Microlensing experiments naturally accumulate huge datasets of images, which cover the same area of sky for years. At the same time, the temporal coverage of the observations should be very good, since the probability of microlensing events to occur is very low and the events are comparably short. This is especially true for pixellensing experiments. The data sets of microlensing experiments therefore are perfectly suited for the study of all types of variable phenomena.
class periods detected pulsating variables δ Scuti stars ≤0.3 - RR Lyrae stars 0.2−1.0 - type I Cepheids 1−200 X type II Cepheids 1−30 X RV Tauri stars 30−150 X Long Period Variables 30−1000 X
erupting variables
novae X
supernovae -
eclipsing binaries X
Table 1.1: Overview of some major classes of variable stars. The second column gives the approxi- mated range of their periods in days. In the last column we mark if the particular type of variable was detected in the WeCAPP survey (X).
all stars evolving from the main sequence enter a stage of variablity at one time or the other. As their frequency and period distribution depends on metalicity (abundance of elements more massive than helium) and age, they can be used to study stellar populations. In Table 1.1we give an overview of some major classes of variable stars and mark if they were detected in the WeCAPP survey.
In Fig.1.9(taken fromGautschy & Saio 1995) we show the position of the evolutionary sequences in the Hertzsprung-Russell-Diagram (HRD), where the luminosityLof stars is plotted against their ef- fective temperature (Teff). In the HRD, the stars are found in distinct regions and sequences, depending
on their mass, metalicity, and evolutionary stage. The main sequence (MS), the core hydrogen burn- ing phase, is the stadium where most of the observed stars can be found, as all stars spend most of their lifetime there. The position in the HRD at which stars originally enter the MS mainly depends on their initial mass (the more massive ones being brighter and having shorter MS lifetimes) and their metal content. After having ended central hydrogen burning the stars leave the MS and move to the red giant branch. During this stage the energy is produced by shell hydrogen burning on top of an inert contracting helium core. With the onset of helium burning in an electron degenerate core at the tip of the RGB, the low mass stars decline in luminosity and form the horizontal branch (for metal poor stars) or the red clump (for metal rich stars). After this stage the stars climb the Asymptotic Gi- ant Branch (AGB) with He and H burning shells, and finally end as white dwarfs. For more massive stars with masses>∼2.3Mthe helium core burning sets in in a non-degenerate state. After an initial
decline in luminosity they perform ‘blue loops’ in the HRD. Finally, they also climb the AGB and, like the lower mass stars, end up as white dwarfs. Massive stars with initial masses >∼8M explode
in a supernova SNII and will end as neutron stars or stellar black holes.
Along their evolutionary sequence (‘track’) stars can enter a region of enhanced instability, the so called ‘instability strip’. In this region, an almost vertical region in the HRD (see Fig. 1.9), stars are unstable against small perturbations of the equilibrium configuration. After a small perturbation the outer envelopes of the stars begin to oscillate, the mean density ¯ρ and the oscillation periodPbeing
Figure 1.9: Hertzsprung-Russell diagram (HRD) showing the distribution of different types of variable stars. The zero-age MS is shown as thick solid line, thin lines show the evolutionary tracks for stars of solar metalicity and different masses (labels given in solar units). From the MS the lower mass stars move to the red giant branch, located at higher luminosity and smaller log(Teff). Due to the lower
effective temperature the stars shift the maximum of their emission to longer wavelength, i.e. they become redder. The post-AGB track towards the white dwarf cooling sequence is shown just for one star. The instability strip is indicated by dashed lines. Thin diagonal lines mark positions with constant radius (labels given in solar units). Among other variable stars likeβ-Cepheids and Luminous Blue Variables (LBVs) the diagram also shows the position of the Mira and semi-regular variables (M, SR).
correlated as s ¯ ρ ¯ ρP=Q , (1.30)
with the mean solar density ¯ρ and the pulsation constant Q. The radial pulsation to be continuous
and not damped requires some energy input at the time of compression and the release in a subsequent rarefaction. Eddington(1926) proposed a valve mechanism which feeds energy at the time of maxi- mum compression. As the oscillating layers are not penetrating deep enough in the stellar body, the nuclear energy generated in the stellar core cannot be the source for the continuous energy input driv- ing the pulsation. Rather, the oscillations are supported by several mechanisms acting simultaneously, the most important ones being theκ andγ mechanisms. Theκ mechanism (Baker & Kippenhahn,
1962) is related to zones of high opacity in the outer layers of the star. After an initial compression these enhanced opacities prevent energy to be released immediately. The energy is stored below the ionization zone and is released later to drive the stellar pulsation.
The opacity of a region of the star is given by the Rosseland mean opacityκ with a temperature dependence according to
κ=κ0ρnT−s , (1.31)
Figure 1.10: Theoretical HRD for stars with metalicity Z=0.2Z (metalicity of the SMC) and a
helium abundanceY =0.25. The evolutionary tracks for stars of different masses are shown as lines
(labels given in solar units). Open circles and crosses (for the first and later crossings of the instability strip, respectively) mark the position of fundamental unstable modes (Baraffeet al.,1998).
no abundant element (i.e. hydrogen or helium) is undergoing ionization, the slope of the energy dependence s≈3.5 and the slope of the density dependencen≈1. Thus, the opacity declines as the
temperature rises and the energy can be radiated away more easily. However, in ionization zones of an abundant element, sgets small or even negative and energy can be stored. In fact, the second helium ionization zone is very efficient to drive pulsations with the κ mechanism. Also the first helium and the hydrogen ionization zones can drive pulsations, their strength depends however on the stellar type, in which pulsation is occurring.
Simultaneously to hindering the energy transport by theκ mechanism, the driving zones also ab- sorb energy during compression to ionize the abundant element. The radiation flow is locally reduced and absorbed by the matter in the ionization region, which leads to a pressure maximum right after the epoch of minimum volume. This driving is called theγmechanism (Coxet al.,1966).
The driving mechanisms get more inefficient the more convection contributes to the energy trans- port. As convection in the envelopes gets more important as the stars get cooler (e.g., Baraffeet al.
1998), it is believed that this process limits the instability strip at the red edge. As stars get hotter and bluer the position of the ionization zone moves outward to less dense regions of the envelope. With the density being too low for an efficient driving, the stars again get stable against small perturbations of their equilibrium configuration (e.g.,Christy 1966).
Model calculations of the stability of stellar envelopes have been widely used to determine the position of the instability strip in the HRD. In Fig.1.10we show the result of a self-consistent model
of Baraffe et al.(1998), showing some evolutionary tracks for stars of different mass. At positions
marked with crosses and circles stars are unstable against radial pulsation in the fundamental mode. As can be seen, the circles and crosses (marking the first and later crossings during an evolutionary track, respectively) are restricted to a small area, the instability strip.
are high mass, young and metal-rich (population I) red giants, whereas the type II Cepheids belong to an older population (also known as population II). Variables residing in the intersection of horizontal branch and instability strip are known as RR Lyrae stars. Due to the nature of the pulsation, period and luminosity are tightly correlated which can be observed in the period-luminosity (PL) relations for the different classes of stars. The logarithm of the pulsation periodPis proportional to the (intensity- or magnitude-weighted) mean magnitudeM(∝−log(L)) of the star
M=alog(P) +b , (1.32)
wherea denotes the slope andbdenotes the zero-point. This type of relations makes variable stars excellent distance indicators. Once the extinction on the line of sight is known, the absolute (in- trinsic luminosity at a standard distance) and apparent (i.e. observed) magnitudes can be linked to determine the distance of the star. Since their discovery (Leavitt & Pickering,1912) the PL-relations for Cepheids were continuously improved. However, the studies regarding a possible metalicity and environmental dependence of the PL-relation are not finished yet.
Many of the stars on the upper red giant branch and AGB are variable. Long Period Variables with periods from about 30 days to over 1000 days can be found there. In these stars with predominantly convective envelopes the mechanisms driving the pulsation are not yet fully understood, although progress is made in the modeling. For example, Xiong et al. (1998) proposed that the dynamical and thermodynamical coupling between convection and oscillation could be the dominant factor for pulsation instability. Using this scheme they succeeded in showing the presence of an instability strip for red variables.
Apart from pulsation, stellar variability can also be due to eruptive processes, novae and super- novae being the best known examples of this class. Novae are caused by a thermonuclear runaway in a hydrogen-rich accretion disk around a white dwarf in a close binary system. In contrast, super- novae explosions of type II mark the final stage of stellar evolution of massive stars. As their type Ia counterparts they are of fundamental importance for the creation of heavy elements as well as for the metal-enrichment of the intergalactic medium.
Finally, variability can also be induced by environmental and geometrical effects. Eclipsing binary systems and ellipsoidal variables belong to this group, the first ones being important systems for the determination of distances.