1 3 How do we “see” the Galaxy? Chapter 02 Astronomical Measurements Astronomical Measurements

Astronomical Measurements

References: CBMB: ch 1 CO: sec 1.3, 24.3 BM: sec 2.1, 2.2, 2.3, 2.5; 3.5, 3.6, 3.7

2 4 How do we “see” the Galaxy? Radiative Mechanisms

Astronomical Measurements Astronomical Measurements

photons

continuum lines

thermal synchrotron molecular atomic emission (rotational, relativistic jets (electronic, ro-vibrational, ionization, etc) electronic, jets, fine-structure, molecular blackbody bremsstrahlung hyper fine-structure, ionization, etc) clouds (graybody) (free-free outflows, stars, dust emission) molecular optically thick ionized thermal clouds emissions jets, HII regions 5 7 Equatorial Coordinates Galactic Coordinates

Astronomical Measurements Astronomical Measurements

(α, δ) = (RA, Dec) (l, b) = (Galactic longitude, Galactic latitude) Declination δ Right ascension α Described by 24 hrs 1 hour = 15 deg Hour circle Local sidereal time: elapsed time since the vernal equinox last traversed the meridian (hour angle of the vernal equinox) Hour angle H

Hour circle Carroll & Ostlie Fig. 24.17 Carroll & Ostlie Fig. 1.13

6 8 Precession & Epoch Radial Velocities

Astronomical Measurements Astronomical Measurements Precession Frequency observed by an observer First observed by Hipparchus moving at a velocity r Slow wobble of Earth’s rotation axis due to its non-spherical shape and its ⌫ =(1 )⌫0 interactions with the Sun and the Moon /c Slow precession ⌘ r Epoch 1 Commonly used: B1950, J2000 ⌘ 1 2 Altair, the brightest star in Aquila p (J2000)(19h50m47.0s,+8°52′6.0″) Doppler e↵ect 1 ! (B1950)(19h48m22.4s,+8°44′24.8″) r = c 0

Signs: Carroll & Ostlie Fig. 1.15

Blueshifted: r < 0, ⌫ > ⌫0 Redshifted: r > 0, ⌫ < ⌫0 Carroll & Ostlie Fig. 1.1 9 11 Magnitudes Stellar Spectral Types

Astronomical Measurements Astronomical Measurements Astronomers measure the brightness of a star with magnitude,which (early) O B A F G K M L T Y (late) is based on human eye response to the light and therefore on a log- Stars (a.k.a. dwarfs): OBAFGKM arithmic scale. In short, 5 magnitude increment corresponds to 100 Brown dwarfs: LTY times dimmer in brightness. Credit: KPNO 0.9-m Telescope, Followed by subclass 0, 1, ..., 9 AURA, NOAO, NSF Apparent magnitudes f m m = k log 1 1 2 f ✓ 2 ◆ f1 0.4(m m ) = 10 1 2 f2 Absolute magnitudes m M = 5 log d 5, distance modulus D 2 f = F, where D = 10 pc d ✓ ◆ f d m M = 2.5 log = 5 log F D ✓ ◆ ✓ ◆

10 12 Filters and Colors Luminosity Classes

Astronomical Measurements Astronomical Measurements U B V R I J H K L M Morgan-Keenan (MK) luminosity class I. supergiants II. bright giants III. normal giants IV. subgiants V. dwarfs = main sequence

Credit: Sky & Telescope 13 15 Colors Interpretation of Stellar Spectra

Astronomical Measurements Astronomical Measurements

With more than one filter bands, we can take the di↵erence in magni- E↵ective temperature Te↵ : tudes measured in two di↵erent bands to form a color, or color index for a selected source. Let X and Y denote two di↵erent filters, a color L 1/4 T . can be obtained with e↵ ⌘ 4⇡ R2 ✓ SB ◆ X Y m m ⌘ X Y Surface gravity: pressure-broadening lines 1 d S(X)f = const. 2.5 log 0 G 1 d S (Y )f g M. R0 ⌘ R2 A color can serve as a measure of stellarR temperature when the amount Chemical compositions: metallicity, Z, or iron abundance of extinction is known. Fe n(Fe) n(Fe) log log . H ⌘ n(H) star n(H)   

14 16

Stellar Astronomical Measurements Astronomical Measurements Spectral Types Hertzsprung- Russell Diagram

22000 stars from Hipparcos catalog and 1000 from Gliese catalog of nearby stars

Credit: Richard W. Pogge 17 19 Interstellar Interstellar Dust

Astronomical Measurements Astronomical Measurements Medium Extinction: measure of absorption Reddening: measure of frequency response, i.e. colors

Display of ISM Absorption: dark clouds Scattering: reflection nebulae Emission: emission nebulae

Credit: Daniel Verschatse (Antilhue Observatory) Credit: Nick Strobel, Astronomy Notes

18 20 Interstellar Extinction Extinction & Reddening

Astronomical Measurements

Extinction, AX

A m m X ⇥ X X,0 where mX,0 is the magnitude that would be observed in the absence of dust.

Reddening or color excess, E(X Y ) E(X Y ) [m m ] [m m ] ⇥ X Y X,0 Y,0 = A A X Y

Measured AX when mX is known

m = M + A + 5 log d 5 X X X 21 23 Standard ISM Extinction Law Color-Color Diagram

Astronomical Measurements Astronomical Measurements

E(X V ) AX Band X E(B V ) A V Reddening vector U 1.64 1.531 B 1 1.324 V 0 1 1989ApJ...345..245C R -0.78 0.748 I -1.6 0.482 J -2.22 0.282 H -2.55 0.175 K -2.74 0.112 L -2.91 0.058 M -3.02 0.023 N -2.93 0.052 Carroll & Ostlie Fig. 3.11

22 24 Extinction Curve Reddening-Free Indices stronomical Measurements A Astronomical Measurements Reddening-free indices: a photometric parameter that depends only on the AV spectral type of a star and is independent of the amount of reddening. : slope of extinction curve AJ RV near the V band E(U B) Q (U B) (B V ) ⌘ E(B V ) A A R V = V V ⌘ A A E(B V ) (U B) 0.72(B V ). B V ' 3.1 ' For early-type stars of spectral types O through A0, Q determines uniquely a star’s intrinsic color from photometric data alone without the need for their Gas-to-dust ratio spectra. One finds that

NH 21 2 1 (B V )0 =0.332Q. =5.8 10 cm mag , E(B V ) ⇥ Spectral Type Q Spectral Type Q equivalent to a mass ratio of 100. O5 -0.93 B3 -0.57 One may infer the column density O8 -0.93 B5 -0.44 2 NH (cm ) by measuring the color ex- O9 -0.9 B6 -0.37 cess E(B V ) (mag). B0 -0.9 B7 -0.32 B0.5 -0.85 B8 -0.27 Cardelli, Clayton, & Mathis, 1989, ApJ, 345, 245 Mathis 1990, ARA&A, 28, 37 B1 -0.78 B9 -0.13 B2 -0.7 A0 0 25 27 Distances for Nearby Stars Moving Cluster Method I

Astronomical Measurements Astronomical Measurements Photometric parallax d = 10 (m-M+5)/5 Trigonometric parallax Moving-cluster method Secular parallax Statistical parallax

Proper motions of Hyades members

Carroll & Ostlie Fig. 24.29

26 28 Trigonometric Parallax Moving Cluster Method II Astronomical Measurements Astronomical Measurements

Require stationary background references Determine the distance of a star cluster with identified members and the con- Hipparcos: accuracy of ~ 2 mas vergent point 1 d tan µ parsec (parallax-second; pc) = r . pc 4.74 km s 1 mas yr 1 ✓ ◆ = 3.09 × 1018 cm ⇣ ⌘

= 3.26 light years t = r tan = 206264.8 AU = µd

d 1 = pc p

Carroll & Ostlie Fig. 24.30 29 31 Secular Parallax I Secular Parallax III

Astronomical Measurements Astronomical Measurements Utilize solar peculiar motion to extend the baseline for trigonometric parallaxes (xˆi ˆ ) ((i xˆi)+(xˆi )) µ i = ⇥ · ⇥ ⇥ -1 -1 k x sin Solar motion = 13.4 km s = 2.8 AU yr i i 2 xˆi ˆ (xˆi ˆ ) (xˆi i) Select stars of same spectral type, same distance (apparent xi = | ⇥ | ⇥ · ⇥ ) µ i sin i µ i sin i magnitude) k k 2 sin i (xˆi ˆ ) (xˆi i) = ⇥ · ⇥ . µ i sin i µ i sin i k k =0 ⇥ d = Taking average over all N stars, we| find that{z the second} term vanishes µ d ⇥ µ since i i = 0 by our choice of reference frame. Given p 1/xi,we obtain h i⌘ P µ i sin i k p = h 2 i h i sin i h i

r = ±

30 32 Secular Parallax II Statistical Parallax

Astronomical Measurements Astronomical Measurements Statistical parallaxes are similar to secular parallaxes but use the per- pendicular component of the proper motions. An additional assumption Choose a frame in which the µ i mean velocity of the stars of µi µi ⇥ˆ is introduced that the velocities vi are isotropically distributed. the · observed radial velocity of the ith star is µ µ i the chosen group is zero, i.e. i xˆ uri = xˆi (i ) i i = 0. The heliocentric ve- i · locity of the ith star is i xˆi = xˆi i cos i. · P i The component of vi perpendicular to the plane congaing the Sun, the ui i ⌘ star, and v is xiµ i, and by the hypothesis that the mean magnitude ? of any component of vi is the same, we have Given the vector identity (a b) c =(b c) a, one can prove that ⇥ · ⇥ · xˆi i = xiµ i . h| · |i h| ? |i (ui xˆi) xˆi ((i ) xˆi) xˆi Therefore, the statistical parallax is µi = ⇥ ⇥ = ⇥ ⇥ . xi xi µ i p = h| ? |i µi ˆ (((i ) xˆi) xˆi) ˆ h i uri + cos i µ i = · = ⇥ ⇥ · h i k sin i xi sin i Note that isotropic assumption is not always valid. A more sophisticated (xˆi ˆ ) ((i ) xˆi) = ⇥ · ⇥ . analysis involves the ellipsoidal shape of the random velocity distribution x sin i i is also available. 33 35 Summary of Methods Biases in Counting

Astronomical Measurements Astronomical Measurements Supergiants Occurs in a survey of fixed solid angle, in which the volume studied increases with distance clusters Malmquist bias (Malmquist 1922, 1936) O-A stars The mean absolute magnitude of observed sample is brighter than clusters, secular and statistical parallaxes the mean absolute magnitude of the population F-M dwarfs Caused by magnitude-limited survey, i.e. brighter stars can be seen trigonometric parallaxes, moving cluster method farther F-M giants Lutz-Kelker bias (Lutz & Kelker 1973, PASP, 85, 573) moving cluster method, clusters, secular and statistical parallaxes Observed parallax to be on average higher than its true value White dwarfs ➫ Underestimate of distance trigonometric parallaxes, binaries, clusters ➫ Underestimate of intrinsic luminosity

34 36 Stellar Luminosity Function Malmquist Bias

Astronomical Measurements Astronomical Measurements A bias caused by a magnitude-limited sample, i.e. usual observations, dN = number of stars with absolute magnitude (M+dM, M) in the comparing to a volume-limited sample, i.e. intrinsic populations. The 3 volume d x around the point x objects in a survey that have a given apparent magnitude, m, will, on the dN = Φ(M, x) dM d3x average, have a higher luminosity than the mean luminosity of the pop- ulation as a whole. In short, the apparent magnitude and the dispersion are General luminosity function Φ(M) dlnA d2 ln A Irrespective to spectral types M M = 2 , and 2 2 = 4 . h im 0 dm m dm2 dN = Φ(M)dM n(x)d3x dN A(m) , star-count function relative fractions of number density of ⌘ dm 2 stars around x 1 (M M0) stars with different (M)= exp , luminosities p 2 22 2⇡ ✓ ◆ 2 M 2 M 2 , m ⌘h im h im where M0 and are the mean absolute magnitude and the dispersion in absolute magnitude of a volume-limited sample. 37 39 Lutz-Kelker Bias I MNRASL 444, L6–L10Lutz-Kelker (2014) Bias III doi:10.1093/mnrasl/slu103

Astronomical Measurements Astronomical Measurements

The Lutz–KelkerMNRASL 444, L6–L10 paradox (2014) doi:10.1093/mnrasl/slu103 For a survey of fixed solid angle, the volume studied increases with dis- tance and causes an observed trigonometric parallax to be on average ‹ Charles FrancisThe Lutz–Kelker paradox higher than its true value, i.e. an underestimate of distance and hence 25 Elphinstone Rd, Hastings, TN34 2EG, UK the luminosity. Let P (p p0)dp be the probability that the true parallax ‹ | Charles Francis 25 Elphinstone Rd, Hastings, TN34 2EG, UK of a given star lies in (p, p +dp) given its measured parallax is p0.The Accepted 2014 June 25. Received 2014 June 20; in original form 2014 April 7 probability is given by Accepted 2014 June 25. Received 2014 June 20; in original form 2014 April 7

P (p p0) P (p0 p) P (p) ABSTRACT

| / | The Lutz–KelkerABSTRACT correction is intended to give an unbiased estimate for stellar parallaxes and Downloaded from @s magnitudes,The but Lutz–Kelker it is shown correction explicitly is intended that to it give does an unbiasednot. This estimate paradox for stellar results parallaxes from theand applicationDownloaded from 2 magnitudes, but it is shown explicitly that it does not. This paradox results from the application P (p0 p)(M) n(s) s , of an argument about sample statistics to the treatment of individual stars, and involves the / | @p m of an argument about sample statistics to the treatment of individual stars, and involves the erroneouserroneous use of use a frequency of a frequency distribution distribution in in the the manner manner of a probabilityof a probability density function density function 2 4 (p0 p) consideredconsidered as a Bayesian as a Bayesian prior. prior. It isIt is shown shown that that the the Bayesian Bayesian probability probability distribution distribution for true for true (M) p exp , parallax given the observed parallax of a selected star is independent of the distribution of http://mnrasl.oxfordjournals.org/ 2 parallax given the observed parallax of a selected star is independent of the distribution of http://mnrasl.oxfordjournals.org/ / 2p other stars. Consequently, the Lutz–Kelker correction should not be used for individual stars.  other stars.This Consequently, result has important the implicationsLutz–Kelker for thecorrection RR Lyrae should scale and not for bethe usedinterpretation for individual of stars. This resultresults has from importantGaia and implicationsHipparcos.TheLutz–Kelkercorrectionisapoortreatmentofthe for the RR Lyrae scale and for the interpretation of where s =1/p and M = m + 5 log(p/10). Note that Lutz-Kelker bias is Trumpler–Weaver bias which affects parallax limited samples. A true correction is calculated results fromusingGaia numericaland integrationHipparcos and confirmed.TheLutz–Kelkercorrectionisapoortreatmentofthe by a Monte Carlo method. strongly biased toward small values of p and moderated by the luminosity Trumpler–Weaver bias which affects parallax limited samples. A true correction is calculated Key words: astrometry – parallaxes. function, (M). using numerical integration and confirmed by a Monte Carlo method. at National Tsing Hua University Library on March 22, 2015 Key words: astrometry – parallaxes. 1998). We can conclude that it is sufficiently close that it would 1BACKGROUND

be the brightest RR Lyrae irrespective of whether it is atypical at National Tsing Hua University Library on March 22, 2015 The issue of statistical bias in astronomical analysis was addressed in any way. Therefore, no bias should be assumed. However, the as long ago as 1913, by Eddington, who considered the bias to selection of further RR Lyrae variables is inevitably biased towards the number of observed stars within a given range of magnitudes 1998bright). We stars can so as conclude to select near that stars it and is sufficiently reduce parallax close errors. that it would 38 1BACKGROUNDcaused by observational errors in magnitude (Eddington 1913). In- Consequently, a Malmquist (1920)correctionshouldbeapplied. deed, whenever a stellar sample is selected either by magnitude or beBecause the brightest of other selection RR Lyrae criteria, irrespective and because of of uncertainties whether in it is atypical R“­ S“ž­ Y­5ž˜“­Z“i­''/ ­.0 0#­#%­ The issue of statisticalby a boundbias in on astronomicalparallax there will analysis be bias to the was statistics addressed of that pop- inthe any Malmquist way. Therefore, correction itself, no itbias is probably should not be correct assumed. to apply However, the as long ago as 1913,ulation. by Consequently, Eddington, a correction who considered will need to thebe made bias dependent to selectionthe full Malmquist of further correction RR Lyrae of 0.014 variables mag assuming is inevitably an intrinsic biased towards upon the sample selection. Distinct from bias to statistics, Lutz & dispersion in magnitude σ 0.1.∼ the number of observedKelker (stars1973,within hereinafter a ‘LK1’)given sought range to of address magnitudes bias resulting brightBecause stars of so the as large to size select≈ of the near Malmquist stars bias and for reduce many stellar parallax errors. R“­Lutz-Kelker S“ž­ Y­5ž˜“­Z“i­''/ ­,]m.0_E@\@m 0#­#%­\@:KKim :m 2c_j0@KJ@\m Bias;I:])m7@@\IRCm II <:KI;\:_IVRm caused by observationalfrom observations errors in ofmagnitude individual stars. (Eddington 1913). In- Consequently,classes, and the adifficulty Malmquist in estimating (1920 it)correctionshouldbeapplied. accurately, it is prefer- Astronomical Measurements eI_Em_\ICVRVP@_\I@\IRCm<:KI;\:_IVRm Hubble Space Telescope (HST;Benedictetal.2011)anditwillbe (2003), the Lutz–Kelker (L–K) bias has∼ been a source of confu- e£¸›½Â°£±ÂÒ¶¥ÒdÁ½¶±¶°Ë(Ò “±©Ç£½Á©ÃËÒ¶¥Òi¯¶½© ›(Ò k›©±£ÁÇ©¯¯£(Òi¯¶½© ›Ò“dÒPLXGGÒ upon the sample selection.vital to the Distinct correct interpretation from bias of todata statistics, from Gaia. Lutz Selection & bias dispersionsion since it in was magnitude originally described.σ 0.1. There has been some doubt eI_Em_\ICVRVP@_\I;FP>"PF>8!P >;$JF->BP;>GP.GP!M.FGFP ; P-"PF>P L+GP.GP.FP; PL+!;P-GPF+>I4 P!P >BB! G! P !B!PL!P B!$J44NP Hubble Space Telescopesamples( shouldHST;Benedictetal. be subject to bias. 2011)anditwillbe (2003serious), thanthe Smith Lutz–Kelker realized. (L–K) bias has been a source of confu- B!!M8.B.*-;5PL>B2PG>PI< !BFG; PL+GPG+!NP  GI55NP - P ; PG+!;P5>>2P vital to the correct interpretationAcorrecttreatmentmaydependupontheparticularpropertiesof of data from Gaia. Selection bias sionApparently since it wasa lack originally of clarity continues, described. because There Feast (2002 has)states been some doubt GP G+!-BP 4G!BP @@!BFP < P F>8!P >G+!BP L>B2FP >;P G+!P FI1! G P +!B!P -FP @B>@!B5NP F@!3-;*P ;>P the population under consideration. RR Lyrae itself is brighter than that ‘when only one object of a class is measured, as in the important 7 can affect observations of individual stars as well as samples. If, in the minds of many astronomers as to whether the bias is real, I;.K!BF4PIGO!52!BP-FP>"P-; .K. I5P@B45M!F P +!B!P.FPP.FP">BPFGBFPG+GPB!P8!8!BFP other RR Lyrae variables by more than a magnitude (Fernley et al. Hubble Space Telescope (HST) observations of the parallaxes of ;PG+!PB! !;FP>"PF>8!P >;$JF->"P F8@4!FP L+- +P -FP -%!B!;GP "B?9P IGP >)!;P +FP G+!P F8!P ">D8P FP ; P -FP *-K!;P G+!P ;8!P for example, the brightest star (or stars) of a given class has been whetherRR Lyrae it and is correctlyδ Cephei (Benedict calculated et al. 2002a in published,b)’ that ‘the treatments, absolute and when L+!G+!BP>BP;>GP.GP!M.FGFP ; P-"PF>P L+GP.GP.FP; PL+!;P-GPF+>I4 P!P >BB! G! P !B!PL!P B!$J44NP >"P G+!P IGO!42!BP -F P ,!P >K!B55P-FP ">BP F8@5!FP F!5! G! P  >B -;*P G>P B!5G-K!P @B55MP chosen, then it may not be representative of the class. Groenewegen it shouldmagnitudes be derived applied. from theWhereas parallaxes Sandage are therefore ¬ Saha subject (2002 to )state‘The B!!M8.B.*-;5PL>B2PG>PI< !BFG; PL+GPG+!NP  GI55NP - P ; PG+!;P5>>2P ⋆ Ò !BB>BP .FP F>9!G-8!FP *-K!;P G+!P ;9!P >"P IGO!52!BP .;P" GP .GP .FP >BP .FP K!BNP ;!5NP G+!P F8!P E-mail: [email protected] Lutz-Kelker bias’, but Smith (2003)describestheapproachtaken GP G+!-BP 4G!BP @@!BFP < P F>8!P >G+!BP L>B2FP >;P G+!P FI1! G P +!B!P -FP @B>@!B5NP F@!3-;*P ;>P (2008) and Francis & Anderson (2014) identified a selection bias in Lutz-Kelker paper was so clear that it soon became the principal FP G+GP .F IFF! P NP BI8@5!BP < P !K!B P +!P IGO !52!BP >BB! G.>"P-; .K. I5P@B45M!F P +!B!P.FPP.FP">BPFGBFPG+GPB!P8!8!BFP calibrations of red clump magnitudes based on a sample for which reference to the problem’, Smith (2003)findsthatC 2014 The Author‘the seeming clar- >; .G.>;FP !PIF! P G>P >IBB! G->

BP 

4G! P ⃝ >"P F8@4!FP L+- +P -FP -%!B!;GP "B?9P IGP >)!;P +FP G+!P F8!P ">D8P FP ; P -FP *-K!;P G+!P ;8!P Published by Oxford University Press on behalf of the Royal Astronomical Society FGBP/; !@!= !=GP>#PF:@6!P@B>@!BG/!FP0FP=P/; >:@7!G!PC!&;!:!;GP?#PG+!P!FG0:G!P>#PF>7IG!P I-band measurements are available. Similarly, one should consider ity of LK1 masks a profound ambiguity surrounding the nature of the >"P G+!P IGO!42!BP -F P ,!P >K!B55P-FP ">BP F8@5!FP F!5! G! P  >B -;*P G>P B!5G-K!P @B55MP 9*;.GI !P 5 I4G! P .B! G5NP (>9PG+!P@B44MP ;>GPP >BB! G.>;P">BP.F P whether other magnitude calibrations from single stars or from small Lutz-Kelker bias’. I will show here that the implications are more !BB>BP .FP F>9!G-8!FP *-K!;P G+!P ;9!P >"P IGO!52!BP .;P" GP .GP .FP >BP .FP K!BNP ;!5NP G+!P F8!P !PB! ?;F- !BPG+!P8>B!P*!;!B4P@B>5!9P>"P 4-BG->;PIF-;*P>;4NPGD-*>;>9!HB. P@D44M!FP samples should be subject to bias. serious than Smith realized. FP G+GP .F IFF! P NP BI8@5!BP < P !K!B P +!P IGO !52!BP >BB! G.>9!P >"P G+!P 8M-8I9P 5.2!5.+>> P 8!G,> FP @B>@>F! P ">BP .GFP F?5IG->< P "P G+!F!P Acorrecttreatmentmaydependupontheparticularpropertiesof Apparently a lack of clarity continues, because Feast (2002)states >; .G.>;FP !PIF! P G>P >IBB! G->

BP 

4G! P F!K!B5P B!P F! P >;P 5.M.8G->;P ; P B!P G+!B!">B!P >"P 5.8-G! P K4- .GN P +B!!P !M GP the population under consideration. RR Lyrae itself is brighter than that ‘when only one object of a class is measured, as in the important FGBP/; !@!= !=GP>#PF:@6!P9!G+> @B>@!BG/!FPFP B!P 54P F!0FP=P P >

:@7!G!P!FF!;G.44NPC!&;!:!;GPG+!P F8!P?#P">B9PG+!P!FG0:G!P>"P G+!P 4.2!4.+>>>#PF>7IG!P P IGP B!P -8@5!8!9PLNF PG+!P@B44MP"P G+!F!P;>GPP.FP >BB! G.>;PFGG-FG- 55NP">BP'L!.F P P FP LFP >B-*.<55NP @>-;G! P >IGP NP I<* P other RR Lyrae variables by more than a magnitude (Fernley et al. Hubble Space Telescope (HST) observations of the parallaxes of !PB! ?;F- !BPG+!P8>B!P!P G!FGP*!;!B4PG+!P>G+!BP@B>5!9PGL>PIF.;*P>"P 4-BG->;PFN;G+!G. PIF-;*PF8@5!FP>;4NP >8@B!PGD-*>;>9!HB. PG+!.BP @!B">B8; !P@D44M!FP ; P .F IFFP G+!.BP RR Lyrae and δ Cephei (Benedict et al. 2002a,b)’ that ‘the absolute < P !M8-;!P F>9!P >"P@@5. G.>; PG+!P 8M-8I9P!P5.2!5.+>>5F>P @@4NP P >;!P8!G,>*B- FP P@B>@>F!8!G+> P PG>P">BPG+!P.GFP!FGG ,@>4!PF?5IG->< P "P G+!F!P+.*+P L!-*+GP F8@5!P >"P magnitudes derived from the parallaxes are therefore not subject to F!K!B5P B!P F! P >;P 5.M.8G->;P ; P B!P G+!B!">B!P >"P 5.8-G! P K4- .GN P +B!!P !M GP n¨··š¼žµÀÒ!@+!- P@B55M!FPFPPG!FG P+!P*B- P9!G,> P-FPG?P!P@B!"!BB! P>K!BPG+!P@@B>M-9G!P ⋆ E-mail: [email protected] Lutz-Kelker bias’, but Smith (2003)describestheapproachtaken 9!G+> FP B!P 54P F! >!FPF8!P<>GP +K!P">B9PP 5-9.G!>"P G+!P P B;*!P4.2!4.+>>>"P K5. P IGP -GNPB!PF+>I5-8@5!8!GP B!AI-B!P P ;NP P @>FG!B->B.P -;P -%!B!;GP LNF P E! G->;P>"P G+!F!P.FPK. !FPP'L!9>B!P >8@4!G!PP FP LFP >B-*.<55NP@- GIB!P>"PG+!P@>-;G!I< !BG.IGP NP I<* P.;PG+!P">B8P>"PP >IBP C 2014 The Author !P G!FGP G+!P>G+!BP GL>P IF.;*P.*B8PFN;G+!G. P>"P5>*P5.2!5.+>>F8@5!FP  P >8@B!P G+!.BP @!B">B8; !P ; P .F IFFP G+!.BP ⃝ @@5. G.>; P !P 5F>P @@4NP >;!P *B- P 8!G+> P G>PG+!P !FGG ,@>4!P +.*+P L!-*+GP F8@5!P >"P Published by Oxford University Press on behalf of the Royal Astronomical Society {¢ÊÒÈ´»Ÿ¿[Ò8!G+> FPFGG.FG. 5  FHB>8!GBNP FGFP .FG; !FP FGFP"I; 8!;G5P@B8!G!BF P n¨··š¼žµÀÒ!@+!- P@B55M!FPFPPG!FG P+!P*B- P9!G,> P-FPG?P!P@B!"!BB! P>K!BPG+!P@@B>M-9G!P >!FP <>GP +K!P P 5-9.G! P B;*!P >"P K5. -GNP F+>I5 P ;>GP B!AI-B!P ;NP P @>FG!B->B.P >E! G->;P < P@B>K. !FPP9>B!P >8@4!G!P@- GIB!P>"PG+!PI< !BG.B8P>"PP >IBP .*B8P>"P5>*P5.2!5.+>>  P 
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