arXiv:astro-ph/0208537v1 29 Aug 2002 bevdprlae ( parallaxes is observed It area. the this on correction. in agreement bias be work of should further question there to that lead important missions will therefore future GAIA and as parallaxes determination such of magnitude use increased absolute been for has there available became paral- from the magnitudes in absolute situation of the derivation laxes. clarify the to of matters. attempts case these paper i in present there misunderstanding the that and quite shows uncertainty are literature these some Eddington Athough current by 1926). the out analysis Dyson straightforward, set also the were see in data 1940, bias (1913, other) with (or dealing astronomical for of methods general The INTRODUCTION 1 eeto)b h ai of ratio the by rejection) ad rvddteei oslcino egtn according weighting other or the selection On no 1973). to is Kelker & there (Lutz provided bias hand, to subject is result ie o ntneTrn&C´ze(97 osdrol the only Cr´ez´e consider en- & (1977) in Turon dealing arise instance none to For likely but situation tice. parallaxes the applicable with reduced bias satisfactorily of tirely of method discussions of the number to a been have There REDUCED OF METHOD THE 2 their from considered. objects be individual will of parallaxes magnitudes method determinati the absolute this of in the question occur of the can this, Following that reduced out. biases set of the are absolut section method to values next the relative the convert then In to known used be are can group parallaxes a in objects

o.Nt .Ato.Soc. Astron. R. Not. Mon. oebr2018 November 7 [email protected] Feast Michael from Determination Parallaxes Magnitude Absolute in Bias c srnm eatet nvriyo aeTw,Rondebosch Town, Cape of University Department, Astronomy π 00RAS 0000 PARALLAXES fojcsaeslce o nlsso h ai ftheir of basis the on analysis for selected are objects If 1997) al. et (Perryman catalogue Hipparcos the Since or π/σ π n fterltv bouemgiue fthe of magnitudes absolute relative the if and π rwihe icuigslcinor selection (including weighted or ) π 000 oissadr error, standard its to 0–0 00)Pitd7Nvme 08(NL (MN 2018 November 7 Printed (0000) 000–000 , other eain r ie o h orcino iswe enaslt ma considered. of also absolute case words: is the mean Key objects in when bias individual The of bias parallaxes. magnitudes of reduced absolute of correction the method the the by for derived given are Relations ABSTRACT σ srmty-sas udmna aaees-Cped star - Cepheids - parameters fundamental stars: - astrometry π hnthe then , prac- on e. s 71 ot Africa. South 7701, , osdrfis h aeo bet ihama bouemag- absolute mean a volume, unit with per objects nitude of case the first Consider h ehdo eue aalxsmyb written: be may parallaxes reduced of method The h bet ntegopudrdsuso aentbe se- ( been not parallaxes have measured discussion their under by group lected the in objects the signifi- are corrections is these this when cant. and arise well 1999) may However al. 1999). occasions (Pont et simulations Groen- Carlo Lanoix Monte (FC, by 2000, small confirmed Oudmaijer negligibly & is gener wegen correction en- now any is none it that forward, case agreed put particular that been In have satisfactory. tirely corrections various 2000) bias Oudmaijer on & views Groenewegen Oud- 1998, (=FC), al. 1997 et Catchpole & maijer Feast (e.g. considered. Cepheids sky of laxes of area magnitude the apparent in certain parallax a for objects to measured the down are all class when relevant to case the (1920) the of Malmquist only of consider refer, analysis they the which and they, the should that It in noted parallaxes. uncertainty be from dis- the derived intrinsic magnitude with absolute the magnitude mean confuse absolute to to relevant in seems completely persion the and not and purpose is magnitude) present Oja the absolute & in Ljunggren dispersion of discusion no magnitude is absolute there identical have (i.e. objects the all when case hr h aalxi esrdi milliarcsec, in measured is parallax the where σ n where and (0 and by; magnitude apparent free sorption 10 T 2 . 0 01 . = 2 ntemto frdcdprlae ti sue that assumed is it parallaxes reduced of method the In paral- Hipparcos the of discussions recent of case the In M σ σ T π 2 = . 10 + X 0 σ b . 2 2 T m π 0 M 2 sgvnby; given is o . 01 ) o 2 A ( π T σ 1 = E 10 m 2 tl l v2.2) file style X o 0 /p . + 2 M m σ o o M 2 .p/ n nitiscdispersion intrinsic an and , o ) X . p π p rteratio, the or ) stewih given weight the is h eiainof derivation the ntdsare gnitudes m :variables: s: o steab- the is π/σ σ ally M (3) (1) (2) π o s . . 2 Michael Feast

Here, by FC. This was to reduce the intrinsic scatter about the b = 0.2loge10 = 0.4605, PL relation by the combined use of a PL and PC relation

πMo is the photometric parallax of an object computed from for reddening correction and the determination of relative (mo − Mo). absolute magnitudes. If this is not done it might be neces-

σmo is the standard error of the reddening corrected appar- sary to consider whether Cepheids are distributed uniformly ent magnitude. throughout a strip in the PL plane, at least at long periods.

Equation 3 is from Koen & Laney (1998). In that case, and assumimg σmo is small and a constant 0.2Mo Equation 1 gives an unbiased estimate of 10 only if σMo space density distribution, equation 5 above would need to is zero. be replaced by; Put x = (mo − Mo). This will differ from the true dis- ebǫ = 3sinh(2b∆)/2sinh(3b∆) (9) tance modulus by ǫ (say), due to observational errors in mo and intrinsic dispersion in Mo. It is then evident that equa- where ∆ is the halfwidth of the strip in magnitudes. If one tion 1 yields an estimate of; adopts 2∆ ∼ 0.7mag at V from the longer period LMC Cepheids (Caldwell & Coulson 1986), the correction factor 100.2M = 100.2(Mo +ǫ) = ebMo .ebǫ. (4) would be ∼ +0.05mag. Consider objects all of the same mo (and x). From The above considers the case when the objects have Malmquist (1920) or more compactly, from Feast (1972) a mean absolute magnitude per unit volume of Mo with a where some of the present nomenclature and sign conven- gaussian scatter σMo . In a number of cases the relative abso- tion is used, we find; lute magnitudes of the objects concerned are known through the measurement of some auxilliary quantity, y, and a rela- 2 2 bǫ 0.5b σt 2 tion of the form; e = e v(x − bσt )/v(x) (5) where; Mo = Ay + B (10) 2 2 2 where A is a known constant. Two separate cases then arise. σt = σmo + σMo (6) if the measuring error in y, σy, introduces an error in Mo and v(x) is the frequency distribution of x which would have which is negligibly small compared with the intrinsic disper- been found if a complete survey had been made. Note that sion in Mo at a given y, then the results given above apply this is the case whether or not the objects under considera- in this case with obvious modifications to equation 1 (see tion actually form a complete survey. Feast 1987). If this is not the case, then the quantities v(x) 0.2Mo Evidently at any mo an unbiased estimate of 10 etc. have to be replaced by P (x) etc., where P (x) is the dis- is obtained by multiplying the r.h.s. of equation 1 by the tribution in x of the objects actually under discussion (i.e. reciprocal of the r.h.s. of equation 5. Summed over all mo, as taking into account that only a fraction, f(x), of all the ob- in a practical case, the final result will depend v(x) which in jects may have been observed at a given x (see Feast 1972). general will depend on x. Furthermore if one were analysing This procedure essentially involves using the inverse solution parallaxes over a significant galactic volume (as may well be of equation 10. In the case of the Hipparcos Cepheids, the possible with GAIA data), v(x) might not be the same in quantity y is the period and this is very accurately known, all heliocentric directions. In such cases the problem would so that the earlier discussion holds in this case. benefit from Monte Carlo simulations (Pont 1999, Sandage Even in the case of Mira variables, the percentage un- & Saha 2002). However if the underlying density distribution 2 2 certainty in the periods probably makes a much smaller con- −2.5b σt is constant then the r.h.s of equation 5 becomes 10 tribution to the uncertainty in Mo than the intrinsic disper- (see for instance Feast (1972) equation 9). Thus provided σt sion. Thus equation 8 can be applied to the discussion of the is constant for the objects studied, this is simply a constant Hipparcos parallaxes of these stars by Whitelock & Feast bias factor. (2000). These workers found an infrared PL zero-point of The relation between natural and logarithmic quantities +0.84±0.14. Using their data one obtains from equation 8 a shows that if σ1 is the standard error of the mean value of correction of ∼ 0.02mag, making the zero-point +0.86±0.14. Mo derived as above, then; There is another source of bias that should be discussed.

0.2Mo 2 In an analysis it is necessary to work with mo, the absorption Mo = 5log(10 ) − 0.23σ . (7) 1 free absolute magnitude. Evidently if absorption is present, Thus for the case of a constant underlying space density the objects of a given mo will have a range of uncorrected ap- best estimate of Mo is given by; parent magnitudes and selection according to apparent mag-

0.2M 2 2 nitude will affect the bias correction. In general absorption Mo = 5log(10 ) + 1.151σ − 0.23σ (8) t 1 increases with distance. Thus if the fraction of objects of a 2 Note that the coefficient of σt in this equation is differ- given apparent magnitude observed decreases with increas- ent from that in the conversion between Mo and Mm, the ing apparent magnitude, this will have the effect of reducing mean value for objects of a given mo (=1.38). This latter the fraction of objects of a given x whose true distance mod- factor is the well-known Malmquist (1920) bias, first given ulus is greater than (mo −Mo). Thus absorption, if it affects explicitly by Eddington (1914). the bias, is likely to do so by reducing the number of objects In the case of the Hipparcos parallaxes of Cepheids dis- in the sample whose lumniosities are greater than the aver- cussed by FC, the bias terms in equation 8 are negligibly age at a given mo. This means for instance that, if anything, small. The total bias correction is 0.010mag and this would the coefficient of the second term of the r.h.s of equation 8 change the PL zero-point from –1.43 to –1.42. The very small will need decreasing. Thus the value of Mo produced from bias correction is due to the method of analysis adopted an unchanged equation 8 would be too faint.

c 0000 RAS, MNRAS 000, 000–000 Bias in the use of parallaxes 3

3 OBSERVATIONS OF INDIVIDUAL OBJECTS an [Fe/H] of –1.39, Mo = +0.64 ± 0.11. It has of course to be realized that, especially in the case of single objects, the Provided there is no systematic error in the measuring pro- uncertainty in any applied bias corrections is large. cess, an observed parallax is not in itself biased. This was, in Obviously the same general arguments apply if more fact, clearly recognized by Lutz & Kelker (1973) who point than one object of a class is observed. Bias of the Lutz- out that the distribution of measured parallaxes about the Kelker type arises when the individual absolute magnitudes true parallax is expected to be gaussian. So provided the of objects derived from parallaxes are combined by weight- objects concerned are not chosen or weighted by their mea- ing them using the observed parallaxes. The effect of this sured π or by σπ/π, an analysis is not subject to bias of the on the weighting is illustrated in a practical case in Feast Lutz-Kelker type. This has been stressed recently by e.g. (1998). In some at least of these cases there will also be Whitelock & Feast (2000) and Groenewegen & Oudemaijer selection by apparent magnitude, so that corrections such (2000). as that given by equation 8 will apply in addition to the It is of interest to consider the case when only one ob- Lutz-Kelker correction. ject of a class is measured, as in the important Hubble Space telescope (HST) observations of the parallaxes of RR Lyrae and δ Cephei (Benedict et al. 2002a, Benedict et al. 2002b). 4 CONCLUSION These stars were presumably chosen as bright members of their class and their HST parallaxes subsequently measured. The bias corrections to the absolute magnitude scales of The absolute magnitudes derived from the parallaxes are Cepheids, RR Lyraes and Miras discussed in this paper are therefore not subject to Lutz-Kelker bias. However, since all relatively small. However, with the forthcoming great im- the method of determining an absolute magnitude from a provements in parallax measurements expected from GAIA single object is effectively the method of reduced parallaxes and other space missions, bias corrections of this type will applied to one member alone of the class, the bias correc- become increasingly significant relative to other sources of tions outlined above will be required, i.e. for the case of uncertainty. an adopted uniform space distribution of the objects of the class, equation 8, if an estimate of Mo is required. If the best estimate of the absolute magnitude of the individual object ACKNOWLEDGMENTS itself is wanted then, obviously, the term in σt in equation 8 is not required. I am grateful to the referee (Dr G.F. Benedict) for his com- In the case of δ Cephei (Benedict et al. 2002b) and ments. adopting their parallax, apparent magnitude and reddening, and with σMo = 0.21 (Caldwell & Coulson 1986), σ1 = 0.1 and assuming σmo can be neglected one obtains an estimate REFERENCES of Mo of −3.41 ± 0.10 and a zero-point of the PL relation adopted by FC of −1.36 ± 0.10. Alternatively one can use Benedict G.F. et al., 2002a, AJ, 123, 473 the PL and PC relations together for the reasons discussed Benedict G.F. et al., 2002b, astro-ph/0206214 above to obtain the reddening and absolute magnitude. This Caldwell J.A.R., Coulson I.M., 1986, MNRAS, 218, 223 Dyson F., 1926, MNRAS, 86, 686 gives Mo = −3.37±0.10 and a PL zero-point of −1.32±0.10 Eddington A.S., 1913, MNRAS, 73, 359 which may be compared with the FC zero-point (corrected − ± Eddingtom A.S., 1914, Stellar Motions and the Structure of the as above) of 1.42 0.12 (where the standard error of FC Universe, MacMillan, London, p.172 has been increase from 0.10 for reasons given in Feast 1999). Eddington A.S., 1940, MNRAS, 100, 354 Evidently the agreement is good, as it is with other Cepheid Feast M.W., 1972, Vistas in Astronomy, 13, 207 zero-point estimates (see e.g. Feast 2002). Note that, al- Feast M.W., 1987, The Observatory, 107, 185 though uncertainty in the reddening is a limiting factor in Feast M.W., 1998, Mem. Soc. Ast. It., 69, 31 these determinations of absolute magnitude, it is not impor- Feast M.W., 1999, PASP, 111, 775 tant in the use of Cepheids as distance indicators so long as Feast M.W., 2001, Odessa Pubs., 14, 141 a consistent reddening scale is used for both calibrators and Feast M.W., 2002, astro-ph/0010590 programme stars. This is most easily accomplished using a Feast M.W., Catchpole R.M., 1997, MNRAS, 286, L1 Freedman W.L. et al., 2001, ApJ, 553, 47 standard PC relation. The application of the Cepheid scale Groenewegen M.A.T., Oudmaijer R.D., 2000, A&A, 356, 849 to derive the distance of the LMC is made somewhat uncer- Koen C., Laney C.D., 1998, MNRAS, 301, 582 tain by the need for rather poorly known metallicity correc- Lanoix P., Paturel G., Garnier R., 1999, MNRAS, 308, 969 tions. However, this is not a problem in their use in the HST Ljunggren B., Oja T., 1965, Arkiv f. Astron., 3, 501 “key” programme on extragalactic Cepheids (Freedman et Lutz T.E., Kelker D.H., 1973, PASP, 85, 573 al. 2001) since the weighted mean metallicity of these galax- Malmquist K.G., 1920, Lund Medd. Ser. II, no.22 ies is close to solar (Feast 2001). Oudmaijer R.D., Groenewegen M.A.T., Schrijver H., 1998, MN- In the case of RR Lyrae, using the adopted parallax, RAS, 294, L55 Perryman M.A.C. et al., The Hipparcos and Tycho Catalogues, apparent magnitude and reddening from Benedict et al. ESA SP-1200 (2002a) and assuming RR Lyrae variables of the relevant Pont F., 1999 in ASP Conf. Ser. 167, Harmonizing the Cosmic metallicity ([Fe/H] = –1.39) uniformly fill an instability strip Distance Scale in the Post-Hipparcos Era, ed. Egret D, Heck of width ∼ 0.4mag, as is the case in globular clusters (see A., San Francisco, p.113 Sandage 1990, especially fig. 16), one obtains as an esti- Sandage A., 1990, ApJ, 350, 603 mate of the absolute magnitude of RR Lyrae variables at Sandage A., Saha A., 2002, astro-ph/0201381

c 0000 RAS, MNRAS 000, 000–000 4 Michael Feast

Turon Lacarrieu C., Cr´ez´eM., 1977, A&A, 56, 273 Whitelock P.A., Feast M.W., 2000, MNRAS, 319, 759

c 0000 RAS, MNRAS 000, 000–000