SDMM) (Renson 1978)

36. Fourier analysis : Fourier transform methods are well-known to be very powerful in identifying and analysing periodic phenomena. Dedicated methods have been specifically developed for incomplete data sets and / or unevenly spaced data, which do not fit into the ordinary, classic, Fourier signal processing techniques. (Deeming, Scargle, Roberts et al. etc…) The « periodogram » is the output of these computations : its peaks identifies periods and pseudo-periods found in the data distribution. It is not always straightforward to reject the pseudo-periods introduced by the temporal discretization of the data. As an example, a pseudoperiod of ~1 day will appear in periodograms of data taken each night at a ground-based observatory. For variable stars which have intrinsic periods close to 1 day, the extraction of the value of the intrinsic period is very difficult !

37. Literature sources for getting (much !!) deeper: Fourier discrete transform method : complete description in Deeming, 1975, Astrophys. & Space Science vol. 36 p. 137 (See also the theory of the CLEAN algorithm in: Schwarz, 1978, Astron. & Astrophys. vol. 65, p. 345 and its application to time series analysis in Roberts et al. 1987, Astron J. vol. 93, p. 968 Reliability of periodograms and false period detections are studied in detail in: Scargle, 1982, Astrophys. J. vol. 263, p.835 Non-parametric methods: the first introduction of the « theta statistic » is Lafler & Kinman 1965, Astrophys. J. Suppl. Ser. vol. 11, p. 216 Phase Dispersion Minimization Method : Stellingwerf, 1978, Astrophys. J. vol. 224, p.953 Signal Variation Minimization Method : Renson, 1978, Astron & Astrophys. vol. 63, p. 125 (unfortunately, this paper is written in French !) A very good synthetic paper analysing the respective qualities of each method: Heck et al. 1985, Astron & Astrophys Suppl. Ser. vol 59 p. 63

38. The search for extrasolar planets with CoRot satellite The CoRot (Convection and ROTation) experiment is a very high accuracy photometry space mission dedicated to the study of asteroseismology of solar-like and turnoff stars and observations of photometric transits of extrasolar planets. We shall focus on the planet search. Telescope : diameter 27 cm CCD camera for planet detection: field of view : 1.5 degree x 3 degrees point spread function: 50 % of the flux in 35 x 23 arc seconds simultaneous observation of ~ 12 000 stars in the field several months of continuous pointing on the same field exposure times : 512 seconds (basic mode) and 32 seconds (alarm mode) a small prism makes a very low resolution spectrum enabling to detect chromaticity in the variations of light.

41. Astronomical Sources of false alarms From Almenara-villa et al. The CoRoT consortium (adapted from Brown 2003) We look for: PLANETS Confusion from: • UNDILUTED BINARIES Eclipses of stellar components with large mass ratio Grazing Eclipsing Binaries • DILUTED BINARIES Eclipsing Binaries with deep eclipses + light from a bright 3rd star → shallow eclipses Eclipsing Binaries + unrelated (fg/bg) star within psf Eclipsing Binaries in triple system

42. Sample All detections from IRa01, LRc01 and LRa01 12 < R [mag] < 16 0.24 < P [days] < 106.4 0.018 < DF/F [%] < 81.64 Run durations: IRa01 55 days LRc01 142 days LRa01 131 days * IRa01 and LRa01 have 1402 targets and 30 detections in common

43. Results from Follow-Up Observations (IRa01,LRa01,LRc01) 122 candidates in Follow-Up lists 49 solved by Follow-Up (40 %)

44. Planet candidate + eclipsing binary in the same mask Found by Aviv Ofir dF/F = 0.9 % Eclipsing binary 1.1 d period Planet candidate 13.5 d period

45. 2 binaries in the same mask 211625668=SRc01_E2_1728 Original LC LC - Binary (1.772 d) Folded (2.628 d) 7 cases in 5 fields (0.015%)

49. Search experiments for gravitational microlenses There you want to find an achromatic amplification of the brightness of an object, as an unique event. The global shape of the light curve is characteristic, with a sharply peaked profile. But in some cases, if the source star crosses the amplification caustics along the right path, there could be spectacular effects of multiplicity inside the main peak. See the OGLE web site for many interesting discussions and beautiful data. These experiments are performed on crowded (highly crowded !!) fields, in order to increase the probability of detection of an event !