The problem of RR Lyrae stars with several periods

The Problem of RR Lyrae Stars with Several Periods L. DETRE Konkoly Observatory, Budapest, Hungary SUMMARY

The observational material concerning multiple periodicities of RR Lyrae stars has rapidly grown in recent years, and some general conclusions as to the nature of the changes of the radial velocity- and light-curve can now be drawn. The results showthe complexityof the problem of the secondaryperiods, which cannot be explained satisfactorilyby present theories.

1. INTRODUCTION THE majority of the short-period Cepheids (RR Lyrae stars) repeat their light-curves from period to period with great regularity, although the periods themselves are subjected to minute and slow changes. Some of them, however, show striking changes especially in the height of maximum of the light-curve, as was first reported by BLAZHKO (1925, 1926) and SHAPLE¥ (1915, 1916) nearly forty years ago. From long series of observations, extending over two decades, BLAZHKOsucceeded in demonstrating in the case of XZ Cyg and RW Dra t h a t the changes in the shape of the short-period light-curve can be fitted by a longer period. Russian authors therefore refer to these fluctuations of the light-curve as the "Blazhko-effect". Similar results have been obtained by HERTZSPRUNG (1922) for R R Lyr. 2. OBSERVATIONAL MATERIAL

The Budapest Observatory started in 1935 with an investigation of the light-curves and period-changes of a larger number of R R Lyrae stars, using a 16-cm photographic camera. I t might be hoped t h a t by scrutinizing both problems a better insight can be gained into the still unknown mechanism of these stars. Furthermore, the analysis of period-changes, if they are proved to contain secular terms, may also be of cosmogonic significance. The periods can be determined with incomparably greater accuracy than any other physical data of these stars. Sooner or later, changes in these periods will be the first sign of changes in other characteristic data on which the periods are dependent. Up to date nearly 80,000 photographs have been taken for forty-six R R Lyrae stars, and the greater part of this material has already been measured and discussed. Since 1950 our 24 in. reflector has been engaged in photo-electric work on SW And, VZ Cnc, RZ Cep, R R Lyr, and RU Psc. With the same instrument also the periodvariations of the variables in the globular clusters M3, M5, M15, M56, and M92 are under observation. In recent years a number of important investigations of this problem have been carried out also at other observatories. Results of great interest have been obtained by the Leiden observers, OOSTERIIOFF (1946a), WALRAVEN (1949), and MULLER (1953), for RS Boo, R R Lyr, XZ Cyg, and especially for the two bright ultra-shortperiod southern variables AI Vel and HD 233065 (WALRAVEN,1952, 1953). Of utmost importance were the co-operative investigations by the McDonald and 1156

537.0

76.0

40.7

1

~

?

1424.0

160.0

153.8:

122.7

1

121.4

122.1

d

1%“.

?

85.2:

91.1

?

2?04:

94.0

84.0

83.7

71.8

67.3

63.7

54.7

4.016

3.508

3.398

1.354

I

-

2.69:

2.92

2.85

?

3.00

?

2.56:

3.05

?

2.9:

0.15

0.32

0.25

0.56

0.65

0.05

0.32

0.53

0.66

0.65

0.44

0.45

0.39

l”37

1.19

1.56

~ 1.16

i 0.90

~ 1.32

~ ?

I i

I ;I;

0.38

--~

0.16

1.34

1.29

~ 0.86

/ O-08: I .40

0.34

-

0’14

-

0.48:

0.32

-

-

II

0.11

0.25

’ 0.29

0.40

0.42

0.05

0.36

0.40

05i

0.69

0.73

1.04

1.70

0.25

0.16

-

-

0.19

0.06:

0.22

1 -

j

~ -

/

! -

-

-

-

(1953)

(1940);

(1938);

(1953)

OOSTERHOPB (1946a)

(1941)

BAL~ZS

DETRF: (1952) (1953);

and

(1954)

and DETRE

BAL.~ZS and DETRE

MULLEK

RALAZS

GUMAN

BAL~ZS

(1954)

and

(1943);

(194tib) DETRE

O~STERHOFF

BALAZS and DETRE (1953)

(1951)

WALRAVEN (1949); BAL~ZS DETRE and GUMAN (1954)

BALAZS and DETRE

iwARTINOFF

BAL.~ZS and DETRE

GUMAN (1954)

WALRAVEN

( 1952)

(1954)

WALRAVEN

GUMAN

References

Notes to Table 1. JIore periodicities have been obtsinrd by us for the stars SW and, P, = 4.6 years; RW Cnc, P, = 25 years; RW Dra, P, = 7.4 years, P, = 41 years; RR Lyr, I’$ = 11 years. The two short periods 0?04439 and 0’!03176 found by WALRAVEN for AI Vel ~eenl to me not warranted. The sign (?) indicates a period, the existence of which is established, although its length could not yet be determined. ZESSEWlTSH arid USTINoV (Trudi&?rnberg Inst. Mowou~, 23, 1953) announced a secondary period of 222 days for RV Cap; apparently this star has still xwther, shorter period. DL IIer has two periods. P, = Of%“ and P, = 49$L according to ZESSEWITSH(12~. _4stro?t.Obs. Odessa, vol. III, ‘757, 1953). P.4KEN.4oO(private connnunication), using O’COSSELL’S observations of VY Pup found for this star P 1= l$OO.

1 0.377

RS Boo

~

57.25

0.467

CYzz

XZ

~ 0.476

41.64

0.443

Dra

RW

XZ Dra

42.95

0.511

RZ Lyr

i

37.0

0.442

SW And

.

0.567

RR Lyr

31.6

0.470

33.4

29.6

AR Her

0.547

0.522

Cnr

0.716

0.193

1

~

0.379

i

0?711

Y L Mi

RW

Cm

vz

0.178

0.055

233065

HD

0!525

0.112

.

AI Vel

AC And

star

] 158

Th(~ l)rohlem of I~R Lyra(~ stars with s(~vc~'al l)erioels

Leiden Observatories, achieving simultaneous measurements of radial velocity and of brightness for R R Lyr (STRUVE and BLAU~rW, 1948) and XZ Cyg (STRUVE and vA~ HooF, 1949). At Cordoba, GRATTO~- and LAVAGh~NO (1953; GRATTON, 1953) determined the variations of the radial velocity curve of AI Vel. All these observations have demonstrated that the velocity curve changes in conformity with the periodic fluctuations in the character of the short-period light-curve. Furthermore, the spectrographic observations by STRUVE (1947, 1949), SANFORD (1949, 1952), and ABT (1952) of R R Lyr and W Vir have thrown new light upon the entire problem of Cepheid variation. 3. D~scussIoN Table 1, which contains much unpublished material, summarizes the present data for those R R Lyrae stars which show variations in their light-curves. The funda2.

i

3,

I i I

q

5.

II I

(

6

I I I

I I

J

i

r

4.

b*

L

I I

7.

i

I

i

Fig. ]. The light-curve in different phases of P1, when P1/Po is large (schematic). The dotted line marks a constant phase of the flmdamental period P0

mental period is denoted b y P0, the period of the light-curve variations by P1. For some stars there exists a second period, P~, with which light-curve fluctuations of the same character are connected as is the case with P1- The quantities Am 1 and Am 2 denote the total amplitude of the variation of maximum light (during P1 and P2, respectively), expressed in magnitudes, z[ is the mean amplitude of the shortperiod (P0) light-variation. The quantities (~1 = Am1/'4 and 62 = Am2/z~ characterize in some manner the strength of the light-curve variations. It is apparent from the table that according to P1 the stars form two groups. In the first, P1 is shorter than a day. The ultra-short-period variables belong to this group, but AC And shows that the latter is not confined to such stars. In the second group, P 1 is much longer than P0. Fig. 1 illustrates how, in this case, the hght-curve is changing during a cycle of P1. It is sufficient to consider only the ascending branch and the maximum of the light-curve, as the changes along the descending branch are scarcely perceptible. The light-curve variations in Fig. 1, especially the appearance of a surge in the lowest maximum, resembles the special type of wave-form arising from the superposition of two waves when one frequency is nearly a multiple of the other. Consequently, the period P1 is in general considered as a beat-period, arising from the

L. DETICi,~"

1159

interference of the fundamental period P0 with a shorter period Pl, the shift of the maximum being from right to left. As the oscillation Pl affects only the neighbourhood of the maximum, and the superposition of Pl with P0 is far from being linear, the commensurability factor of the frequencies, and consequently Pl itself, cannot be unambiguously determined from the observations. As can be seen from Fig. 1, the changes of the light-curve are essentially composed of oscillations in phase and height of the maximum of the curve. The variations are

d + 0-04

/

+ 0'02

0.00

-0.02

-0.04

In

f ~-

,0.5

=

0 "5

0"0

-*

0-5

0-0

0.5

Fig. 2. Above." V a r i a t i o n of p h a s e of m a x i m u m light of I4W Dra. Below: V a r i a t i o n of m a x i m u m b r i g h t n e s s d u r i n g t h e s e c o n d a r y period P1. Left: in 1937 ; right: in 1944 (photographic o b s e r v a t i o n s b y BAL~ZS a n d DETRE). ~ d e n o t e s t h e p h a s e s of t h e period P1 ~ 41~.16

much less pronounced for minimum light. As the phase-shift of the minimum is opposite to that of the maximum, a strong variation can be observed in the slope of the rise to maximum. The phase-shift curve of maximum light generally stands in the same phase-relation to the brightness of the maximum as shown in Fig. 1. But there are exceptions to this rule and the phase-relation mentioned can vary with time even for one and the same star (Fig. 2). For stars in the first group the light-curve obviously does not take during one cycle of P1 all the forms shown in Fig. 1. But Fig. 3 shows that essentially the same things are going on here, too.

The iu'oblem of RH Lyrae stars with several periods

1160

A different picture has been obtained for SW And (Fig. 4). Here the maximum brightness shows only insignificant changes, if any, but a hump periodically appears and disappears on the ascending branch. Now, SW And is an anomalous RR Lyrae star, probably of BAADE'S Population I, while most RR Lyrae stars are typical representatives of Population II. I f there also exists another period P~, the system of light-curves will be different in different cycles of the period Pl. As we know from the observations, the oscillations P1 and P2 are combined in a very nearly linear manner. Very curiously the Am

W=

0"25

0'50

0.75

I

[

1

-0.5

J.D.2434476

00

J D. 2434393

I

1

+0. ¢

I

O.35

I

0.40

I

045

I

0.40

I

0.45

t

0..50

I

0-55

I

O.bO

065

Fig. 3. Changes of the light-curve of the ultra-short-period RI~ Lyrae star VZ Cnc during half a cycle of the secondary period P1. (Photo-electric observations by GroAN)

ratio P2/P1 for all stars is closely the same: 2.9. It is customary to consider also P2 as a beat-period of two oscillations Px and P2. But the constancy of P2/P~ and the fact that for RW Cnc, I~Z Lyr, and Y LMi the period Pe surely cannot be interpreted as a beat-period makes it probable that Pe is a real period for all stars. 4. T H E CHANGES OF THE PERIODS

After the observations have been extended over some years, the periods P0 and P1 turn out not to be constant. The same is true for P2 as was shown for RW Cnc. After some time long-period terms can be established in the variations of P0 and P1. In the case of P1 this was demonstrated for R W Dra and R R Lyr. In the lightvariation of RW Dra we established until now not less than five periods and a sixth is suspected (see Note to Table 1). As an illustrative example I may take the star RI~ Lyr. The light-curve of the principal period P0 = 0-d5668 changes with a period P1 = 40"d7- The upper part of Fig. 5 shows how the phase of median magnitude on the ascending branch is oscillating in successive cycles of the secondary period P1. As can be seen, the amplitude of this oscillation varies in a longer period, P2 = 122d, found first by WALRAVEN (1949). The variations of P0 and / ° 1 a r e shown in the lower part of Fig. 4 by the so-called (O-C)-diagrams, on which are plotted the residuals in the observed times of a fixed phase in P0 and P1, compared with time computed with a constant period.

0

L

6

"C

',,...

o

0

2

®

6 ~r

~6

C

N~

6

0

0

--

m

0

0

~o ~,° o~

--

.\

0 ~

0

0

~o 6

%

0

6 ~0 ~r

\

o

o

|

o

0'

0

S

L

?

\

0

6

0 0

6

r~

o

~o

~o

o 0 0 0

~ o

~ 0

0

0

0

0

'~0

..9. 0

~9 0

0 ~.

1162

T h e p r o b l e m of R R Lyrae stars with several periods

In the diagram for P0 our photographic and photo-electric observations, extending over 15 years, demonstrate the presence of a period P3 of about 4000 days superposed upon a much longer cycle. If we now construct the same diagrams for different points of the ascending branch of the short-period light-curve, they will (while showing the 4000-days cycle) not run exactly parallel to each other. That is, the slope of the rise to maximum varies not only with P1 and P2, but also with P3. These variations are of the same character as those associated with P1 and P2. We have obtained J.D. 2434560

O-C d O-O10

6OO

650

- O'O15

-0"020

-0025 d +Oq

d

+ 40-c,

0.0

f /

4- 20

/

-0.1

S/

o",

/ / / --

\

/

I

f

-0"2

J.D.

2420000

-

2430000

/

/

/

/

20

2420000

2430000

Fig. 5. The periods of R R Lyrae and their variations. Above." Variations of phase-shift of the epoch of median brightness, in 1953, c o m p u t e d from the elements: C -- J D 2414856.580 -- 0.d5668340E (photo-electric observations by DETRE). Below left: (O-C)-diagram for 1' 0 after eliminating the P1- a n d P2-oscillations. Right: (O-C)-diagram for P1 with C -- J D 2414905 ~ 40~77n. D o t s represent photographic, circles photo-electric observations obtained at Budapest, the other s y m b o l s s t a n d for observations b y W~'NI)ELL, HERTZSPRt:NG, KUKARKIN, ZACHAI¢OV, DE SITTER, RYBKA, and MERGENTALER

the same results for RW Dra by discussing the effects on the form of the light-curve of the long periods P3 --~ 7.4 and P4 ---- 41 years. We have thus demonstrated that all cycles, short or long, are connected with light-curve variations of the same type. The period P3 of RR Lyr is conspicuous also in the (0-C)-diagram for P1. Here, it causes beside changes in the length of P1 also changes in the amplitude of the lightcurve oscillations connected with P1 itself. In the case of RW Dra this holds likewise for Pa and P4. The differences between the left and right halves of Fig. 2 can be explained by the presence of these periods. As far as the radial velocities are concerned, only changes connected with P1 could hitherto be studied. One would think that the composition of two periods is

n . DETRE

1163

much simpler for the radial velocities than for the light-variations, where a large degree of intermingling must be taken into account. But this is only partly true. As GRATTONand LAVAG~INOhave shown for AI Vel, although the amplitudes of the changes in the maximum and minimum radial velocity are equal, the composition of P0 and P1 is otherwise not linear. 5. THEORETICAL CONSIDERATIONS The theoretical side of the problem of RI~ Lyrae stars showing multiple periodicities is very similar to that of the Beta Canis Majoris stars, which were recently reviewed by STaUVE (1952), though the physical differences between the two groups of stars 6i I-8

I.b I-4 1.2 I.O 0.8 O.b AI,

0.4 0.2 O'O

a[~Oo o

@

i

0

1.0

2.0 LOG P.._[i

3"0

Po Fig. 6. Correlation b e t w e e n t h e q u a n t i t i e s bi (i = 1, 2), c h a r a c t e r i z i n g t h e s t r e n g t h of t h e l i g h t - c u r v e c h a n g e s d u r i n g P i , a n d t h e ratio Pi/Po. D o t s for i = l, circles for i = 2. T h e a n o m a l o u s R R L y r a e s t a r S W A n d is n o t included in t h e d i a g r a m

may be very large. Multiple periodicities can be present in a pulsating star if, besides the fundamental mode, some higher modes are also excited. This can be caused by resonance, if the frequency of a higher mode is nearly a multiple of that of the fundamental mode. KLUYVER (1936, 1938, 1947) and WOLTJER (1937, 1946) have suggested that the period P1 is due to a coupling between pulsations in the fundamental mode and in another mode, the period of which is nearly half that of the principal mode. SCHWARZSCI-IILD( 1 9 4 1 ) has shown for the standard model that for a reasonable value of the ratio of the specific heats there exists a close 2 : 1 commensurability between the first and second modes. Now, there exists (see our Table 1) a remarkable statistical correlation between the lengths of the secondary periods and the quantities 51 and 52, which are proportional to the relative variations of maximum light, and in this way characterize the strength of the light-curve variations. This correlation is represented in Fig. 6, and if P1 is interpreted as the beat-period of two commensurable periods, it can be expressed as follows: the closer the commensurability between the interfering

1164

The p r o b l e m of R R L y r a e s t a r s w i t h s e v e r a l pe ri ods

periods, the smaller the light-curve changes caused by this interference. This is actually contrary to what would be reasonably expected. So we must abandon the interference of radial modes as an explanation of the changes of the light and radial velocity curves. Multiple periodicities can come into being also by rotation. The general effect of the rotation is, as was shown by LEDOUX (i 951) and by COWLING and NEWING (i 949) that, corresponding to each free period of a non-rotating star, there are several free periods of the rotating star, some greater, some smaller than the original period. By identifying PI as the beat-period of the two simplest oscillations, LEDOUX (1949) derived for the period of rotation the neat expression Prot ~ l'6P1' This would give reasonable data for stars with large P1/Po, but the effect of rotation on the broadening of spectral lines should be well observed in such stars as AI Vel. However, no such effect was observed. For AC And the hypothesis was proposed (LuRJE, 1950) that this is a close double, with each component an RR Lyrae star. But the spectrum of this star, studied by MfrNcH (195 i), excludes this possibility. I think the solution of the problem must be sought in the light of the abovementioned results of STRUVE, SANFOI~D, and ABT. According to these, the atmosphere of an RI~ Lyrae star receives pulses or shocks from the interior in intervals Po, but the shock waves have a pulsation period 2P 0 in the atmosphere. The atmosphere thus consists of two layers, one falling into the star, the other moving outwards. The oscillation of these layers can be nearly harmonic, the steep fall in the radial velocity from maximum to minimum, and the corresponding steep rise in the brightness, arises from the overlap of the two cycles : the component belonging to maximum velocity tends to predominate in the early part of the overlap, and the conlponent accompanying the curve at minimum does so at the end of the overlap. Now, the variations in the velocity- and light-curve are essentially confined to this overlap stage, which may probably suffer periodic disturbances through the interaction of the two waves. Before discussing this problem further, however, one must first be convinced that the phenomenon of the overlapping of two layers is not a freak occurrence for RR Lyr and W Vir. VZ Cnc will now give a good opportunity for further study, since it is nearly as bright as l~R Lyr. For smaller instruments the search for new RR Lyrae stars having multiple periodicities and the determination of these periods continues to be an attractive and valuable field of investigation.

REFERENCES ABT, A . . . . . . . . . . . . BAL~,ZS, JULIA a n d DETRE, L . . . . .

BAL~ZS, JULIA . . . . . . . . . DETRE, L. a n d GUMXN, I . . . . . . BLAZHKO, S . . . . . . . . . .

1952 1938 1941 1943 1951 1952 1953 1954 1954 1925

Astron. J., 57, 158. Budapest Ob8. Mitt., Astron. Naehr., 271, Budapest Obs. Mitt., Budapest Obs. Mitt., Budapest Obs. Mitt., Budapest Obs. Mitt.,

1926

Ann. de l'Ob8. Astron. de Moscou S~r. 2, vol.

1949

Ap. J., 109, 149.

No. 8. 231. Nos. 18-19. No. 23. No. 27. No. 33. B u d a p e s t Obs. ( u n p u b l i s h e d d a t a ) . Budapest Obs. Mitt., Nos. 34-35.

Ann. de l'Obs. Astron. de Moscou, S(~r. 2, vol. V I I I , livr. No. 1. V I I I , livr. No. 2.

COWL~N(~, T. G. a n d NEWING, R . A .

D. ,L K . O'CoNNELL, S.J.

1165

1953 1953

Bull. Astron. Inst. Netherlands, X I I , 31. Z. Astrophys., 32, 69.

LEDOUX, P . . . . . . . . . . .

1954 1922 1936 1938 1947 1950

LURJE, M. A .

1951 1950

B u d a p e s t Obs. ( u n p u b l i s h e d data). Bull. Astron. Inst. Netherlands, I, 139. Bull. Astron. Inst. Netherlands, V I I , 313. Bull. Astron. Inst. Netherlands, V I I I , 211. Bull. Astron. Inst. Netherlands, X , 251. Inst. d'Astrophys. Univ. Liege Coll., 4 °, 5 pp., No. 25. Ap. J., 114, 373.

L. . . . . . . . . . G R A T T O N , L. and L A V A G N I N O . . . GRATTON,

GUMAN, I .

.

.

.

.

.

.

.

.

.

.

.

~IERTZSPRUNG,E . . . . . . . . . KLUYVER, H. A .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Peremennye Zvezdy, Akad. Nauk USSR, Moscow (Variable S t a r Bull.), 7, 182.

MARTINOFF, D. J . . . . MULLER, A. B . . . . . MtiNCH, G . . . . . . OOSTERHOFF, TH . . . .

. . . . . . . . . . . .

. . . . . . . .

SANFORD, It. F .

.

.

.

.

.

.

.

.

SCHWA~ZSCRILD, M . . . . . . . . SHAPLEY, H . . . . . . . . . . STRUVE, O .

.

.

.

.

.

.

.

.

.

STRUVS, O. a n d BLAUUW, A . . . . . STRUVE, O. a n d VAN HOOF, A . . . . . WALRAVEN, TH . . . . . . . . . WOLTJER, J .

.

.

.

.

.

.

.

.

.

1940 1953 1951 1946a 1946b 1949 1952 1941 1915 1916 1947 1949 1952 1948 1949 1949 1952 1953 1937 1946

Bull. Engelhardt Obs. Kazan, No. 17. Bull. Astron. Inst. Netherlands, X I I , 11. Ap. J., 114, 546. Bull. Astron. Inst. Netherlands, X , 101. Bull. Astron. Inst. Netherlands, X , 123. Ap. J., 109, 208. Ap. J., 116, 331. Ap. J., 94, 245. Ap. J., 42, 148. Ap. J., 43, 217. Publ. Astron. Soc. Pacific, 59, 192. Astron. J., 54, 50. Ann. Astrophys., 15, 157. Ap. J., 108, 60. Ap. J., 109, 214. Bull. Astron. Inst. Netherlands, X I , 17. Bull. Astron. Inst. Netherlands, X I , 421. Bull. Astron. Inst. Netherlands, X I I , 57. Bull. Astroa. Inst. Netherlands, V I I I , 193. Bull. Astron. inst. Netherlands, X , 125.

Photographic Light Curve of Eta Carinae D. J. K. O'CoNNELL, S.J. Specola V a t i c a n a , Castel Gandolfo, V a t i c a n City S t a t e

SUMMa_BY T h e h i s t o r y of t h e u n u s u a l s t a r E t a Carinae is reviewed. P h o t o g r a p h i c m a g n i t u d e s o b t a i n e d f r o m plates t a k e n a t R i v e r v i e w O b s e r v a t o r y , 1935-52, a n d f r o m s o m e early S y d n e y A s t r o g r a p h i c plates, are listed. I t is s h o w n t h a t t h e s t a r h a d c o m m e n c e d to b r i g h t e n b y April, 1941. T h e b r i g h t e n i n g h a s c o n t i n u e d since t h e n , w i t h fluctuations. T h e colour of t h e s t a r is discussed.

1. INTRODUCTION

IN 1837 Sir JoHn H E R S C H E L , during his stay at the Cape of Good Hope, noticed that Carinae had increased greatly in brightness (HERSCHEL, 1847). HALLEY had observed it in 1677 as a fourth magnitude star, and LACAILLEin 1751 classed it as of the second magnitude. HERSCHEL found it to be nearly zero magnitude. Since that time, ~ Carinae has attracted attention as one of the most remarkable stars in the sky. IN~ES (1903) gives a history of the star and the visual light curve to 1902. By 1843 it had reached --1 m and was second only to Sirius in brightness. B y 1886 it had faded to about 8m. I t then brightened again and reached 6m in 1889. By 1897 it had again faded to about 8m and remained at about that level for m a n y years.