Abstract
Cold, dense clouds of gas have been proposed to explain the dark matter in Galactic haloes, and have also been invoked in the Galactic disc as an explanation for the excess faint submillimetre sources detected by SCUBA. Even if their dust-to-gas ratio is only a small percentage of that in conventional gas clouds, these dense systems would be opaque to visible radiation. We examine the possibility that the data sets of microlensing experiments searching for massive compact halo objects can also be used to search for occultation signatures by such clouds. We compute the rate and time-scale distribution of stellar transits by clouds in the Galactic disc and halo. We find that, for cloud parameters typically advocated by theoretical models, thousands of transit events should already exist within microlensing survey data sets. We examine the seasonal modulation in the rate caused by the Earth's orbital motion and find it provides an excellent probe of whether detected clouds are of disc or halo origin.
1 Introduction
Over the last decade several workers have suggested that the Galaxy may contain a large population of cold (T≲10 K), dense, low-mass clouds of molecular material. Several authors have argued that the mass contained in such clouds could be responsible for keeping galaxies’ rotation curves flat beyond the edges of optical discs. Pfenniger & Combes (1994) argued that the clouds would be an integral part of a fractal interstellar medium and be confined to a thin disc. By contrast, de Paolis et al. (1995), Gerhard & Silk (1996), Draine (1998) and Walker & Wardle (1998) considered the case of cold clouds that are distributed in an approximately spherical halo. De Paolis et al, Draine and Gerhard & Silk argued for clouds of approximately solar mass, while Walker & Wardle pointed out that ionized winds from clouds with masses around a Jupiter mass (MJ) could explain the extreme scattering events that are frequently observed in the light-curves of pulsars seen at low Galactic latitudes.
Clouds that are an integral part of the interstellar medium (ISM) would be expected to be dusty, and radiate strongly at submillimetre wavelengths. Lawrence (2001) argues that a significant fraction of the reddest SCUBA sources could be dusty clouds in the Galactic disc. Clouds with M∼MJ would have temperatures T∼10 K, and if they were in virial equilibrium, they would be ∼14 au in diameter and totally opaque at optical wavelengths. Their number density on the sky has to be comparable to that, ∼1000 deg−2, of unidentified SCUBA sources, so they would cover a fraction ∼10−4 of the sky. Consequently, at any given time, of order one star in ten thousand should be occulted by one of these clouds. In this letter we consider the feasibility of detecting such occultations in the data bases that have been produced in searches for microlensing events (for a recent overview of microlensing results see Kerins 2001, and references therein).
2 Cloud characteristics and detectability
In order to occult background stars cold clouds must meet certain criteria. First, the clouds must be sufficiently diffuse that they do not appreciably gravitationally lens the starlight. The case of gravitational lensing by gas clouds has been studied previously by Henriksen & Widrow (1995). A necessary condition is therefore that the cloud radius R exceeds the Einstein radius, leading to the requirement that
where M is the cloud mass and D≡Dc(Ds-Dc)/Ds, with Dc (Ds) the distance between the observer and cloud (source).
A major issue with the cold cloud proposal is to understand why the clouds do not collapse to form stars or brown dwarfs. Sciama (2000) has argued that cosmic-ray heating may be able to balance molecular cooling in a stable way for dust-free clouds, whilst Lawrence (2001) has presented similar arguments in the case of dusty clouds. If opaque clouds can be stabilized against collapse, their size would be of the same order as their virial radius (cf. Lawrence 2001)
R∼Rvir would also satisfy the lensing condition in equation (1) provided
This condition is satisfied for almost all plausible clouds since, regardless of their formation mechanism, hydrogenous clouds less massive than ∼10−7 M⊙ evaporate away over a Hubble time (de Rújula, Jetzer & Massó 1992).
If they are transparent (Walker & Wardle 1998, Draine 1998), diffuse clouds will still refract, rather than occult, starlight. How opaque might we expect the clouds to be? The average hydrogen column density through a virialized cloud is
Assuming typical dust properties, the visual extinction through a cloud is therefore (e.g. Binney & Merrifield 1998)
where f is the factor by which the cold cloud dust-to-gas ratio exceeds that of typical Galactic molecular clouds.
We should expect disc clouds to have a similar dust-to-gas ratio to that of the interstellar medium and therefore be completely opaque. However, equation (5) also implies strong limits on halo cloud properties for them to escape detection by occultation. This can be seen by considering how halo clouds may have interacted with the interstellar medium over the lifetime τgal of the Galaxy. Over this time one would expect a typical halo cloud to have passed through the Galactic disc N times, where
with rc and σ the halo core radius and velocity dispersion. A lower limit on the mass swept up by these clouds on each passage through an exponential disc is
where ρ is the typical density of gas in the plane of the disc, H the disc scaleheight and ɛ the efficiency with which the clouds retain swept-up gas. Adopting the parameter values in equations (6) and (7) implies that the present mass fraction of swept-up disc gas in a typical halo cloud is NMsweep/M∼0.007ɛ for a 10−3 M⊙ cloud. Substituting this quantity for f in equation (5) gives a limit ɛ>0.008 required to produce at least a one magnitude diminution in the light from background stars.
For isothermal clouds with a mean extinction given by equation (5) the extinction at their periphery would be lower. However, even at 99 per cent of the projected radius it would only be 5 mag below the mean value. Therefore, provided f>6×10-4, the extinction would be at least 5 mag across almost the entire cloud, and the resulting light-curve would exhibit a sharp flux cutoff close to the onset of transit. In the case f=0 the clouds would be transparent and, whilst not producing an occultation signal, could be detected by refraction or lensing effects considered by previous authors. More exotic light-curve profiles are possible for clouds of modest extinction due to a combination of refraction at the cloud periphery and extinction towards the centre. However the above considerations imply that f must be finely tuned for this intermediate case to apply.
Lastly, we need to know how easily the flux change from transit events can be detected. An important consideration is source resolvability. For Galactic microlensing experiments it is often the case that more than one star occupies the seeing disc, leading to a detection bias referred to as blending (Alard 1997). The situation is more extreme for experiments looking towards M31 where there are many stars per detector pixel. Transit events could escape detection by these surveys if the occulted star lies so close to other stars that the effect of its occultation is masked by light from other stars. Quantitatively, if a fraction f∗ of the flux in the seeing disc comes from the target star, then the event will be detected only if f∗≳ΔMT, where ΔMT≪1 is the smallest measurable magnitude variation. In the context of gravitational microlensing searches, experiments to detect microlensing events in our Galaxy typically adopt the very conservative value ΔMT∼0.3 as a means of excluding variable stars. The M31 surveys typically use a much lower threshold ΔMT∼0.02. If similar thresholds are adopted for the detection of transit events then they should be detectable within our Galaxy provided blending effects are not too severe. It may even be possible to detect transit events towards M31 provided the occulted stars ordinarily contribute upwards of 2 per cent of the flux in a seeing disc.
3 Transit rate and time-scale distribution
From here onwards we assume that the cold clouds are opaque and exceed their Einstein radius, so that they give rise to observable occultation events. We calculate the rate at which occultations of a given duration should be observed at any season of the year and along any line of sight. Similar calculations of microlensing rates are described by Griest (1991) and Kiraga & Paczyn´ski (1994).
Let there be n(v)d2v spherical clouds of radius R per unit volume and with velocities in the plane of the sky in d2v. Then in a sheet of thickness dDc that extends perpendicular to the line of sight, the rate at which clouds pass a given background star with impact parameters in the range (b,b+db) is
where v⊥=|v|. In an obvious notation we have d2v=v⊥dv⊥dθ, so
An impact with b<R causes an occultation of duration
so we may obtain the distribution over durations by eliminating b in favour of t. One finds
This equation gives the occultation rate of a single source star. To get the rate that can be measured observationally, we have to average it over all detectable source stars (cf. Kiraga & Paczyn´ski 1994). If the probability that a source star at distance Ds has its velocity on the sky in d2vs is p(vs)d2vs, then the required average of equation (11) is
v⊥ is the magnitude of the cloud's transverse velocity with respect to its local point on the line of sight from the source to the Sun; consequently it is a function of vs. The argument v of the velocity distribution n(v) must relate to a fixed frame. In the frame in which the Earth is at rest
where Ds is the distance to the source star, so, for fixed v⊥, v is a function of vs and, to make further progress analytically, we assume that both v and vs have Gaussian distributions. Since we are working in the Earth's rest frame, these Gaussians will be centred on non-zero velocities v and vs, respectively. Consequently we have
where nc is the volume density of clouds. Now θ is the direction of v-(Dc/Ds)vs, so with (13) we have
where
When we insert (15) into n(v) and integrate over θ we find
where I0 is the usual modified Bessel function. Inserting this result in (12) we find
This result simplifies significantly if the sources have negligible random motions (i.e. σsDc/Ds≪σ). In this case we can approximate the Gaussian distribution over vs by δ2(vs-vs), giving
with vs replacing vs in (16). The rate of occultations integrated over all durations ∫dt d2Γ/dDcdt is readily obtained from either (18) or (19):
Since the central velocities vs and v are relative to the Earth, equations (18) to (20) yield occultation rates that depend on observing season.
From equation (20)Γ∝R/M (because nc=ρ/M), so for virialized clouds equation (2) indicates that the transit rate is controlled simply by the cloud temperature T, i.e. Γ∝T-1, independent of the scale size of the cloud. However, for fixed dimensionless impact parameter b/R the transit time t∝R from equation (10) so dΓ/dt∝R-1T-1 when integrated over all impact parameters 0≤b/R≤1.
In some applications the sources are widely distributed in distance Ds. In this case we follow Kiraga & Paczyn´ski (1994) in evaluating weighted averages over Ds of the rates (18) to (20), with the weight factor taken to be 
. Here ns is the source volume number density and β parameterizes the effects of the luminosity function upon the distance distribution of sources. In our calculations we adopt β=-2 for Ds>100 pc and β=-1 at smaller distances.
4 Model predictions
We explore two locations for potential cloud populations, the Galactic disc and halo, and we compute the rate of occultations for lines of sight to three targets: the Large Magellanic Cloud (LMC) at Galactic coordinates (l,b)=(280°,-33°), the Small Magellanic Cloud (SMC) at (l,b)=(303°,-44°) and Baade's Window towards the Galactic Centre (GC) at (l,b)=(1°,-4°). These are all regions which have been intensively monitored by several experiments looking for microlensing events. As discussed in Section 2, a fourth possibility is the Andromeda Galaxy (M31), though in that case the formulae of the previous Section must be modified to take account of the fact that the sources in M31 are mostly unresolved. We therefore confine our attention in the remainder of this paper to the GC, LMC and SMC only. The LMC and SMC source stars are assumed to lie at fixed distances of 50 kpc and 60 kpc, respectively, whilst background sources towards Baade's Window are assumed to be distributed throughout the Galactic disc and bar.
Since we are not concerned with the precise details of the halo structure, it is sufficient for our purposes to model the distribution of halo clouds as a simple softened isothermal sphere with a core radius of 5 kpc and a local density of 0.01 M⊙ pc−3. The halo velocity distribution is taken to be Gaussian with σ=156 km s-1. For both clouds and sources in the Galactic disc we model their density distribution as a sech-squared profile with a scalelength of 2.5 kpc and scaleheight of 190 pc. We assume the clouds contribute up to a third of the disc density and so have a local density of 0.03 M⊙ pc−3. Strong local kinematical and dynamical constraints rule out the existence of a more significant population (Crézé et al. 1998). We adopt a velocity dispersion of σ=25 km s-1 and a rotation speed vrot=220 km s-1 for both disc clouds and sources, independent of radius.
For calculations towards the LMC and SMC we neglect random stellar motions in the clouds themselves though we do include the bulk motion of the Clouds (u,v,w)=(60,-155,146) km s-1 (Jones, Klemola & Lin 1994), where the (u, v, w) coordinate system is centred on the Sun and the velocity components point, respectively, towards the Galactic Centre, the direction of rotation, and the North Galactic Pole. Equations (19) and (20) are therefore sufficient to compute the halo cloud time-scale distribution and rate towards these targets. For bulge sources we adopt the bar model of Bissantz et al. (1997) with exponential scalelength am=3.82 kpc, power-law scalelength a0=100 pc, axis ratios η=ζ=0.3 and bar angle φ0=25°. We assume a bar pattern speed of 70 km s−1 kpc−1 and a velocity dispersion σ=100 km s-1. Whilst our disc model is simpler than the double-exponential profile of Bissantz et al. (1997) its scalelength is the same and the chosen scaleheight corresponds to the weighted mean of the scaleheights in their model. Our overall disc-bulge mass normalization is therefore in good agreement.
The observer's motion is modeled as the sum of three distinct components: the LSR motion (u,v,w)=(0,vrot,0) km s-1; the Sun's motion in the direction of the Solar Apex (u,v,w)=(10,5.2,7.2) km s-1 (Binney & Merrifield 1998); and the motion of the Earth around the Sun, where we adopt a circular orbit with v⊕=30 km s-1.
4.1 Rate and time-scale distribution
Fig. 1 shows seasonally-averaged distributions of occultation time for both halo and disc clouds towards the LMC, SMC and GC. We have adopted a radius R=7 au and virial temperature T=10 K for both cloud populations, corresponding to a cloud mass of M=10-3 M⊙ from equation (2). For these parameters we see that the halo cloud distributions (solid lines) are peaked at shorter durations than for disc clouds (dashed lines), with the peak occurring around 60 d for all three target directions. For disc clouds the peak occurs at around 100 d in the direction of the GC and around 300 d towards the LMC and SMC.
Seasonally averaged cloud transit time-scale distributions towards the LMC, SMC and GC for halo clouds (solid lines) and disc clouds (dashed lines). The figure assumes the clouds are virialized, with R=7 au and T=10 K. The scaling for other values is d〈Γ〉/dt∝R-1T-1 and t∝R.
Seasonally averaged cloud transit time-scale distributions towards the LMC, SMC and GC for halo clouds (solid lines) and disc clouds (dashed lines). The figure assumes the clouds are virialized, with R=7 au and T=10 K. The scaling for other values is d〈Γ〉/dt∝R-1T-1 and t∝R.
For fixed cloud temperature T=10 K the highest transit rate comes from disc clouds observed towards the GC and halo clouds towards the SMC, both producing ∼1500–3000 events per yr per million stars. Halo clouds towards the SMC have the largest rate for transit durations below 500 d. Halo clouds observed towards the LMC and GC have smaller, but comparable, rates. These large rates in comparison with microlensing are due to the size of the clouds. For 10−3 M⊙ clouds their virial radius is 1.5–2 orders of magnitude larger than the typical Einstein radius of compact objects of the same mass, so the rate is correspondingly higher. Both the time-scales and the expected rate are easily within range of current microlensing experiments. Towards the LMC and SMC the disc cloud transit rate is lower by more than an order of magnitude at ∼100 events per year per million stars. Even when detection efficiencies have been taken into account, microlensing data sets should contain hundreds or even thousands of transit events, so it is clear that much of the parameter space for both halo and disc cloud populations can be probed.
4.2 Discriminating between disc and halo clouds
Can we discriminate between halo and disc cloud populations? A rate of ≳1000 events per year per million sources towards the LMC and SMC would argue strongly for a halo origin. Another obvious diagnostic is the transit duration. Fig. 1 indicates that we should expect few halo cloud events with transit durations above 100 d, whilst more than three-quarters of the rate of disc clouds towards the GC comes from events with durations above 100 d. Of course larger halo clouds can produce longer transits so this is not a robust test.
A potentially cleaner test is the seasonal modulation in the observed rate resulting from the Earth's motion around the Sun. The modulation is most pronounced for nearby clouds, which will typically be disc clouds. Fig. 2 shows the modulation in the instantaneous transit rate Γ, as a ratio of the seasonally averaged rate 〈Γ〉. The modulation is plotted as a function of Solar Ecliptic Longitude, λ⊙, where λ⊙=0 at the Vernal Equinox which occurs around March 21. The dark grey region of the figure, spanning from September to March, indicates the six-month period when microlensing experiments monitor the Magellanic Clouds. For the rest of the year the GC is observed, as shown by the light grey region. Consequently, only half of the modulation period is accessible for any of the targets, so the modulation curves are shown as bold lines whilst the relevant target is observed and as thin lines when not.
Seasonal modulation of the occultation rate, displayed as a function of Solar Ecliptic longitude, λ⊙. Line coding is as for Fig. 1. The modulation curves are shown as thick lines during the relevant observing season and as thin lines otherwise (see text).
Seasonal modulation of the occultation rate, displayed as a function of Solar Ecliptic longitude, λ⊙. Line coding is as for Fig. 1. The modulation curves are shown as thick lines during the relevant observing season and as thin lines otherwise (see text).
The largest modulation occurs for disc clouds observed towards the LMC and SMC. During the LMC/SMC observing season the rate swings from 0.8 to 1.2 times the seasonal average, a modulation of 20 per cent. Assuming 100 per cent detection efficiency, a 20 per cent modulation could be detected at the 3σ level after four seasons of data for a million sources. Current experiments are monitoring closer to 107 sources towards the LMC, though their detection efficiency to transits will be somewhat less than 100 per cent. It's nonetheless clear that this modulation should be detectable already. Halo clouds show only a small (few per cent) variation in their rate towards the LMC and SMC, however their transit rate is much larger than for disc clouds. A sample of upwards of 20 000 transit events is required to detect a modulation of 2 per cent at greater than 3σ significance. Experiments monitoring 107 sources would amass such a sample within four years (assuming 50 per cent detection efficiency), so even this small modulation should be evident within existing data sets. Towards the GC the modulation of the disc and halo cloud rate is similarly small but the rate is comparably large, so again the modulation should be apparent. In the case of halo clouds, observations towards the GC should show a maximum rate during mid-season, whilst disc clouds should exhibit a minimum. The modulation signal should therefore provide a good test for the origin of detected events.
5 Discussion
Cold opaque clouds can be detected by searching data sets already compiled by groups searching for gravitational microlensing events. Rather than the flux excess that is the signature of microlensing, cold cloud events are characterized by a transient dimming of background sources. Like microlensing, this signal should be non-periodic and one should expect events to involve a representative population of background source stars. In selecting microlensing events one problem is to reject variable stars which may mimic a microlensing signal. The equivalent ‘background’ for cloud transit events is potentially much smaller since there are fewer classes of objects which involve a significant dimming of flux. One such class, eclipsing binaries, is unlikely to present much of a problem. For short-period systems their true nature would be evident over the lifetime of the surveys. Binary systems with longer periods for which only one flux dropout is detected, should statistically comprise more massive stars, so we should expect the source stars of these ‘events’ to be unrepresentative of the target stellar population as a whole, contrary to the expectation for true cloud transit events. It is also likely that the light curve of cloud transit events would have a form that generally would be inconsistent with that of eclipsing binaries.
The previous section demonstrates that the transit rate and time-scales of cold clouds in the disc and halo is well within the range of detectability if they constitute a significant population and are of a Jupiter mass scale. Whilst the total rate depends only on the temperature of the clouds, virialized clouds more massive than ∼0.05 M⊙ would have transit durations exceeding the present baseline of microlensing experiments.
The seasonal modulation of the rate provides a promising method to distinguish whether the clouds are in the disc or the halo. There are already strong arguments from the consideration of submillimetre sources (Lawrence 2001) that opaque cold clouds do not contribute much to the halo dark matter budget, though these arguments are sensitive to the precise dust properties of the clouds. Existing microlensing data sets represent a significant corpus of data which can provide an independent line of approach. Though originally conceived as a probe only of compact objects, microlensing surveys may also turn out to be one of the most sensitive probes of diffuse cold clouds.
