NGC 1850

Introduction

NGC 1850 is a young cluster situated in the bar of the LMC. After the 30 Doradus complex, it is the brightest of the LMC clusters, and is quite compact and very rich. The cluster has an unusual morphology, in that it appears to have several smaller sub-components of different ages in close proximity to the main cluster. This aspect of the cluster has been studied in detail by Vallenari et al. (1994) and Fischer, Welch and Mateo (1993). The cluster also appears to be associated with an emission nebula (N103B), which is easily seen in a narrow-band Hα image of the cluster.

This chapter reports the results of CCD photometry of NGC 1850 and the surrounding 10^′ field.

Observations

Observations of NGC 1850 were obtained on the 1-m and 2.3m telescopes at Siding Spring Observatory using a variety of CCD detectors. The observations presented in this chapter were acquired during the period October 1988 to March 1994, a time span of nearly 2000 days. This long time span enables accurate periods to be determined for short period objects such as Cepheids and RR-Lyrae variables, and also allows monitoring of LPVs over multiple cycles. A total of 86 frames were taken in each of the Cousins' V and I bands, with typical exposures of 300 and 150 seconds, respectively, yielding a limiting magnitude of V $\approx$20.5.

The choice of the V and I passbands as opposed to the more traditional B and V was motivated by several factors. Photometry of LPVs is more accurate, since they are several magnitudes brighter in I than in B. This is particularly helpful in the extremely crowded cores of clusters. The atmospheric seeing in I is usually better than in B, due to both a steadier atmosphere and the shorter exposures required in the I band. Again, this factor is very important in the cluster cores. Another advantage is that extinction corrections are much smaller in I than in B. Finally, model atmosphere calculations for V-I colours are more reliable than those for B-V colours, due to large metal line blanketing effects in B.

Infrared J and K photometry was also obtained on one night for selected LPV candidates, using the CASPIR near-infrared array camera on the 2.3m telescope at Siding Spring Observatory. The camera uses a 256x256 InSb detector, at an image scale of 0 .^′′25/pixel. This data was obtained in order to derive more accurate bolometric corrections for these objects than is possible with optical photometry.

Data Reduction

The VI CCD observations were bias subtracted and flat fielded using the IRAF data reduction package. The DoPhot photometry package (Mateo and Schechter 1989) was used to determine relative photometry for each frame. DoPhot was allowed to find its own starlist for each frame, rather than adopting a common starlist based on a good-seeing frame. Whilst the latter approach usually produces improved photometry in crowded fields, the early version of DoPhot used in the analysis was not able to do this. Colour terms appropriate to the particular CCD-filter-telescope combination of each frame were removed from the photometry at this stage. A reference frame in each bandpass was then chosen, and every other frame had an empirically determined zero point correction applied to it in order to transform it to the magnitude system of the reference frame.

On three separate photometric nights, standard star fields of Landolt (1992) were used to calibrate NGC 1850 frames in each of V and I. On the three nights, 3, 5 and 1 standard fields, respectively, were observed and used to reduce the photometry. Each standard field typically had 5 to 7 stars with a colour range of 0 <V-I< 2. These frames were then used to calculate the zero point of the reference frame in each bandpass. The mean zero point obtained by averaging the three separate zero points in each bandpass has a formal error of 0.002 magnitudes for V, and 0.022 magnitudes for I. Comparison of our V photometry with that published by Fischer et al. (1993) shows a zero point difference V(this chapter)-V(Fischer et al.) = -0.04 while comparison with Alcaino and Liller (1987) gives a zero point difference V(this chapter)-V(Alcaino & Liller) =  0.04. Our zero point in I differs from that of Alcaino and Liller (1987) by an amount I(this chapter)-I(Alcaino & Liller) = -0.04.

The JK CASPIR data was also reduced using IRAF. Dark frames were subtracted from the raw images. A median sky image for each colour was then computed based on all the frames. This was normalised and divided into the dark subtracted images to flat field them. Due to the pixel-to-pixel dependence of the dark current rate on the illumination of the frame (mostly sky), this procedure left some residual structure. To overcome this problem, a median sky frame was computed, normalised and subtracted from all the frames. This step successfully removed virtually all the remaining dark-current residuals.

Aperture photometry was performed on the frames in order to derive J and K magnitudes for the LPV candidates. The results of this photometry were checked by converting the known V-I colour of some non-variable field stars in the frames to V-K colours using the transformations of Caldwell et al. (1993), and comparing this to the instrumental V-K colour. Based on this comparison, it is estimated that the photometric zero point for the JK data is good to within 0.03 magnitudes.

Data Analysis

The analysis of this dataset was performed in a similar way to the procedures described in Chapter 2. Briefly, variable star candidates were selected by two methods: blinking pairs of frames, and examining objects with anomalously high magnitude scatter. Stars lying in regions of the CMD where variability is common, in particular the Cepheid Instability Strip and stars redder than the giant branch, were also examined individually for variability. We estimate that we have identified almost all objects brighter than V=19 which have variability on timescales of 0.4-1000 days and amplitudes > 0.15 mag.

The interactive pdm task in IRAF was used to determine the true period of the objects, where possible. Classifications for the variables were made, based on the period, shape of the light curve and the position of the star in the colour-magnitude diagram. The error in the period was determined using the width of the dip in the pdm curve. The long time-span over which the data was taken means that period errors are very small. Periods are listed with a number of significant digits such that the uncertainty is in the last digit.

Results

The variables found in a ∼10^′×10^′ region centred on NGC 1850 are identified on Figure 3.1, and photometric properties are listed in Tables 3.1, 3.2, and 3.4. The positions of the variable stars have been measured using the HST Guide Star Catalogue. The B1950.0 coordinates of each variable are shown in Table 3.5 to facilitate easier cross-referencing with other publications. The positions should be accurate to ±2^′′.

Figure 3.1: A finding chart for the variable stars in the NGC 1850 field. North is to the right and East is up. This is a V band image, approximately 10^′ on a side

Table 3.1: Cepheid Variables.

	Var		Period		<V >		<V >- <I >			MPuls				ID
									V		I		0.18		0.09		0.18		0.09		0.18		0.09
			(days)		mag.		mag.		mag.		mag.		M$\odot$		M$\odot$		M$\odot$		M$\odot$		M$\odot$		M$\odot$
	9		30.40		13.519		1.022		1.2		0.8		7.64		8.43		10.1		9.6		9.0		8.3		HV 904
	17		18.66		13.899		0.867		0.8		0.5		5.63		6.22		9.2		8.6		8.1		7.5
	110		11.858		14.475		0.852		1.0		0.7		5.23		5.81		7.9		7.4		7.0		6.6		HV 905
	58		8.558		14.717		0.770		1.0		0.6		4.70		5.23		7.4		6.9		6.6		6.2		Robertson
	269		7.010		15.192		0.877		0.9		0.6		5.36		5.98		6.6		6.1		6.0		5.6		Robertson
	341		3.5930		15.952		0.751		0.8		0.4		3.76		4.21		5.3		4.9		5.0		4.7
	679		2.7105		16.413		0.773		0.8		0.6		3.65		4.10		4.7		4.4		4.5		4.2

Table 3.2: Long-Period Variables - VI data.

Name	Period	<V >	<V >- <I >	AV	AI	ID
	days	mag.	mag.	mag.	mag.
120		14.68	2.20	0.25	0.16
347	109.5	16.01	1.80	1.0	0.6	SHV050826-684447
395		16.26	2.36	0.18	0.12
502		16.03	2.20	0.30	0.18
504	23?	15.78	3.00	0.5	0.30
558	56	16.89	1.98	0.35	0.18
574		16.47	2.43	0.32	0.15
587	145.6	17.8	3.1	4.5	1.9	SHV050828-684840
670	330	17.01	2.89	0.67	0.45
1038	597	17.7	4.6	5.9	2.5	HV11984,SHV050943-685131
1078		17.05	2.35	0.62	0.25
1084	269?	16.84	2.05	0.4	0.15
1135	∼450	18.45	2.3	>2	>2	SHV050828-685326
1847		17.22	2.04	0.69	0.27
1906	230	17.13	2.73	0.9	0.5
2093	208	17.81	3.1	1.9	1.0	SHV050934-684934
2199	900?	17.27	2.86	0.74	0.29
2202	870?	17.5	3.1	1.1	0.6
2268	134.5	17.46	3.6	0.60	0.20

Table 3.3: Long-Period Variables - JK data

Name	Period	<K >	<J >- <K >	Mbol
	days	mag.	mag.	mag.
347	109.5	12.47	0.74	-4.00
504	23?	9.81	1.26	-5.90
587	145.6	12.32	1.23	-3.43
670	330	10.46	1.84	-4.84
1038	597	8.72	1.15	-7.12
1084	269?	12.35	1.15	-3.49
1135	∼450	10.45	2.87	-4.70
1906	230	11.22	1.77	-4.10
2093	208	11.60	1.77	-3.96
2202	870?	10.41	1.76	-4.92
2268	134.5	10.86	1.27	-4.84

Table 3.4: Other Variables

Name	<V >	<V >- <I >	P(days)	AV	AI	Type
	mag.	mag.	days	mag.	mag.
7254	18.40	0.57	0.50972	0.55	0.35	Cepheid?
9276	19.01	0.46	∼0.55	0.90	0.70	RR-Lyrae
10753	19.11	0.63	0.52202	0.84	0.64	RR-Lyrae
39	14.51	-0.10	1.4978	0.40	0.30	Eclipsing
1956	17.48	1.60	106.3	0.5	0.4	Eclipsing?
652	16.47	0.13	>2200	0.32	0.32
441	15.55	0.12	502	0.36	0.51
349	15.91	0.17		0.60	0.95

Table 3.5: B1950.0 positions of the NGC 1850 variables

Name	R.A. (1950.0) Dec.		Name	R.A. (1950.0) Dec.
558	05h08m18.6s	-68o45′57 ′′	7254	05 09 12.9	-68 50 59
341	05 08 20.7	-68 44 53	1847	05 09 14.9	-68 50 43
574	05 08 23.2	-68 47 06	1906	05 09 20.0	-68 50 10
347	05 08 26.3	-68 44 47	441	05 09 23.5	-68 52 33
587	05 08 28.5	-68 48 39	1956	05 09 23.8	-68 45 18
1135	05 08 29.0	-68 53 26	110	05 09 30.1	-68 48 10
349	05 08 31.0	-68 52 22	2093	05 09 34.8	-68 49 33
9	05 08 32.7	-68 50 30	120	05 09 38.9	-68 51 03
10753	05 08 33.6	-68 48 28	1038	05 09 43.0	-68 51 30
652	05 08 42.4	-68 48 47	2202	05 09 44.4	-68 46 05
39	05 08 42.8	-68 52 07	2199	05 09 46.6	-68 52 30
670	05 08 46.5	-68 48 13	502	05 09 49.6	-68 45 21
679	05 08 50.1	-68 52 41	504	05 09 53.1	-68 51 45
395	05 08 53.5	-68 47 28	2268	05 09 54.0	-68 49 13
58	05 08 57.4	-68 49 15	1078	05 09 55.9	-68 49 04
17	05 08 58.3	-68 49 10	1084	05 09 56.8	-68 48 54
269	05 09 08.8	-68 46 51	9276	05 10 02.2	-68 48 45

The variable stars are plotted on the CMD in Figures 3.2 and 3.3. Figure 3.2 is the CMD for the full 10^′ field, obtained by averaging magnitudes from all frames. Figure 3.3 is the CMD for the region restricted to within 1^′ of the cluster core. The magnitude scales are the apparent magnitudes uncorrected for interstellar reddening.

Figure 3.2: The Colour-Magnitude Diagram for the 10^′×10^′ field surrounding NGC 1850. Variable stars are shown by special symbols as defined on the figure.

Figure 3.3: As for Figure 3.2, but restricted to stars within a radius of 1^′ of the core of NGC 1850

Cepheids

Periods for 7 definite classical Cepheid variables and one probable anomalous Cepheid of very short period (P $\approx$ 0.5 d, see Section 3.5.3) have been determined. Two of the classical Cepheids were previously discovered (Hodge and Wright 1967), two were discovered to be variable by Robertson (1974), but with unknown periods, and four are newly discovered. Light curves of the 7 definite Cepheids variables in V and I are shown in Figure 3.4. Two complete cycles are plotted to more clearly show the shape of the light curve. The epoch of zero phase is Julian Day JD=2,440,000.0. All the Cepheids have fairly large amplitudes, and asymmetric light curves. These properties are characteristic of Cepheids pulsating in the fundamental mode. One Cepheid has a very distinct bump at maximum light.

Figure 3.4: Light curves for the Cepheid variables. The upper curve for each object is the I light curve, and V is the lower curve. The epoch of zero phase is Julian Date 2,440,000.

Variable 9 (=HV 904) is a bright, long-period Cepheid which was discovered on Harvard plates at the beginning of this century. The light curve is unusual in that the I light curve is very flat around maximum light, whilst the V light curve is not. This corresponds to a fairly rapid change in colour around the time of maximum light.

Variable 110 (=HV 905) is another Harvard variable, and a fine example of a bump Cepheid. The bump only appears in Cepheids with periods in the range of 8-12 days, with the phase of the bump slowly decreasing along the light curve as the period increases. In this case, the location of the bump is close to the time of maximum light, giving the light curve an unusual appearance. The bump is believed to be the result of a 2:1 resonance between the fundamental and second overtone modes of pulsation (Simon and Schmidt 1976).

Variable 269 was found to be variable by Robertson (1974). This star also has an I light curve with a fairly flat top around maximum light akin to 9, but to a lesser extent.

Variables 341 and 679 are lower-mass, short-period Cepheids, typical of the LMC field population (Payne-Gaposchkin 1974).

Variables 17 and 58 are both located about 15 ^′′ from the center of NGC 1850 (Figure 3.5). It is for this reason that the light curves are somewhat noisier than for other variables of comparable brightness - they lie in extremely crowded fields. Only magnitudes from frames for which the seeing was better than 2 ^′′ have been shown in Figure 3.4. Fischer et al. (1993) suspected variable 17 to be a Cepheid, based on its luminosity, colour, and two measurements indicating radial velocity variability. The location of these variables on the CMD is shown in Figure 3.3. The large difference in period and luminosity of these two stars suggests that they cannot both be cluster members. Variable 58, the fainter object, has a luminosity consistent with it being on the cluster helium-burning loop, while variable 17 is too bright (unless this star has had its mass increased by binary mass exchange). Examination of an HST WFPC2 image of NGC 1850 from the HST archive shows that both these Cepheids appear to be single objects at 0 .^′′1 resolution.

Figure 3.5: A finding chart for the two variable stars near the core of NGC 1850.

One additional Cepheid is known to exist close to the core of NGC 1854 (Robertson 1974). NGC 1854 is the smaller cluster in the upper left edge of Figure 3.1. This Cepheid is located at the extreme edge of our field, and consequently there were insufficient observations available to determine a period.

Long-Period Variables

Nineteen long-period variables were identified in the field. Five of the brightest, highest-amplitude LPVs were previously known (Hodge and Wright 1967; Hughes 1989) while the remainder are new discoveries. Eight of the LPVs have well-defined light curves and these are shown in Figure 3.6. The remainder of the LPVs vary somewhat erratically, or are near the edge of the field and have too few points to define the period well. All are red and exhibit variability on timescales of hundreds of days. Light curves for these LPVs are shown in Figure 3.7.

Figure 3.6: Light curves for the Long-Period Variables which show regular behaviour. The upper curve for each object is the I light curve, and V is the lower curve.

Figure 3.7: Light variations for the LPVs for which periods could not be determined. The upper curve for each object is the I light curve, and V is the lower curve.

RR Lyraes and an anomalous Cepheid

Two RR Lyrae stars, variables 9276 and 10753 were found and their properties are listed in Table 3.4 along with the probable anomalous Cepheid, variable 7254. The rate of detection of RR Lyraes is similar to that of Nemec, Hesser and Ugarte (1985) for the field around the remote LMC cluster NGC 2257. The true density of RR Lyraes in the bar is presumably much greater than the 3 per 10^′×10^′ field which has been found here. This low detection rate is undoubtedly due to the faintness of these objects (V∼19) combined with the difficulties in obtaining good photometry from faint stars in extremely crowded fields.

The anomalous Cepheid and one of the RR Lyraes have certain periods (7254 and 10753 with periods of 0.50972 and 0.52202 days, respectively, and light curves in Figure 3.8), whilst the other RR Lyrae (9276) had too few observations to make an unambiguous period determination. The periodogram has many aliases for periods of 0.54-0.56 days, and it is almost certain that the true period lies in this range.

Figure 3.8: Light curves for the RR Lyraes, Eclipsing and unusual variables. The upper curve for each object is the I light curve, and V is the lower curve.

The reason that variable 7254 is assumed to be an anomalous Cepheid and not an RR Lyrae is that, with <V >= 18.45, it is about 0.5-1.0 magnitudes brighter than expected for an RR Lyrae. Given the <V > and <V >- <I >in Table 3.4, and a reddening E(B-V)= 0.15 and LMC distance modulus of 18.5 (see Section 3.6, we estimate M_B=-0.25. This magnitude puts variable 7254 in the region of the (M_B,log P) plane occupied by the anomalous Cepheids (Zinn and Searle 1976). Comparison of the period, luminosity and T_eff of this star with the theoretical calculations of Chiosi, Wood and Capitanio (1993) shows that these properties are those expected for a Cepheid of mass ∼3M_$\odot$. As an alternative, variable 7254 has an amplitude approximately half that of the other two RR Lyraes. It is therefore possible that this object is in fact an unresolved combination of an RR Lyrae and a non-variable star of similar magnitude.

Eclipsing and other variables

Two possible eclipsing binaries and several unusual variables were also identified in the NGC 1850 field. They are listed in Table 3.4 and light curves are shown in Figure 3.8.

Variable 39 is a typical upper main-sequence eclipsing binary with orbital period 1.49775 days. Many such binaries are known in the Magellanic Clouds (Payne-Gaposchkin 1974; Chapter 2).

Variable 1956 has a light curve which resembles that of an LPV with period 53.15 days. However, there is some evidence that it may be a rare type of eclipsing variable consisting of two similar giant stars with an extremely long orbital period of 106.3 days. The main reason for suspecting that this variable is an eclipsing binary is that it shows no perceptible colour variation through its cycle. It is also bluer than any of the other LPVs, suggesting that it may not be in the region of long-period instability.

The symmetry of the light curve suggests that the two components in the system are similar. The dereddened colour of <V >- <I >=1.40 yields T _eff∼ 4100 K (Bessell 1979) for each star. The dereddened maximum I magnitude of 15.6 suggests I∼ 16.37 for each star, or M_bol = -1.57 using the bolometric correction to I from Bessell and Wood (1984) with an LMC distance modulus of 18.5. From the definition L = 4πσR²T _eff⁴, we derive a radius of 36 R_$\odot$ for each component. Since the system is on the old giant branch in Figure 3.2, a mass of 1M_$\odot$ is assumed for each star. The orbital period of 106 days then yields a separation of 118 R_$\odot$. The radii and orbital separation just derived are consistent with the suggestion that variable 1956 is a close binary consisting of two red giants. If this interpretation is correct, evolution up the giant branch should see the two components soon merging.

The remaining three variables in Table 3.4 and Figure 3.8 do not fit into any well-known groups, but they all lie on the upper main sequence. Variable 441 appears periodic with a very long period of 502 days while variable 349 remained virtually constant in brightness except for a period over ∼200 days when it brightened significantly. Both these stars have higher amplitude in I than in V, which is the opposite sense to normal pulsating variables. This may be caused by a strong hydrogen emission line component in the variability with the Paschen continuum playing a prominent role in the I band. Hydrogen emission shifts the V-I colour of Be stars well to the red of the main sequence (Bessell and Wood 1993; Chapter 2). Typical colour shifts are up to 1 magnitude in V-K, and ∼0.25 magnitudes in V-I.

Finally, variable 652 has a very unusual light curve. Both colours show a slow, almost linear decrease in brightness by around 0.3 mag over the entire length of this study, and virtually no change in colour. If it is periodic, the period must be longer than 2200 days.

Reddening, distance modulus and metallicity

In order to compare the observations of the Cepheids with theory, we need estimates for the cluster reddening, metallicity and distance modulus. Estimates for the reddening of the NGC 1850 region have been published by several authors. Fischer et al. (1993) calculated a reddening E(B-V) = 0.17±0.03 based on isochrone fitting of the very young subcluster (region B) to the west of the main cluster. However, reddening determinations based on ground-based photometry of region B must be viewed as approximate due to the extreme crowding in this region and the difficulty of obtaining accurate photometry under such conditions. Examination of Hubble Space Telescope images of NGC 1850 show region B to be crowded even with the 0 .^′′1 resolution of WFPC2, and considerably more crowded than the core of NGC 1850 itself. Vallenari et al. (1994) obtained E(B-V)=0.18±0.06 from UBV photometry of main-sequence stars. Earlier determinations are slightly lower, ranging from 0.09 (Cassatella 1987) to 0.15 (Persson et al. 1983, Caplan and Deharveng 1985, Caplan et al. 1986). Given the significant scatter in the published reddening determinations, calculations will be made using a mean reddening of NGC 1850 of E(B-V)=0.09 and E(B-V)=0.18. For other colours, the reddening law A_V=3.1E(B-V) and E(V-I_c)=1.31 E(B-V)  (Taylor 1986) has been adopted. A LMC distance modulus of 18.50 has been assumed. A wide variety of independent determinations seem to average around this value, some recent examples are the Cepheid-independent measurements of the distance to SN 1987A (Panagia et al. 1991), who estimate a distance modulus of 18.50, and a reanalysis by Gould (1994) who give and upper limit of 18.63. Simon and Clement (1993), derive a distance modulus of 18.54 based on the LMC RR Lyrae variables.

For the metallicity of the Cepheids, Z = 0.008 is adopted, based on metallicity determinations for young LMC objects by Russell and Bessell (1989) and Russell and Dopita (1990) and Y = 0.25 based on determinations of He abundances in HII regions (Dufour 1984).

Evolution and pulsation masses for Cepheids

Pulsation masses have been derived for the Cepheids and are shown in Table 3.1. The masses were calculated from the P-M-M_V-(V-I) relation of Chiosi, Wood and Capitanio (1993) assuming a distance modulus of 18.5, reddenings E(B-V)=0.18 and 0.09, metallicity Z=0.008 and that the Cepheids are pulsating in the fundamental mode.

We next consider the evolution mass for the Cepheid 58 which appears to be a member of NGC 1850. Vallenari et al. (1994) derived a cluster age of 70 Myr by fitting UBV photometry to the main-sequence turn-off. The isochrones of Bertelli et al. (1994) show that an age of 70 Myr corresponds to a Cepheid (helium-burning loop) mass of 6.2M_$\odot$ for a reddening E(B-V)=0.18 or mass 5.7M_$\odot$ for E(B-V)=0.09.

Using the isochrones of Bertelli et al. (1994), we now derive a reddening and age of NGC 1850 from our VI photometry. We note that the calculations of Bertelli et al. incorporate the new OPAL radiative opacities (Rogers and Iglesias 1992) and the intermediate mass isochrones include convective core overshoot with overshoot parameter Λ = 0.5, corresponding closely to the parameter d_ov/H_P=0.25 in Maeder and Meynet (1989) or f_ov=0.25 in Chiosi et al. (1993).

The CMD for stars within 1^′ of the core of NGC 1850 is plotted in Figure 3.9, where each point is an average of the 86 VI frames. Examination of Figures 3.3 and 3.9 clearly shows the main sequence and two distinct clumps of evolved stars. The stars in the region centred on (V,V-I) = (15.3,1.3) belong to the red end of the core helium-burning loops while the stars near (14.6,0.4) are on the blue end of the loops. The region in between these two clumps is the region where the Cepheid Instability Strip lies. The few very blue, luminous stars at (14.0,-0.1) originate from a bright, young, compact subcluster located about 40 ^′′ from the main cluster, designated as region B in Vallenari et al. (1994). It is believed that this subcluster is the source of the ionising radiation for the emission nebula N103B.

Figure 3.9: The dereddened cluster CMD with isochrones overlaid. The age is given in log(T) (years), and the mass M_HeB is that of the stars evolving along the helium-burning loops on each isochrone.

Overlaid are the isochrones for ages log T(years) = 6.6, 7.7, 7.8, 7.9 and 8.0 assuming a LMC distance modulus of 18.50. The cluster stars have been dereddened assuming E(B-V)=0.15. Plotting the CMD with E(B-V)=0.18 and 0.09 clearly shows that these two reddenings bracket the true value, as the higher reddening produces a main sequence that is too blue, while it is too red for the lower reddening. We estimate the mean cluster reddening to be 0.15±0.03.

The cluster is well fit by the log(T) = 7.9 isochrone, corresponding to an age of 80 Myr for NGC 1850. This may be compared to the ages of Fischer et al. (1993), who derived 90 Myr, and Vallenari et al. (1994) who derived 70 Myr. The age derived here is different from that of Vallenari et al., despite the fact that the same isochrones were used. This difference is probably caused by the fact that Vallenari et al. determined their cluster age mostly from fitting the main-sequence turn-off and luminosity function, whilst the age derived here is based on the luminosity of the helium-burning loop stars.

The two Cepheids located in coincidence with NGC 1850 are also plotted in Figure 3.9. Evolution masses can readily be derived by fitting the Cepheids to the isochrones and reading the appropriate mass from the isochrone tables. These masses have been derived for all seven Cepheids in our frames, assuming reddenings of E(B-V) = 0.09 and 0.18. Evolution masses have also been derived for evolutionary tracks computed without convective core overshoot (Alongi et al. 1993). These latter models were made with the older Los Alamos opacities which yield loop luminosities ∼ 0.06 brighter in log L than the OPAL opacities used by Bertelli et al. (1994) (see Figure 7 of Alongi et al. 1993). The Alongi et al. loop luminosities have been corrected by this amount when deriving evolution masses from the non-overshoot models. All evolution masses, for overshoot and non-overshoot models and E(B-V)=0.09 and 0.18, are given in Table 3.1.

The evolution and pulsation masses of the seven Cepheids are compared in Figure 3.10. Each Cepheid is plotted 4 times, corresponding to core overshoot parameter of Λ=0 and 0.5 and E(B-V)=0.09 and 0.18. A set of arrows indicate the effect on a 5M_$\odot$ Cepheid of changing reddening, amount of core overshoot and distance modulus. It is clear that the evolution mass is still higher than the pulsation mass. Using the mass corresponding to the age derived by Vallenari et al. (1994) gives a slightly better agreement between pulsation and evolution mass. Increasing the amount of core overshoot to Λ∼ 1.0 or decreasing reddening to E(B-V)∼ 0.03 would bring evolution and pulsation masses into alignment. However, for our preferred reddening of E(B-V)=0.15 and overshoot parameter Λ=0.5 (d_ov/H_P=0.25) favoured by Bertelli et al. (1994) and Maeder and Meynet (1989), evolution masses are ∼ 20% larger than pulsation masses. Changing the distance modulus by acceptable amounts (one or two tenths of a magnitude) will make very little difference to the mass ratio (Figure 3.10).

Figure 3.10: Evolution mass plotted against pulsation mass in M_$\odot$ for the seven Cepheids in out field centred on NGC 1850. Each Cepheid is plotted 4 times according to the amount of core overshoot and reddening. The Cepheid (58) which is a member of NGC 1850 is circled. The triangle shows this star plotted using the evolution mass derived from the cluster age of Vallenari et al., overshoot models and E(B-V)=0.09 and 0.18. The arrows show the effect of changing the amount of overshoot, distance modulus and reddening.

The effect of changing metallicity has little effect on the pulsation masses, but a small effect on the evolution masses. If the LMC abundance was Z=0.02 (solar), the ratio of evolution mass to pulsation mass is only 2% higher than for the Z=0.008 case. If the LMC abundance was Z=0.004 (SMC), the ratio is around 7% lower. In either case, the evolution masses are still substantially higher than the pulsation masses. It is also far more likely that the metallicity of young stars such as Cepheids would be enhanced rather than decreased. Thus, it seems that the results are insensitive to even large increases in metallicity.

The bump Cepheid HV905

In this section, theoretical models are made for the bump Cepheid HV905 (variable 110) in order to derive a bump mass for comparison with the evolution and pulsation masses. As a byproduct of these calculations, it is found that only a very small range in luminosity is allowed for this star. Hence, an accurate distance modulus to the LMC can be derived, where the main uncertainty is the extinction correction.

Simon and Schmidt (1976) showed that the existence of a bump on the light curve is the result of a 2:1 resonance between the fundamental and second overtone modes of pulsation. Moskalik et al. (1992) have suggested that this condition can then be used to constrain the properties of bump Cepheid models. The requirement of having two periods known accurately (P₀ and P₂) while simultaneously demanding that the fundamental mode be unstable provides strong constraints on possible models. We note that satisfactory models for bump and beat Cepheids (Moskalik et al. 1992) have only become possible since the advent of the OPAL opacities (Iglesias et al. 1990; Iglesias and Rogers 1993).

The linear pulsation models described here were done with the linear pulsation code described in Chiosi et al. (1993) except that the opacity routine has been modified to use the tables of Iglesias and Rogers 1993 (with a molecular contribution to opacity as detailed by Chiosi et al. 1993). The nonlinear pulsation calculations were performed with the code of Wood (1974), using the updated opacity calculations.

Initial guesses at physical parameters log L/L_$\odot$ and log T_eff for HV905 were derived from the mean V and V-I given in Table 3.1. A reddening E(B-V)= 0.15 was assumed along with a true distance modulus to the LMC of 18.5. The bolometric correction and the transformation from V-I₀ to log T_eff were derived from equations (20) and (21) of Chiosi et al. (1993). The abundance adopted was Y=0.25 and Z=0.008 to agree with the mean abundances of HII regions and young stars in the LMC (Dufour 1984; Russell and Bessell 1989; Luck and Lambert 1992).

The fundamental requirement demanded of the linear pulsation calculations was that P₀ = 11.858 days and that P₀/P₂ = 2. Given a value for M_bol, a number of models were computed with different masses. At each mass, T_eff was varied until the condition P₀/P₂ was satisfied. The loci of models satisfying P₀/P₂ = 2, for four luminosity values, are shown in Figure 3.11. The bump mass is displayed in the lower panel and the corresponding value of log T_eff is shown in the top panel. The thick line segments correspond to those models that have positive growth rate for the fundamental mode. Clearly, real bump Cepheids must lie on the thick parts of the curves in Figure 3.11. The estimated log T_eff of HV905 is indicated in the top panel of Figure 3.11 along with a range of log T_eff values corresponding to our estimated uncertainty in V-I of ±0.03 magnitudes.

Figure 3.11: Lower panel: Loci of Cepheid models with P₀/P₂ = 2 in the M-P plane. Each curve is labelled by its bolometric luminosity. The thick line segments correspond to models with unstable fundamental modes. The thin vertical line is the period of the bump Cepheid HV905 (variable 110). Upper panel: Curves corresponding to those in the lower panel but in the log T_eff-P plane. The solid point is the estimated temperature of HV905 using the (V-I,T_eff) relation in Chiosi et al. 1993. The error bars correspond to uncertainty in reddening of ±0.03 magnitudes. The open circle shows the position of the model used for the nonlinear pulsation calculations.

From Figure 3.11, it can be seen that models with M_bol < -4.61 do not develop a fundamental mode period as short as that of HV905 (the period is indicated by the vertical line in the lower panel of the figure). On the other hand, unstable models with the required period which are more luminous than M_bol = -4.57 have T_eff values below our estimated temperature range for HV905. It would appear from these models that the bolometric luminosity of HV905 is constrained to a very small range -4.57 > M_bol > -4.61. Similarly, log T_eff is constrained by the instability requirement to be <3.762 and by observations to be >3.753. The corresponding bump mass lies in the range 5.75 < M_bump/M_$\odot$ < 6.1. This is only slightly more than the pulsation mass of 5.23 to 5.62M_$\odot$ given in Table 3.1 (for reddening E(B-V) = 0.15±0.03). Both the bump and pulsation masses are considerably less than the estimated evolution mass of 6.7 to 7.9M_$\odot$ for E(B-V) = 0.15±0.03 and overshoot parameter 0 < Λ < 0.5 (Table 3.1). Once again, an overshoot parameter Λ∼ 1.0 would give agreement between the pulsation, bump and evolution masses.

The small allowable bolometric luminosity range for HV905 allows us to derive an accurate distance modulus for the LMC. Adopting E(B-V) = 0.15 and using the bolometric correction noted above, an LMC distance modulus of 18.48 to 18.54 is derived. If an uncertainty of ±0.03 in reddening is allowed, the distance modulus range is 18.39 to 18.63.

As a further test of the bump Cepheid theory, a nonlinear pulsation calculation was performed to see if the theoretical light curve shape resembled the observed one. The model chosen has M, M_bol and T_eff near the middle of the allowable range (M = 5.95M_$\odot$, M_bol = -4.595, log T_eff = 3.757). The model was initiated in the fundamental mode with a surface velocity amplitude of 25 km s^-1 and was allowed to develop a regular pulsation. The light and colour curves after 25 cycles are shown in Figure 3.12 where a distance modulus of 18.5 and the reddening, bolometric corrections and colour transformations noted above were used to convert L/L_$\odot$ and log T_eff of the models into observed quantities. We note that the amplitude of the light curve was still decreasing after 25 cycles so that a slightly cooler, more unstable model would be required to reproduce the amplitude of pulsation. Apart from this factor, the theoretical light curve reproduces the bump at maximum light and kinks on the ascending and descending branches, although the overall ascending branch is not well reproduced.

Figure 3.12: Light and colour curves for HV905 (points) along with the light and colour curves of the nonlinear pulsation model of this star (see text for details).

In summary, the theory of bump Cepheids tightly constrains the possible parameters of HV905. The theoretically allowable T_eff agrees well with the observed value, the bump mass and the pulsation mass are in reasonable agreement and the nonlinear pulsation calculations produce a light curve that closely follows the observed light curve. All these factors suggest that the study of bump Cepheids can potentially yield very accurate luminosities, masses, temperatures and distance moduli to these stars. However, the above calculations should be considered preliminary in that the effect of assuming a P₀/P₂ ratio not identical to 2.0 has not been investigated, and neither has the effect of metal abundance. A more detailed study will be presented elsewhere.

The Cepheid Period-Luminosity-Colour Relation

We can use the theoretical PLC relation of Chiosi et al. (1993) to derive a distance to the LMC independent of other calibrations. Figure 3.13 shows the PLC relation for the 7 Cepheids in this study, where the PLC relation is in the form

M_V = α  logP + β(V - I) + γ   (PLC)

We use the theoretical values of α=-3.68, β=3.85 and γ=-3.21 appropriate to fundamental mode LMC Cepheids from Chiosi et al. (1993), with Z=0.008 and overshoot parameter f_ov=0.25 (Λ=0.5). The data is plotted in Figure 3.13 for reddenings E(B-V)=0.09 and 0.18. By comparing the observed value of γ to the theoretical value, we derive distance moduli of 18.66 and 18.48 for E(B-V)=0.18 and 0.09 respectively. For our preferred reddening E(B-V)= 0.15 in this field, the distance modulus is 18.60.

Figure 3.13: The PLC relation for the Cepheids compared with the theoretical relation of Chiosi et al.. The crosses show the data plotted assuming E(B-V)=0.09 and circles are for E(B-V)=0.18. The line is the fitted theoretical relation assuming a reddening E(B-V)=0.15.

Long-Period Variables

The JK photometry of the LPVs was dereddened using E(J-K)=0.59E(B-V) and A_K=0.38E(B-V) (Lee 1970). Only one J and K measure was available for each star. These were converted to mean JK magnitudes by phasing the VI light curves and assuming the J and K amplitudes were 1.5 and 2.5 times, respectively, smaller than the I amplitude. These amplitude ratios were derived from a sample of LMC LPVs for which J, K and I amplitudes were available (Hughes and Wood 1990; Wood, Bessell and Fox 1983; Feast et al. 1989). The exact amplitude ratios are not critical, as amplitudes in the infrared tend to be fairly small. Table 3.3 gives the <K >  and <J >- <K >  values derived.

The LPVs with non-questionable periods are plotted on the K-logP diagram for LMC LPVs (Feast et al. 1989; Hughes and Wood 1990) in Figure 3.14. We note that the scatter in <K >  about the K-logP relation for the large amplitude LPVs with P <420 days is ∼ 0.2 mag. The LPVs found in this survey generally fit the K-logP relation well. The exceptions lie on the short period side of the Mira relation. It is well known that the bright LPVs with P >420 days, such as variable 1038, do not fit the Mira K-logP relation (Feast et al. 1989; Hughes and Wood 1990). The other point on the short period side of the mean relation (variable 2268) could be an overtone pulsator (as could variable 504 which has a very short period, probably around 23 days). Since the ratio of first overtone to fundamental mode pulsation is ∼ 2.5 (Fox and Wood 1982), and if the Mira relation corresponds to fundamental mode pulsation, it might be expected to find a sequence of low-amplitude pulsators in a band parallel to the Mira relation and shifted by an amount similar to that found for variable 2268. Surveys such as this one can potentially pick up these variables, whose existence could provide definitive evidence that the Miras are fundamental mode pulsators (Wood and Bessell 1985). Earlier surveys which are almost exclusively photographic, are not sensitive to these low-amplitude variables.

Figure 3.14: The K-logP relation for LPVs in the NGC 1850 field. The line is the mean relation for all Miras given by Feast et al. (1989).

Summary

Seven large-amplitude Cepheid variables have been found around NGC 1850. Only one of the Cepheids appears to be a cluster member. An age of 80 Myr has been derived for NGC 1850 based on isochrone fitting of the cluster CMD. This corresponds to a mass of ∼6.0M_$\odot$ for the evolved stars in the cluster. The isochrone fits indicate a reddening of E(B-V)=0.15±0.03 in the direction of NGC 1850.

Pulsation and evolution masses have been derived for the Cepheids, and they still show a mass ratio M_ev/M_Puls∼1.2 when a moderate degree of convective core overshoot (Λ = 0.5) is included in the evolutionary models. Agreement of the masses requires a large amount of overshoot ( Λ∼ 1.0) or an implausibly low reddening E(B-V)= 0.03.

The bump mass derived for a bump Cepheid in the field agrees reasonably well with the pulsation mass, further emphasising the continuing disagreement between pulsation and evolution masses. Analysis of the bump Cepheid data shows that these stars can potentially lead to derivation of very accurate distance moduli. A distance modulus of 18.51 is derived for the LMC, with a theoretical error of ±0.03 magnitudes, or a combined theoretical and observational error of ±0.12 magnitudes.

Comparison of photometry of the Cepheid variables around NGC 1850 with the theoretical PLC relation yielded a distance modulus to the LMC of 18.60, assuming a moderate amount of convective core overshoot during main-sequence evolution.

Finally, the LPV population in this field is shown to consist of both fundamental mode pulsators and lower-amplitude overtone pulsators, none of which appear to be cluster members.

The raw data (magnitudes, epoch) upon which this chapter is based are available through the ApJ/AJ CD-ROM series, Volume V, December 1995.