Period, Color,

and Luminosity for

Cepheid Variables

Cecilia H. Payne-Gaposchkin

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SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS • NUMBER 17

Period, Color, and Luminosity for

Cepheid Variables

Cecilia H. Payne-Gaposchkin

SMITHSONIAN INSTITUTION PRESS

City of Washington

1974
ABSTRACT

Payne-Gaposehkin, Cecilia H. Period, Color, and Luminosity for Cepheid

Variables. Smithsonian Contributions to Astrophysics, number 17, 10 pages,

8 tables, 1974.—The Cepheids of the Galaxy and the Magellanic Clouds display a gradation in properties, but when amplitude as well as period is

taken into account, their intrinsic colors are sensibly similar. The periodluminosity relations in the three systems are probably parallel, but not necessarily coincident.

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Period, color, and luminosity for Cepheid variables.

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Period, Color, and Luminosity for

Cepheid Variables

Cecilia H. Payne-Gaposchkin

The Period-Color Relation

The well-known relationship between period

and color for classical Cepheids has usually been

expressed in terms of a mean color. Magnitude at

mean intensity is rightly considered an appropriate measure of luminosity. It is certainly more

significant than mean magnitude, though it must

differ from true (bolometric) luminosity by

terms dependent on effective wavelength and intrinsic color. An extension of this principle has

led to the use of mean colors, either (B) — (V)

where both magnitudes have been reduced to

mean intensity, or mean (B —V) expressed in

magnitudes. These quantities are of course not

synonymous, though their numerical values do

not differ greatly. But the mean colors thus expressed have no obvious physical relation to the

light curve: they are not characteristic of a particular phase, or indeed of the same phase for

stars with differing light curves.

It therefore seems worthwhile to consider separately the colors (B —V) at maximum and at

minimum. These colors are associated with physically recognizable stages of the star's variation

and are the same for all stars. We shall choose

maximum and minimum in V light as reference

points. By considering the colors that correspond

to the extremes of light variation, we include the

effects of light amplitude on brightness and

color, significant parts of the data that are rejected by the use of mean colors.

Cecilia H. Payne-Gaposchkin, Smithsonian Institution

Astrophysical Observatory, Cambridge, Massachusetts

02138.

Oosterhoff (1960) has shown that the ranges in

V and (B —V) are correlated, and that thus

maximal or minimal color (or both) must be

functions of light amplitude as well as of period. We therefore evaluate a relation of the

form

(B-V) = x + ylogP + zA, (l)

where (B —V) is the intrinsic color at maximum

or minimum, P the period, and A the light amplitude in magnitudes. We select the amplitude

in V light, Av, but the results could be converted

to an expression involving AB, the amplitude in

B light, if desired.

To evaluate the constants, we make use of 60

Cepheids whose color excesses have been determined spectroscopically by Kraft (1961) and by

Banner, Hiltner, and Kraft (1962), which were

standardized by means of 5 known Cepheids that

are members of galactic clusters.

The photoelectric observations of Magellanic

Cepheids by Gascoigne and Kron (1965) and

Gascoigne (1969) have been similarly treated.

The maximal and minimal colors are those tabulated by Gascoigne, corrected for reddening by

0.m05 for the Large Cloud and by O.mO2 for the

Small Cloud, in accordance with his determinations. These colors are more uncertain than

those of galactic Cepheids on account of the

smaller number of observations for each star.

Gascoigne's colors have been used as they stand;

although slightly different values might be deduced from his light curves, the effect on the

final results would be small. Two solutions were

made for each Cloud, one using all the published
SMITHSONIAN CONTRIBUTIONS TO A8TROPHYSIC8

data (except those for the highly reddened HV

2749 in the Large Cloud), and another excluding

stars whose minima are not adequately covered

by the observations (HV 2432, 2447, and 883 in

the Large Cloud; HV 1425, 1328, and 1342 in the

Small).

The results of these least-squares solutions are

summarized in Table 1; successive columns give

the following: the system; the number of stars

used; the values of the constants x, y, and z in

equation (1) for maximal color; and the values

of the corresponding constants x', y', and z' for

minimal color.

The number of stars discussed for the Galaxy

is large enough to justify a least-squares solution, but there are too few stars for either Cloud

to give much weight to the results. The values of

z and z' do not seem to differ sensibly in the

three systems. The agreement of y and y' is less

good. The values of x and x' are the least concordant and suggest a difference in the zero

points of the colors such that the Cepheids of the

Large Cloud are slightly redder, and those of the

Small Cloud slightly bluer, than are those of the

Galaxy. We note that the means of the determinations for the two Clouds (x = 0.364, y =

0.223, z = -0.238, x' = 0.340, y' = 0.352, z' =

0.122) do not differ sensibly from the solution

for the Galaxy. Although the accidental error of

the zero point of the colors may be greater for

the Clouds, their high galactic latitudes make the

colors less vulnerable to systematic error than are

those of galactic Cepheids, all of which are appreciably reddened. The zero point of the colors

for galactic Cepheids depends on the intrinsic

colors of the stars in the clusters that contain the

five calibrating Cepheids. These considerations

are of fundamental importance in examining the

justification for using a period-luminosity relation deduced from Magellanic Cepheids in discussing galactic Cepheids.

Fernie (1970) has carried out a similar discussion for 27 galactic Cepheids, of which all but 2

(IT Aql and « UMi) were included in our own

list. He obtains relationships of similar form to

those just presented, but with different numerical

coefficients:

(B-V) m a x = 0.297 + 0.307 log P - 0.194 Av,

(B-V) m i n - 0.238 + 0.373 log P + 0.373 Av.

Table 2 contains the photoelectrically observed

galactic Cepheids for which color excesses have

been determined spectroscopically by Kraft

(1961) and by Bahner et al. (1962), with multicolor photometry by Kron and Svolopoulos

(1959) and by Williams (1966). Successive columns give the name of the star; the logarithm of

its period; the amplitude Av ; the color fexcesses

published by Kraft (K), by Bahner, Hiltner,

and Kraft (BHK), by Kron and Svolopoulos

(KS), and by Williams (W); color excesses at

maximum and minimum predicted by means of

the coefficients given in Table 1 for galactic Cepheids; color excesses predicted by Fernie from

his relations quoted above; and Fernie's adopted

color excess. The data for EU Tau are taken

from Guinan (1972); the observed color excess

given in the column headed K,BHK and marked

with an asterisk is his value.

In comparing the observed and the predicted

color excesses of Table 2, we recall that the coefficients of Table 1 were derived from the K and

BHK color excesses, so that the agreement of the

individual values is a measure of the closeness

with which the observations have been represented. The average algebraic residuals are

-0.002 and -0.005 at maximum and minimum;

the average arithmetic residual is 0.06 for both.

We conclude that the representation is satisfactory.

Since the color excesses of Kron and Svolopoulos and of Williams were not used in deriving

TABLE 1.—Coefficients of equation (1)

System

Galaxy

LMC, all

LMC, set.

SMC, all

SMC, oel.

All, Ml.

No.

60

20

17

27

2-f

101

X

0. 357 ± 0.029

0.455

0.447 ±0.060

0.261

0.280 ±0.083

0.353

0.

0.

0.

0.

0.

0.

y

292 ± 0. 034

16.1

170 ± 0. 053

264

276 ± 0.073

269

z

-0 282 ± 0.038

-0. 249

-0.264 ±0.083

-0. 194

-0.213 ±0.115

-0. 263

0

0

0

0

0

X'

374 ± 0.

373

441 ±0.

186

240 ± 0.

0.353

027

083

116

y'

0.327 ± 0.033

0.268

0.292 ±0.075

0404

0.412 ±0.042

0.342

z'

0.190 ± 0

0.231

0.156 ± 0

0. 142

0. 087 ± 0

0.161

036

115

116
NUMBER 17

TABLE 2.—Galactic Cephcids: Observed and predicted color excess

Star

SUCas

EUTau

DTCyg

AY Cas

EVSct

SZ Tau

BY Cas

DWPer

SSSct

RT Aur

DFCas

SUCyg

STTau

SYCas

Y Lac

CGCas

T Vul

FF Aql

DF Lac

CE Cas t

XYCas

V482 Sco

UXPer

CFCas

V Lac

DWCas

APSgr

CECasa

V386 Cyg

CRSer

TV Cam

BGLac

UYPer

6Cep

SWCas

X Lac

VYPer

CZ Cas

YSgr

ABCam

FM Cas

VWCas

RVSco

WCa s

CRCep

RSCas

RE Lac

logP

0.290

0.323

0.398

0.458

0.490

0.498

0.508

0.562

0.565

0.571

0.583

0.585

0.602

0.610

0.636

0.640

0.647

0.650

0.651

0.651

0.653

0.656

0.667

0.688

0.698

0.699

0.704

0.711

0.720

0.724

0.724

0.727

0.730

0.730

0.736

0.736

0.743

0.753

0.761

0.762

0.764

0.778

0.783

0.793

0.795

0.799

0.807

A V

0.39

0.32

0.36

0.68

0.30

0.36

0.39

0.71

0.49

0.76

0.59

0.72

0.82

0.81

0.69

0.84

0.61

0.36

0.63

0.68

0.58

0.63

0.95

0.55

1.00

0.57

0.83

0.52

0.69

0.80

0.97

0.63

0.91

0.83

0.67

0.41

0. .9

0.83

0.72

1.01

0.58

0.70

0.81

0.85

0.39

0.76

0.83

Observed

K, BHK

0.34

0.23*

0.79

0.58

0.95

0.65

0.33

0.615

0.53

0.135

0.76

0. 34

0.695

0.43

0.43

0.51

0.555

0.385

0.84

0.29

1.02

0.60:

0.33:

0.985

0.11

0.55

0.40

1.06

0.785

0.69

0.32

0.63

0.40

0.54

0.78

0.78

0.31

Color

KS

0.09

0.33

0.14

0.20

0.26

0.15

0. 27

0.34

0.14

0.23

Excess

W

0.61

0.36

0.08

0.17

0.32

0.09

0. 31

0.063

(0.54)

0.73

0.54

(0.54)

1.28

0.76

0.11

0.17

0.97

Max

0.31

0.21

0.11

0.81

0.65

0.36

0.80

0.69

0.44

0.09

0.63

0.08

0.39

0.48

0.21

0.67

0.10

0. 2£

0.66

0.59

0.60

0.45

0.50

0.64

0.38

0.90

0.26

0.61

1.03

1.08

0.64

0.43

1.03

0.10

0.56

0.37

1.12

0.89

0.27

0.67

0.45

0.66

0.41

0.56

0.87

0.89

0.36

Predicted Color Excess

Min.

0.32

0.19

0.06

0.83

0.63

0.38

0.78

0.65

0.44

0.10

0.64

0.01

0.34

0.47

0.19

0.70

0.18

0.64

0.59

0.59

0.43

0.44

0.64

0.35

0.88

0.27

0.65

0.99

0.57

0.41

1.00

0.12

0.55

0.36

: 1.01

0.84

0.32

0.65

0.47

0.61

0.42

0.55

0.85

0.91

0.34

Ad.

0.315

0.20

0.085

0.82

0.64

0.37

0.79

0.67

0.44

0.095

0.635

0.045

0.365

0.475

0.20

0.685

0.14

0.65

0.59

0.595

0.44

0.47

0.64

0.365

0.89

0.265

0.63

1.01

1.08

0.605

0.42

1.015

0.11

0.555

0

N

.365

1.05

0.865

0.295

0.66

0.46

0.635

0.415

0.555

0.86

0.90

0.35

Feraie

0.31

0.13

0.39

0.07

0.10

0.64

0.08

0.27

0.30

Ad.

0.30

0.14

0.37

0.13

0.15

0.52

0.14

0.26

0.29

Star

USgr

APCas

XSgr

7, Aql

AKCep

CKSct

WSgr

CD Cas

WGem

VYCyg

RXCam

DLCas

SSge

V500Sco

FN Aql

SNor

DDCas

BZCyg

YSct

Z Lac

TYSct

AAGem

RXAur

RYCas

Z Set

SZ Cms

TT Aql

CYCas

TXCyg

UZ Set

RWCas

SZCyg

SVMon

XCyg

RWCam

CD Cyg

YOph

SZ Aql

CPCep

YZ Aur

RUSct

VXCyg

WZ Sgr

T Mon

AQPup

SV Vul

logP

0.829

0.836

0.846

0.856

0.859

0.870

0.881

0.892

0.898

0.898

0.898

0.903

0.923

0.969

0.977

0.989

0.992

1.006

1.018

1.037

1.043

1.053

1.065

1.084

1.110

1.134

1.138

1.158

1.167

1.169

1.170

1.179

1.183

1.214

1.221

1.233

1.234

1.234

1.252

1.260

1.294

1.304

1.339

1.432

1.477

1.6S4

Ay

0.78

0.64

0.65

0.84

0.67

0.52

0.84

0.80

0.81

0.81

0.86

0.6U

0 8 2

0.76

0.70

0.66

0.59

0.55

0.83

0.95

0.86

0.65

0.71

1.03

1.02

0.40

1.13

1.10

1.15

0.87

1.18

0.89

1.09

1.06

1.07

1.21

0.54

1.28

0.72

0.76

1.10

1.05

1.03

1.06

1.16

1.02

Observed

K, BHK

0.55

0.915

0.18

0.84

0.83

0.31

0.69

0.62

0.50

0.205

0.56

1.00

0.49

0.74

0.88

1.11

1.23

0.53

0.74

0.68

0.85

1.08

0.41

0.66

Color

KS

0.40

0.20

0.13

0.43

0.28

0.58

0.56

0.27

Exce

W

0.45

0.14

0.15

0.94

0.12

0.61

0.56

0.17

0.62

0.64

0.87

1.28

0.60

0.40

0.73

0.68

1.02

0.59

1.13

1.49

1.18

0.58

0.59

0.30

0.36

1.01

0.58

0.74

0.71

0.71

0.76

1.22

0.89

0.53

0.32

0.70

Predicted Color Excess

is

Max.

0.53

0.81

0.18

0.22

0.73

1.01

0.16

0.88

0.31:

0.65

0.66

0.59

0.20

0.72

0.65

0.35

0.58

0.98

0.94

0.51

1.09

0.42

0.33

0.71

0.69

0.79

0.60

1.04

1.18

1.18

0.48

0.80

0.35

0.43

0.75

0.58

0.69

0.68

0.92

0.63

0.97

1.00

0.58

0 4 1

0.51

0.56

Min.

0.52

0.81

0.16

0.23

0.74

0.98

0.19

0.89

(0.36)

0.66

0.64

0.59

0.23

0.68

0.61

0.35

0.65

0.95

0.98

0.58

1.08

0.46

0.28

0.69

0.71

0.81

0.67

1.08

1.24

1.21

0.60

0.87

0.43

0.54

0.61

0.67

0.66

0.80

1.03

0.68

0.99

1.01

0.72

0.46

0.68

0.66

Ad.

0.525

0.81

0.17

0.225

0.735

0.995

0.175

0.885

0.31:

0.655

0.65

0.59

0.215

0.70

0.63

0.35

0.615

0.965

0.96

0.545

1.085

0.44

0.315

0.70

0.70

0.80

0.635

1.06

1.21

1.195

0.54

0.835

0.39

0.485

0.68

0.625

0.675

0.74

0.975

0.655

0.98

1.005

0.65

0.435

0.S95

0.61

Ferme

Prod. Ad.

0.48

0.16

0.20

0.14

0.62

0.58

0.17

0.58

0. 18

0.46

0.54

0.39

0.67

0.59

0.33

0.48

0.20

0.21

0.17

0.57

0.57

0.20

0.60

0.15

0.42

0.53

0.36

0.74

0.58

0.37

Gulaan (1972).

the coefficients of Table 1, they furnish an independent check. The comparison with the KS

color excesses shows systematic trends; the KS

color excesses tend to be larger than the K,BHK

color excesses for short periods and smaller for

long periods. Thus, the KS system differs systematically from the K,BHK system. The W

color excesses show no evident systematic trend,

but a much larger scatter. The comparison with

Fernie's adopted color excesses shows no sensible

difference for periods under 10 days, but for

longer periods his adopted color excesses are systematically smaller than ours. A comparison between our formulas and his shows that the predicted color indices (in the sense of ours minus

Fernie's) will differ at maximum by

0.060-0.015 log P-0.088 Av,

and at minimum by

0.136-0.046 log P-0.183 AT.
SMITHSONIAN CONTRIBUTIONS TO ASTROPHTSIC8

The greatest differences therefore occur for long

periods and large V ranges (which are also associated with the longest period). The difference

between our formulation and Fernie's must be

ascribed to the difference in our selection of data.

Our own was based on the system of K,BHK,

which has the merit of being based on many

stars and is quite homogeneous. A discussion

based on the KS system would lead to results

close to those obtained by Fernie.

The color excesses of Kraft and his collaborators were based on mean colors, and in using

them to calibrate our conversions, we have tacitly

assumed that the color excess of a Cepheid will

be the same at maximum and minimum. The entries of Table 2 show in fact that color excess is

not independent of intrinsic color. At longer periods, the predicted color excesses tend to be too

large at minimum. The effect has been pointed

out and correctly analyzed by Wildey (1963).

The effect is to render the color excesses (assumed independent of intrinsic color and based

on the O and B stars) too small for redder stars,

and the intrinsic colors derived from them are

accordingly too red. At maximum, the excess

redness for galactic Cepheids is about 0.03; at

minimum, it ranges from 0.06 to 0.12.

We are now in a position to attack the basic

problem: Are the Magellanic Cepheids identical

in properties with galactic Cepheids? The legitimacy of adopting a Magellanic period-luminosity

relation in discussing galactic Cepheids hinges

on the answer to this question.

Observable photometric properties that can be

used in comparing galactic and Magellanic Cepheids are the following:

1. Light curves in V and B (yielding properties such as amplitude, asymmetry, interval from

minimum to maximum, and interval between

maxima where two are observed).

2. Data obtained by differencing light curves

in V and B (yielding (B-V) m a x and (B-V) m l n

as defined above).

3. Data obtained by integrating light curves at

mean intensity (yielding (V) and (B) and also

(B)-(V)).

4. Data obtained by integrating the (B —V)

color curve (yielding (B —V)).

A comparison of the properties enumerated

under 1 has been published elsewhere by the

writer (1969). When considered as functions of

period, most of them differ between the Galaxy,

the LMC, and the SMC; only the interval between two maxima is related to period in the

same way in all three systems.

The properties enumerated under 2 are summarized in Table 1. Here again, the coefficients

of x, y, z, x', y', z' differ in the three systems.

But it is to be noted that in each system the relationship is of the same form: x, y, x', y', z' are

always positive, z always negative; the same is

true of Fernie's formulation.

The numerical values, however, strongly suggest a real difference. The zero point of the intrinsic colors (expressed by the coefficients x and

x') shows that stare of similar period and amplitude in the Large Cloud are slightly redder, and

in the Small Cloud, slightly bluer, than in the

Galaxy. All studies of the Small Cloud have emphasized this blueness of the Cepheids in the

Small Cloud.

The small number of stars on which the results

for the Magellanic Clouds are based makes the

actual values of the coefficients uncertain, and in

order to examine the difference between the galactic and Magellanic colors, we therefore use the

galactic relations (equations (2) and (3)) to predict colors for the Magellanic Cepheids from the

published periods and V ranges. The comparison

of observed colors with those thus predicted is

given in Tables 3 and 4.

TABLE 3.—Large Magellanic Cloud: Observed and

predicted colors

Star

W44

HV 5541 W 24

HV 12225 W 33

W25

HV 12747 W 10

HV 12226 W 29

W 22

(HV 2432)

HV886

HV 1003

HV 902

HV 2251

HV 1002

HV2294

HV 909

HV 900

HV953

HV2369

(HV 2447)

(HV 883)

log P

0.408

0.429

.478

.529

. 556

.569

.669

.038

.380

.387

.421

.447

.483

.564

.575

.677

.680

.684

.074

. 127

A v

0.80

1.20

0. f>0

0.52

0.37

0.54

0.82

0.74

1.28

1.05

1.27

1.25

1.46

1. 10

0.92

1.02

1 00

1.08

0.64

1.25

(B-V),

ObiT

0. 30

0.30

0.53

0.30

0.40

0.44

0.32

0. 50

0.33

0.33

0.25

0.45

0.33

0.40

0.43

0.58

0. 30

0. 52

0.85

1.02

Pred. 7

(B-V)m

0.25 0.95 C

0.14 0.67 C

0.36 0.66 (

0.36 <

0.42 (

0.37 (

0.32 (

0.45 (

0.40

0.47

0.41

0.43 (

0.38

0.50

0.56

0.56

0.57

0.54

0.78

0.63

.48 (

.72 (

in

'red.

.66

.74

.62

.65

.63

. 47 0.66

.95 ( .75

.49 0. 85

.13

.02

.08

.90

).97

.10

). 96

.03

.10

1.29

1.43

L.31

.07

.03

.08

.09

.14

.09

.06

. 12

.11

.13

. 17

.33
NUMBER 17

TABLE 4.—Small Magellanic Cloud: Observed and

predicted colors

Star

HV 1897

HV 1871

HV 1907

HV 1779

HV 1869

HV 11114

HV2015

HV 1906

HV 11216

HV 11113

HV212

HV214

(HV 1425)

HV 1492

HV 1400

HV 11112

HV827

(HV 1328)

(HV 1342)

HV 817

HV823

HV2195

HV837

HV824

HV834

HV829

HV821

logP

0.094

0.114

0.216

0.251

0.391

0.433

0.458

0.486

0.494

0.517

0.591

0.624

0.658

0.799

0.823

0.826

1.129

1.200

1.254

1.276

1.514

1.621

1.629

1.818

1.866

1.942

2.104

(B-V)

Av Obs.

0.54 0.20

0.80 C .18

1.33 0.08

0.60 0.25

0.64 ( .32

1.22 0.24

0.82 (

1.07 (

1.03

0.96 (

0.93 (

1.08 (

0.91

0.87

0.84

0.71

0.96

0.68 (

0.78

0.86

1.22

1.06

1.05

0.97

0.66

0.73

0.58

.23

.20

.20

.19

.18

.18

.22

). 12

1.40

).33

.32

.36

).35

.36

1.48

).4O

1.64

).49

).58

).7O

1.85

max

0.23

0.16

0.04

0.26

0.29

0.14

0.26

0.20

0.21

0.23

0.27

0.23

0.29

0.34

0.36

0.30

0.42

0.60

0.50

0.49

0.45

0.53

0.54

0.61

0.72

0.72

0.81

(B-V),

Obs.

0.36

0.43

0.40

0.38

0.56

0.47

0.58

0.67

0.53

0.48

0.55

0.38

0.63

0.55

0.64

0.62

0.74

0.63

0.57

0.82

1.06

1.14

1.13

1.12

0.87

0.97

1.29

ain

PnxL

0.51

0.56

0.70

0.57

0.62

0.75

0.68

0.74

0.73

0.72

0.74

0.78

0.76

0.80

0.80

0.78

0.92

0.90

0.93

0.95

1.10

1.11

1.11

1.15

1.11

1.15

1.17

The comparison between the observed colors

of Magellanic Cepheids and the colors predicted

with the aid of equation (1) shows what appear

to be real differences of intrinsic color. At maximum, the average residual (O —C) is +0.017 for

the Large Cloud and -0.029 for the Small. It is

especially noteworthy that our "galactic" relationship predicts maximal colors for Small

Cloud Cepheids that are far bluer than any observed in the Galaxy, a consequence of the small

periods and large amplitudes. At minimum, the

average residuals are —0.062 for the Large

Cloud and —0.112 for the Small. However, as

pointed out in a preceding paragraph, the color

excesses have been underestimated, and the intrinsic colors implied by equation (1) for galactic Cepheids are too red—by about 0.03 at maximum and by 0.06 to 0.12 at minimum. These

differences are within the uncertainties of the

quantities measured, and there is no longer any

reason to insist on real physical differences between Magellanic and galactic Cepheids as a consequence of their observed colors. A similar conclusion has been reached by Sandage and

Tammann (1971).

There remain the striking differences in period

frequency between the Galaxy and the Clouds,

which have been long known and are undoubtedly real. No doubt they provide a clue to evolutionary history, in which the timing of active

star formation, the factors that have governed

the mass frequency of the stars formed, and the

condition and composition of the prestellar matter must all be taken into account. Are we justified in using the Magellanic Cepheids to establish the form of the period-luminosity relation

and then applying this period-luminosity relation to the study of galactic Cepheids and of Cepheids in other galaxies? As an introduction to

this problem, we now reexamine the period-luminosity relation for the Magellanic Clouds.

The Period-Luminosity Relation for the

Magellanic Clouds

It has long been known that (excluding stars

with sinusoidal light curves) the Cepheids of

largest amplitude tend to lie above the average

period-luminosity curve. It should therefore be

possible to express the period-luminosity relation

in the form

M - x + ylogP + zA, (2)

where M is the absolute magnitude, P the period

in days, and A the amplitude. In what follows,

we shall use the V magnitudes and amplitudes in

applying this relation. As has already been

shown, the stare of large amplitude are also the

bluest for their periods, so that there will also be

a relationship of the form

M - x' + y'logP + z' (B-V)o. (3)

Equation (3) requires a knowledge of intrinsic

colors; equation (2) does not, and is applicable if

A has been determined.

Least-squares solutions for the selected stars in

the two Clouds lead to the following results (apparent magnitudes, m, are taken from the data

published by Gascoigne (1969) to evaluate the

coefficients for the apparent period-luminosity relations).

I,arge Cloud:

at maximum:

m¥ -17.588 - 2.852 log P - 0.733A,

at minimum:

m -17.590 - 2.852 log P - 0.224A,

(4)

(5)
6

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

at mean intensity:

<V) = 17.640-3.003 log P-0.266A ;

Small Cloud:

at maximum:

mv = 17.903 - 2.828 log P - 0.763A,

at minimum :

m »17.931 - 2.838 log P - 0.221 A,

at mean intensity:

<V) =17.982-2.858 log P-0.317A.

(6)

(?)

(8)

(9)

The coefficients of A are so similar in equations (4) and (7), (5) and (8), and (6) and (9)

that we feel justified in adopting mean values.

The coefficients of log P in equations (4) and

(5) and in (6) and (7), that here again we adopt

a mean. The first term expresses the zero point,

which differs for the two Clouds. We adopt

means from equations (4) and (5) for the Large

Cloud and (7) and (8) for the Small. We thus

obtain the following:

uimax = 17.589-2.840 log P -0.773A (Large Cloud)

17.916-2.840 log P-0.773A (Small Cloud), (10)

mmin = 17.589-2.840 log P+0.223A (Large Cloud)

17.916 - 2.840 log P+0.222A (Small Cloud), (11)

< V) -17.640-2.930 log P-0.292A (Large Cloud)

17.982-2.930logP-0.292A (Small Cloud), (12)

The difference in zero point (SMC —LMC) is

O.m327 for the maximum-minimum relation and

O.m342 for the mean-intensity relation. The mean

of these values, O.m334, is adopted to represent

the difference in true modulus. A value of 0.m27

was derived by the writer (1969) from direct

comparison of the photographic period-luminosity curves. The present value, based on more accurate (though less) material and a more sophisticated approach, is to be preferred.

Tables 5 and 6 compare the observed maximal,

minimal, and mean-intensity magnitudes with

the predictions from equations (10), (11), and

(12). Two final columns give the magnitudes

predicted from the relations given by Gascoigne:

Large Cloud: (V) -17.58-3.072 log P,

Small Cloud: (V)-17.73-2.879 log P,

(13)

(14)

and the corresponding residuals, (O —C), which

have been calculated with an allowance for absorption of O.m

15 in the Large Cloud and O.m06 in

the Small, in accordance with the color excesses

derived by Gascoigne (O.mO5 and O.mO2) for the

two systems.

Our predicted magnitudes in Tables 5 and 6

were derived by using the same factors for slope

and amplitude in the two Clouds, whose predicted magnitudes thus differ only in zero point.

Gascoigne (equations (13) and (14)) determined

period-luminosity relations separately for the

two Clouds, and these differ in slope as well as

TABLE 5.—Large Magellanic Cloud: Observed and predicted

magnitudes

Star

W 4 4

HV 5541

= W 24

HV 12225

= W 33

W 2 5

HV 12747

= W 10

HV 12226

= W 29

W 2 2

(HV 2432)

HV886

HV 1003

HV902

HV2251

HV 1002

(HV 2294)

(HV909)

HV 900

HV953

HV2369

HV2447

HV 883

logP \

0.408 0.80

0.429 . 2 0

0.478 0.50

0.529 0.52

0. 566 0. 37

0.569

0.669

.038

.380

.387

.421

.447

.483

.564

.575

.677

.680

.684

2.074

2. 127

(.54

.82

. 7 4

. 2 8

. 0 5

. 2 7

1.25

.46

1.10

. 9 6

. 0 2

. 0 0

1.08

1.64

1.25

Observed

M

15.75

15.40

15.89

15.55

15.33

15.41

14.98

13.72

12. 4T

12.55

12.40

12.45

12.07

11.95

12.07

12.17

11.60

11.98

11.55

11.47

m

16.55

16.60

16.39

16.07

15.70

15.95

15.80

14.46

13.75

13.60

13.67

13.70

13. 53

13.05

12. 99

13.19

12.60

13.06

12.19

12.72

V

16.21

16.06

16.06

15.88

15.61

15.65

15.40

14.08

13.15

13.10

13.07

12. 95

12.77

12.54

12.56

12.61

12. 12

12.42

11.82

11.97

P

M

15.78

15.41

15.81

15.65

15.69

15.52

15.02

14.04

12.65

12.81

12.54

12.49

12.22

12.27

12.37

12.01

12.02

11.94

11. 17

10.55

redicted

m

16.57

16.60

16.31

16.17

16.06

16.06

15.84

14.77

13.92

13.85

13.80

13.72

13.67

13.36

13.29

13.02

13.00

13.01

11.81

11.79

V

16.21

16. 03

16.09

15.94

15.90

15.82

15 44

14.38

13.22

13. 27

13. 10

13.04

12.87

12.74

12.76

12.43

12.43

12.39

11.38

11.04

Res

M

-0.03

-0.01

+0.08

-0.10

-0.36

-0.11

-0.04

-0.32

-0.18

-026

-0.14

-0.04

-0. 15

-0.32

-0.30

+0.16

0.42

+0.04

+038

+0.92

lduals (O-C)

m

-0.02

0.00

+0.08

-0. 10

-0.36

-0. 11

-0.04

-0.31

-0. 17

-0.25

-0. 13

-0.02

-0. 14

-0.32

-0.30

+0. 17

-0.40

+0.04

+0.38

+0.93

V

0.00

+0.03

-0.03

-0.06

-0.29

-0.17

-0.04

-0.30

-0.07

-0.17

-0.03

-0.09

-0.10

-0.20

-0.20

+0.18

-0.31

+0.03

+044

+0.93

Predicted

(O

16.33

16.26

16.11

15.95

15.87

15.83

15.52

14.39

13.34

13.32

13.22

13. 13

13.02

12.78

12.74

12.43

12.42

12.41

11.21

11.05

-C)

+0 03

-0.05

+0

+0

- 0

- 0

- 0

- 0

- 0

- 0

0

10

08

11

03

0 3

16

04

07

00

-003

- 0 10

-0.09

- 0

+0

03

33

-0.15

+ 0

+ 0

+1

16

76

05
NUMBER 17

TABLE 6.—Small Magellanic Cloud: Observed and predicted

magnitudes

Star

HV 1897

HV 1871

HV 1907

HV 1779

HV 1869

HV 11114

HV 2015

HV 1906

HV 11216

HV 11113

HV 212

HV214

(HV 1425)

HV 1492

HV 1400

HV 11112

HV 827

(HV 1328)

(HV 1342)

HV 817

HV 823

HV 2195

HV 837

HV 824

HV834

HV 829

HV821

log P

0.094

0.114

0.216

0.251

0.391

0.433

0.458

0.486

0.494

0.507

0.591

0.624

0.658

0.799

0.823

0.826

. 129

.200

.254

.276

.504

.621

.629

.818

.866

.942

. 104

A V

0.54

0.80

1.33

0.60

0.64

1.22

0.82

1.07

1.03

0.96

0.93

1.08

0.91

0.87

0.84

0.71

0 96

0.68

0.78

0.86

1.22

1.06

1.05

0.97

0.66

0.73

0.58

Observed

M

6.69

L6.70

6.17

5.80

6.69

5.86

16.00

5.66

5.72

5.94

5.33

5.03

5.38

4.86

5. 17

5.30

3.98

3.70

3.93

3.30

3.19

2.46

2.86

1.97

1.83

1.63

1.53

m

17.54

17.57

7.50

6.63

7.34

7.00

16.85

6.80

6.74

6.89

6.31

6.07

6.29

5.64

6.00

6.00

4.88

4.38

4.71

4. 10

4.39

3.51

3.70

2.94

2.32

2. 11

V

6.71

7.10

6.S6

6.04

6.98

6.44

6.37

6.21

6.21

6.46

5.79

5.61

5.87

5.21

5.46

5.59

4.39

4.07

4.27

3.76

3.83

2.92

3.22

2.29

2.11

1.91

1.86

Predicted

M

17.24

16.98

16.28

16 74

16.32

15.75

15.99

15.72

15.72

15.74

15.52

15.32

15.35

14.98

14.94

15.03

13.97

13.99

m

7.77

7.77

7.60

7.34

6.95

6.96

6.80

6.78

6.74

6.69

6.44

6.38

6.25

5.84

5.77

5.73

4.92

4.66

13.76 14.53

13.63 14.48

12.71

12.50

3.92

3.55

12.48 13.52

12.01 12.97

12. 11

11.84 12.56

11.50 12.07

V

17.55

17.41

16.96

17.11

16.64

16. 35

16.40

16. 24

16. 23

16.21

13.97

15. S3

15.78

15.38

15.32

15.35

14.39

14.26

14.07

13.99

13.21

12.92

12.90

12.37

12.32

12.07

11.64

Residuals (O-C)

M

-0.55

-0.28

-0.11

-0.94

+0.37

+0.11

+0.01

-0.06

0.00

+0.20

-0.19

-0.29

+0.03

-0.12

+0.23

+0.27

+0.01

-0.29

+0.17

-0.33

+

-

+

-

•4

*

). 35

).O4

(.33

. 0 4

. 2 7

).21

).O3

m

-0.34

-0.20

-0.10

-0.71

+0.39

+0.04

+0.05

+0.02

0. 00

+0.20

-0.13

-0.31

+0.04

-0.20

+0.23

+0.27

-0.04

-0.29

+0.18

-0.38

+0.47

-0.04

+0.18

-0.03

-0.24

+0.04

V

- 0 . *<4

-0.31

-0. 10

-1.07

+0.34

+0.09

-0.03

-0.02

-0.02

+0.25

-0.18

-0.22

+0.09

-0. 17

+0.14

+0.24

0.00

-0.19

+0.20

-0.23

+0.62

0.00

+0.32

-0.08

-0.21

-0. 16

+0.22

Predicted

(O-C)

17.46

17 40

17.11

17.01

16.60

16.48

16.41

16.33

16.31

16.27

16.03

15.93

15.84

15.43

15.36

15.35

14.48

14.28

14. 12

14. 08

13.40

13.06

13.04

12.50

12.36

12.14

11.67

-0.69

-0.26

-0.19

-0.91

+0.44

+O.02

+0.02

-0.06

0.04

+0.25

-0. 18

-0.26

+0.09

-0.16

+0.16

+0.30

-0.03

-0. 15

+0.21

-0.24

+0.43

-0.08

+0.24

-0.15

-0.19

-0.17

-0.27

in zero point. The average residual for our predictions in V, without regard to sign, is 0.m

18 for

each of the Clouds. The average residuals from

Gascoigne's formulas are 0.ml7 for each of the

Clouds. Thus, we have represented the data very

nearly as well by using a single period-amplitude-magnitude relation for both Clouds as Gascoigne has done by using two separately determined period-magnitude relations.

If these relationships are applicable to galactic

Cepheids, we can refine the color excesses and

luminosities and rediscuss the galactic distribution. The practical advantage of our formulas is

that they do not require a knowledge of intrinsic

color. The theoretical advantage of a period-luminosity-color relation is counterbalanced by the

fact that its application requires a knowledge of

color excess.

Our success in representing the (V) magnitudes of the Cepheids in the two Clouds by the

same period-amplitude relation suggests that, in

spite of the differences of period-frequency, amplitude, and color, the period-luminosity relations for the Cepheids in the two Clouds are sensibly the same, at least as regards dependence on

period and amplitude. It seems legitimate to consider them parallel; but without an independent

check on the moduli, we cannot be sure that they

are coincident. When we recollect the period-amplitude-color relations of the two Clouds and the

Galaxy, discussed in the previous section, we

must admit that this restriction holds with even

greater force to any application of a period-luminosity relation derived from Magellanic Cepheids to a discussion of galactic Cepheids. In

what follows, we shall assume that the periodamplitude-luminosity relation for the Galaxy has

the same form as equations (10), (11), and (12).

We can then determine its zero point by reference to galactic Cepheids for application to the

Galaxy.

For this purpose, we follow the classical procedure of using the color excesses and distance

moduli of the open clusters in which Cepheids

have been recognized. Two selections of data,

which lead to almost identical results, are represented in Tables 7 and 8. Although we do not

consider that the four Cepheids assigned by Sandage and Tammann (1968) to the Perseus Association are temporally associated with either the
8 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

double cluster or the Association (and the assumption that they are spatially associated rests

on the ad hoc assumption that they are at the

same distance), their inclusion makes little difference to the final result. We have assumed that

the color excess of the Cepheid is the same as

that of the cluster in which it is situated. The

only star for which our discussion of color excess

(under "The Period-Color Relation") might require a modification of this assumption is RS

Puppis.

Table 7 contains material for deriving the zero

point from the photometric moduli and color excesses given by Johnson et al. (1961) for the relevant clusters. Successive columns give the name

of the Cepheid; the logarithm of its period; apparent V magnitude at maximum; (V) averaged

for mean intensity; V amplitude; cluster containing the Cepheid; color excess and true modulus of the cluster as given by Johnson et al.;

absolute magnitudes at maximum and mean

light, derived from columns 3, 4, 7, and 8; values

of -2.840 log P-0.773A (equation (10)) and of

-2.930 log P-0.292A (equation (12)); and the

arithmetic differences between the entires in columns 9 and 11 and in 10 and 12. The means of

these differences should furnish numerical values

for the first terms of the galactic equivalents of

equations (10) and (12).

Table 8 contains similar material for deriving

the zero point from the data used by Sandage

and Tammann (1969). Their list contains more

stars, and their adopted values for color excess

and modulus are not the same as those of Table

7. Despite these differences, the mean values of

the entries in columns 13 and 14 are nearly the

TABLE 7.—Zero paint of pcriod-amplitude-luminosity relation (Johnson data)

1

Star

EV Set

CE Cas b

CF Cas

CE Cas a

USgr

Di Cas

RS Pup*

2

logP

0.490

0.651

0.687

0.711

0.828

0.903

1.617

3

Apparent

Max.

10.00

10.61

10.82

10.62

6.32

8. 64

6.52

4 5

V Magnitude

Mean

10. 12

10.99

1 1 . 12

10.92

6.76

8.96

7.08

Amp.

0.28

0.68

0.57

0.52

0.78

0.60

1. 12

6

Cluster

NGC 6664

NGC 7790

NGC 7790

NGC 7790

H 25

NGC 129

Velam

7

Johnsor

EB-V

0.60

0.52

0.52

0.52

0.50

0. 58

0.62

8

et al.

<m-M)0

11.03

12.80

12.80

12.80

8 . 9

11. 1

11.30

9 10

Absolute Magnitude

Max.

-2.83

-3.75

-3.54

-3.73

-4.08

-4.20

-6.44

Mean

-2.71

-3.37

-3.24

-3.44

-3.64

-3.88

-5.98

11

-2. 840 log P

-0.773 A y

-1.62

-2.38

-2.38

-2.42

-2.96

-3.03

-5.44

12

-2. 930 log P

13 14

Difference

-0.292 Ay Max.

-1.52

-2.00

-2. 17

-2. 15

-2.65

-2.84

-5.06

Means:

. 2 1

37

. 1 6

1.31

. 12

. 17

. 0 0

. 19

Mean

1. 19

1.37

1.07

1.29

0.99

1.04

0.92

1.12

Data from Westerlund (1963).

TABLE 8.—Zero point of period-amplitude-luminosity relation

(Sandage and Tammann data)

1

Star

SU Cas

EVSct

CE Cas b*

CF Cas

CE Cas a*

UYPer

VY Per

f Sgr

DL Cas

SNor

VXPer

SZ Cas

RS Pup

2

logP

0.290

0.490

0.651

0.687

0.711

0.730

0.743

0.828

0.903

0.989

1.037

1.134

1.617

3

Appar

Max.

5. 76

10.00

10.61

10.82

10.63

10.83

10.76

6.32

5.64

6.11

8.97

9.60

6.52

4 5

ent V Magnitude

Mean

3.95

10.12

10.99

11.12

10.92

11.34

11.19

6.76

8.96

6.46

9.33

9. 83

7.08

Amp.

0.40

0.28

0.68

0.57

0. 52

0.91

0.79

0.78

0.60

0.65

0.75

0.40

1. 12

6

Cluster

Adopted

NGC 6664

NGC 7790

NGC 7790

NGC 7790

h,XPer

h,X Per

M 25

NGC 129

NGC 6087

h,XPer

h, X Per

Vela ni

7

EB-V

0.33

0.58

0.555

0.555

0.555

0.98

1.06

0.55

0.50

0.23

0.75

0.88

0.55:

8

m-M)0

7 . 5

1.03

2.53

2.53

2.53

1.90

1.90

8.98

1.28

9.76

1.90

1.90

LI. 30

9 10

Absolute Magnitude

Max.

-2.75

-2.77

-3.585

-3.375

-3.565

-4.00

-4.30

-4.27

-4.09

-4.34

-4.68

-4.93

-6.41

Mean

-2.54

-2.62

-3.205

-3.075

-3.275

-3.54

-3.91

-3.92

-3.84

-4.03

-4.34

-4.71

-5.95

11

-2. 840 log P

-0.773 A y

-1.62

-1.62

-2.38

-2.38

-2.42

-2.78

-2.72

-2.96

-3.03

-3.32

-3.52

-3.53

-5.44

12

-2.930 Ing P

13

Differet

14

c e

-0.292 A y Max. Mean

-1.46

-1.52

-2.00

-2.17

-2.15

-2.40

-2.41

-2.65

-2.84

-3.08

-3.26

-3.44

. 1 3

. 15

. 2 0

. 0 0

. 1 4

. 2 2

. 5 8

. 3 1

. 0 6

.02 (

. 16

. 4 0

. 0 8

. 10

. 2 1

. 9 1

. 1 3

. 14

. 5 0

. 2 8

. 0 0

. 9 5

. 0 8

. 2 7

-5.06 0.99 0.89

Means: . 1 8 .12

Data from Sandage and Ta
NUMBER 17

same for both choices of data. The fact that the

numbers in each of these columns in Tables 7

and 8 are so nearly the same for all the stars is

an admittedly rough verification of our assumption that a period-amplitude-luminosity relation

derived from the Clouds only is also applicable

to galactic Cepheids.

If this assumption is adopted, we obtain for

galactic Cepheids, by combining the results from

Tables 7 and 8,

M{v) 1.12-2.930 log P-0.292A, (15)

Mmax - -1.185-2.840log P-0.774A. (16)

Equation (15) invites comparison with the period-luminosity-color relation derived by Gascoigne for Magellanic Cepheids:

M, -2.33-3.609logP+2.52 (B-V). (17)

Small Cloud: M<v> -17.98-2.930logP-0.292A,

°-17.03-3.609log P+2.52 (B-V).

We arrive at a comparison by means of our relation (equation (12)) and Gascoigne's formulas

(equations (18) and (19)) for the two Clouds:

Large Cloud :M<v) -17.64-2.930log P-0.292A, (12)

-16.40-3.609log P+2.52 (B-V) ;

(18)

(12)

(19)

We can relate our formulations with Gascoigne's

by expressing A in terms of (B —V) by means of

equation (1):

Maximum:

Vo-const-3.640 log P+2.74(B-V)m»ji, (20)

Minimum:

Vo-const.-3.2221ogP+1.17(B-V)min. (21)

The coefficients of log P and of (B —V) in equation (17) should lie between those of equations

(20) and (21), and it is seen that they do so. Our

relations are therefore not inconsistent with Gascoigne's.

The period-amplitude-luminosity relation and

the period-color-luminosity relation are in fact

complementary. The former states that for a

given period the stars of greatest amplitude are

the brightest; the latter, that for a given period

the bluest stars are the brightest. The coefficient

of log P in equation (12) expresses a period-luminosity relation for zero amplitude; those of

equations (17), (18), and (19) express a periodluminosity relation for zero (B-V). The slope

of the period-luminosity relation for (V) light

therefore lies between 3.609 and 2.930.

If we consider that the zero point of the period-luminosity relation is the same for the Galaxy and both the Clouds, we can use equations

(12) and (15) to determine distance moduli for

the latter. Thus, we obtain moduli of 18.77 and

19.10 for the Large and Small Clouds. From

equations (17), (18), and (19), Gascoigne has derived the values 18.73 and 19.36. It should be emphasized that there is nothing in our data to justify the assumption that the period-luminosity

relations in the three systems are coincident. Because of the similarity of the deduced relationships for the two Clouds (except for the difference of apparent zero point), we have argued

that they are probably parallel. But all the differences between the Cepheids in the three systems (period-frequency, amplitude, and color, for

example) require us to leave the coincidence of

the absolute zero point an open question.

We have determined a true modulus of 18.77

for the Large Cloud, but this value was derived

via the galactic Cepheids, on the assumption of

coincident period-luminosity relations. An independent method of determining the distance

modulus is required. A more direct determination

of the moduli of the Clouds, based on the eclipsing stars, has been made by Gaposhkin (1972).

His values, 18.36 for the Large Cloud and 19.15

for the Small, when compared with the values

18.77 and 19.10 given above, imply differences in

zero point of O.m41 and O.Hib for the two Clouds,

the Large Cloud being nearer and the Small

Cloud a little more distant than our own moduli

imply. It would of course be better to use a direct method of determining the distance, such as

that involving the eclipsing stars, than to assume

that the period-luminosity relation has everywhere the same zero point, even though the form

of the relationship has been shown to be the

same everywhere.

In summary, the Cepheids of the Galaxy and

the Magellanic Clouds display a gradation in

properties, but when amplitude as well as period

is taken into account, their intrinsic colors are

sensibly similar. The period-luminosity relations

in the three systems are probably parallel, but

not necessarily coincident.
10 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

References

Bahner, K., W. A. Hiltner, and R. P. Kraft

1962. Colors and Magnitudes for 45 Cepheids in the

Northern Milky Way. Astrophysical Journal

Supplement, 6:319-355.

Fernie, J. D.

1970. A Method for Determining the Intrinsic Colors

of Cepheids from Their Periods and Light Amplitudes. Astrophysical Journal, 161:679-684.

Gaposhkin, S.

1972. The Large Magellanic Cloud: Its Topography

of 1830 Variable Stars. Smithsonian Astrophysical Observatory Special Report No. 310, 52

pages + figures.

Gascoigne, S. C. B.

1969. Further Observations of Magellanic Cloud Cepheids. Monthly Notices of the Royal Astronomical Society (London), 146:1-36.

Gascoigne, S. C. B., and G. E. Kron

1965. Photoelectric Observations of Magellanic Cloud

Cepheids. Monthly Notices of the Royal Astronomical Society (London), 130:333-360.

Guinan, E. F.

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the Astronomical Society of the Pacific,

84:56-60.

Johnson, H. L., A. A. Hoag, B. Iriarte, R. I. Mitchell,

and K. L. Hallam

1961. Galactic Clusters as Indicators of Stellar Evolution and Galactic Structure. Lowell Observatory Bulletin, 5(8) :133-147.

Kraft, R. P.

1961. Color Excesses of Supergiants and Classical Cepheids, V: The Period-Color and PeriodLuminosity Relations : A Revision. Astrophysical

Journal, 134:616-632.

Kron, G. E., and S. Svolopoulos

1959. Color Excesses of 24 Galactic Cepheids Derived

from Six-Color Photometry. Publications of the

Astronomical Society of the Pacific,

71:126-144.

Oosterhoff, P. Th.

1960. Three-Color Photometry in the U, B, V System

of 51 Northern Cepheids. Bulletin of the Astronomical Institutes of the Netherlands,

15:199-228.

Payne-Gaposchkin, C.

1969. Comparison of the Cepheid Variables in the

Magellanic Clouds and the Galaxy. Presented

at the Symposium on the Magellanic Clouds,

Santiago, Chile, on the occasion of the dedication of the Euroi>ean Southern Observatory.

Also pages 34-46 in A. B. Muller, editor, The

Magellanic Clouds. Dordrecht, Holland: D. Reidel Publishing Co., 1971.

Sandage, A. L., and G. A. Tammann

1968. A Composite Period-Luminosity Relation for

Cepheids at Mean and Maximum Light. Astrophysical Journal, 151:531-545.

1969. The Double Cepheid CB Cassiopeiae in NGC

7790: Tests of the Theory of the Instability

Strip and the Calibration of the Period-Luminosity-Color Relation. Astrophysical Journal,

157 :683-708.

1971. Absolute Magnitudes of Cepheids, III: Amplitude as a Function of Position in the Instability Strip: A Period-Luminosity-Amplitude Relation. Astrophysical Journal, 167 :293-310.

Westerlund, B.

1963. An OB Association in the Region of RS Puppis.

Monthly Notices of the Royal Astronomical

Society (London), 127:71-81.

Wildey, R. L.

1963. Optimum Use of Interstellar Reddening Data

to Obtain Intrinsic Stellar Luminosities and

Colors. Astronomical Journal, 68:190-194.

Williams, J. A.

1966. Metal-to-Hydrogen Ratios in the Galaxy as Indicated by Narrow-Band Photometry of Cepheids. Astronomical Journal, 71:615-634.
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