The possiblity of detection of ultracool dwarfs with the UKIRT Infrared Deep Sky Survey

Abstract

We present predictions for the numbers of ultracool dwarfs in the Galactic disc population that could be detected by the WFCAM/UKIDSS Large Area Survey and Ultra Deep Survey. Simulated samples of objects are created with masses and ages drawn from different mass functions and birthrates. Each object is then given absolute magnitudes in different passbands based on empirically derived bolometric correction versus effective temperature relationships (or model predictions for Y dwarfs). These are then combined with simulated space positions, velocities and photometric errors to yield observables such as apparent magnitudes and proper motions. Such observables are then passed through the survey selection mechanism to yield histograms in colour. This technique also produces predictions for the proper motion histograms for ultracool dwarfs and estimated numbers for the as yet undetected Y dwarfs. Finally, it is shown that these techniques could be used to constrain the ultra-low-mass mass function and birthrate of the Galactic disc population.

1 INTRODUCTION

The precise nature of the physical mechanisms which result in the stellar initial mass function (IMF) is one of the most fundamental open questions left in astrophysics at the start of this century. One of the main constraints that can be set on models of star formation is that the spectrum of masses (mass function) predicted is in good agreement with that observed in the local universe. The mass function ξ(log m) at any given time is defined to be the number of stars dn per unit volume in a mass interval d log m. The definition of the mass function is, therefore,

 

Note that the mass spectrum defined by Scalo (1986) as dn/dm is often referred to as the mass function but we shall stick with the original definition in Salpeter (1955). In this paper, we will only consider the IMF below m∼ 0.1 M_⊙ as it is relatively poorly constrained in this region. The two most common functional forms for the IMF are the power-law form and the lognormal form. The power-law form is

 

where above masses m∼ 1 M_⊙, α≈ 1.35 (e.g. Salpeter 1955) while at lower masses it is seen to flatten off. At the high mass end of our mass range, Chabrier (2001) finds α≈ 0.55 while Kroupa (2001) and Reid, Gizis & Hawley (2002) derive α≈ 0.3. Kroupa also fits a power law in the range 0.01 < m < 0.08 M_⊙ finding α=−0.7. Finally, Allen et al. (2005) use a series of assumptions about the birthrate and a Bayesian method to yield a value of -0.7 in the range 0.04 < m < 0.1 M_⊙. In order to explore this range in power-law exponents, here we examine IMFs with values of α of 0, −0.5 and −1. The lognormal form of the IMF is

 

Here we use a lognormal function with the parameters log₁₀m_c=−1.1 and σ= 0.79 (Chabrier 2003).

Regarding observational studies of the IMF, those in open clusters benefit from all the stars in the cluster being of the same age. Hence, to derive the IMF it is simply a matter of converting a measured luminosity function (LF) using a single mass–luminosity relation. However, objects in the field will have a range of ages, and since brown dwarfs (objects below the hydrogen burning limit of 0.075 M_⊙) lack any internal energy source, they cool and decrease in brightness with time. In addition, the scaleheight of any stellar population within the Galactic disc evolves with time. Hence, both the photometric and kinematic properties of these objects are affected by their ages. In order to measure the mass function in the field we must therefore first consider the creation function and use models for the luminosity evolution of stars to convert it to an LF. This simulated LF can then be compared to the observed LF to see if the creation function assumed is viable. We define the creation function C as the number of objects created per unit time, per unit log m such that,

 

where T_G is the age of the Galaxy and b(t) is the stellar birthrate rate relative to the average birthrate such that b(t) = (dn/dt)/(n_tot/T_G), i.e. the birthrate is the relative number of objects formed in the Galactic disc per unit time (note we assume that the IMF is time-independent). The Miller & Scalo (1979) study of the birthrate suggested that it does not depend strongly on the density of gas in the Galactic disc and is approximately constant. More recent studies such as Rocha-Pinto et al. (2000) have shown that stars appear to form in a series of bursts. In this study we model the stellar birthrate as constant or as an exponential,

 

We employ four different values of the scale time τ: three decreasing birthrates with τ= 10, 5 and 1 Gyr and one increasing with τ=−5 Gyr. When generating age distributions in simulations we use 10 Gyr as the maximum age for an object in the Galactic disc.

In order to study the birthrate and the IMF below the hydrogen burning limit a large sample of ultracool dwarfs is required. Ultracool dwarfs are the observed L and T dwarfs and the as yet unobserved Y dwarfs. Infrared surveys are ideal for discovering large samples of cool dwarfs as they are brightest in the near infrared. The first major modern infrared surveys came with the Deep Near Infrared Survey (DENIS; Epchtein et al. 1994) and the two-Micron All-Sky Survey (2MASS; Kleinmann et al. 1994). DENIS covers the southern sky in I, J and K_s down to limits of I= 18.5, J= 16.5 and K_s= 14.0 while 2MASS is an all sky survey in J, H and K_s down to limits of J= 15.8, H= 15.1 and K_s= 14.3. Both have been useful for discovering ultracool dwarfs with DENIS finding several late M and L dwarfs and 2MASS finding countless L dwarfs and a large sample of T dwarfs. In addition several tens of L and T dwarfs (Chiu et al. 2006) have been found using the primarily optical Sloan Digital Sky Survey (SDSS; Adelman-McCarthy et al. 2006) The next generation of infrared surveys will be undertaken with large format imagers such as WFCAM (Henry et al. 2003) and WIRCAM (Puget et al. 2004). WFCAM is a wide-field quasi-Schmidt camera mounted at the Cassegrain focus of the UK Infrared Telescope (UKIRT). A quadruple detector array provides (with mosaiced observations) a field of view of 0.77 deg². The UKIRT Infrared Deep Sky Survey (UKIDSS; Lawrence et al. 2006) is employing WFCAM for a series of Galactic and extragalactic surveys, and two of these surveys are ideal for the detection of ultracool dwarfs in the field. The Large Area Survey (LAS) is a wide-field, high-latitude survey in Y (Hewett et al. 2006) and JHK in the Mauna Kea Observatories (MKO) system (Simons & Tokunaga 2002; Tokunaga, Simons & Vacca 2002); the Ultra Deep Survey (UDS) is a deep, narrow-field survey in JHK. The initial two-year program for the LAS will be 2000 deg² to a depth of J= 19.7. The seven-year plan will include a second J-band epoch to allow proper motion measurements and will cover 4000 deg² to a depth of J= 20. The UDS will cover 0.77 deg² in both the two and seven-year plans. In the two-year plan, only J and K will be observed to a depth of J= 24.0, while the full seven-year plan also includes H and will go one magnitude fainter in J. The number of objects of different effective temperatures and colours (which we shall employ as a detection function) found in these surveys can provide information on the IMF (e.g. Pinfield et al. 2006) and birthrate.

As well as having important statistical properties, ultracool dwarfs are also interesting objects in themselves. One particular area that UKIDSS hopes to study is the transition between the L and T spectral types. Here, the dramatic onset of methane absorption around 2 μ m and the removal of dust clouds high in the photosphere cause a sudden shift from red near infrared colours to blue near infrared colours. The exact details of this transition are still a matter for debate with many theories such as patchy cloud clearing (Burgasser et al. 2002), sudden downpours (Knapp et al. 2004) and runaway dust growth (Tsuji 2005) seeking to explain it. Also of interest are the as yet unobserved Y dwarfs. It is not yet known where there will be a significant shift in atmospheric chemistry causing the use of the label Y. Hence, we define Y dwarfs as cooler than the temperature for the coolest subclass of T dwarf given by Vrba et al. (2004), 770 K. There are many different sets of models – e.g. Burrows, Sudarsky & Lunine (2002), Baraffe et al. (2003), Marley et al. (2002) and Tsuji (2005)– which predict a range of near infrared colours for these objects (Leggett et al. 2005). Clearly Y dwarfs discovered with UKIDSS can help to constrain these atmosphere models.

In the rest of this paper, we go on to describe our simulation method for predicting the number of ultracool dwarfs that will be found in the UKIDSS LAS and UDS given the various assumptions described above concerning the form of the underlying birthrate and IMF.

2 SIMULATION METHOD

We implement a simulated population method to predict the possible results of the UKIDSS LAS and UDS. Simulated (very low-mass) stellar and substellar populations based on the different IMFs and birthrates discussed previously are created.

Each simulated object is given an age based on the birthrate and a mass based on the IMF, and these dictate the photometric and astrometric properties assigned. Each object is subsequently passed through the survey selection mechanism to yield the simulated results of the survey. The local number of very low-mass stars simulated is anchored using the same method as Burgasser (2004) with the mass spectrum in the range 0.1–0.09 M_⊙ set to 0.0055 pc⁻³. Using this method, histograms binned by observables such as proper motions and colours can be produced.

2.1 Positional and velocity simulations

In order to effectively model the positions and proper motions of our synthetic stellar population we must first consider their Cartesian positions and space velocities. We generate heliocentric Galactic Cartesian coordinates out to a maximum distance of 2 kpc for the LAS and 10 kpc for the UDS. Our simulations indicate that L dwarfs are not detectable in either survey at distances greater than these. In our coordinate system x and y lie in the Galactic plane while z is perpendicular to the plane. These have velocities U, V and W associated with them. The x and y positions are randomly generated from a flat distribution while the z position is generated from an exponential distribution with scaleheight z₀. It is well established that the older populations have larger scaleheights, hence we must incorporate this age dependency into our model. It is also clear that due to disc heating the velocity dispersion of a population increases with age, and this is taken into account. Hence, we generate space positions and velocities based on the age of the object. From these we calculate the sky positions, distance and proper motions. Any object whose angular position falls within the survey area passes the positional survey selection mechanism.

2.2 Photometric simulation

We utilize evolutionary models from Baraffe et al. (2003) of ultracool dwarfs to provide look-up tables of quantities such as T_eff and absolute magnitudes in a given passband, versus mass and age. The current picture of substellar models is one of division between two different model sets: those in which dust is suspended in the photosphere and those where the dust has largely settled below the photosphere. The spectrum of T dwarfs is best fitted by the models where the dust has settled, but the infrared colours of such models do not match well those observed for L dwarfs. We use temperatures and bolometric magnitudes from the dust settled (COND) models1 across the M, L, T and Y regimes in a manner similar to Burgasser (2004) (who noted that this would not produce a temperature error greater than 10 per cent). However, in order to properly model the photometric detectability for all these spectral types we must utilize an effective temperature versus bolometric correction relation of each passband. Hewett et al. (2006) calculated the colours of a series of ultracool objects from their spectra. We combined these with absolute J magnitudes (converted into the MKO system) along with absolute magnitudes and effective temperatures2 from Vrba (2004) and bolometric magnitudes from Golimowski et al. (2004). We then fitted two separate polynomials in the T regime and in the M & L regime (Fig. 1) since it is clear that in some passbands the relationship is discontinuous across the L–T boundary. We utilize the Baraffe et al. (2003) models in which the dust has settled below the photosphere to provide an effective temperature and a bolometric magnitude for each of the simulated stars. The temperature is then used to calculate the bolometric corrections (and hence infrared magnitudes) for each simulated object. For objects cooler than T_eff= 770 K we have no observational data on which to base bolometric corrections so we rely on the magnitudes predicted by the models. Finally, the photometric survey selection method is simply detection above the quoted depths for the survey in all passbands used except K. This is because cool T dwarfs will be very faint in the K band due to methane and water absorption.

2.3 Simulated T_eff distributions

The simulated T_eff distributions shown in Figs 2 and 3 illustrate how altering the IMF and birthrate, respectively, affects such histograms. Mass functions which increase at lower masses will increase the number of low luminosity objects and hence will raise the height of the peak around 500 K. A birthrate where most objects were created at early star formation epochs will deepen the trough around 1700 K as most objects will have evolved past the mid L spectral types into the T and Y domains. These peaks and troughs are similar to those found by Burgasser (2004) but, due to the varying ages of the objects, the complicated system of peaks and troughs found by Allen et al. (2003) for single-age populations is not seen. Since the underlying form of the IMF and birthrate affects markedly the number counts in T_eff and colour, those histograms should be a useful probe of the IMF and birthrate.

3 THE LARGE AREA SURVEY

The LAS is designed to search for ultracool dwarfs, high redshift quasars and cool subdwarfs. It uses J, H and K in addition to the Y filter (Hewett et al. 2006). The Y filter lies in between the I and J bands, is centred around 1 μm and is slightly redder than the Z filter. It is specifically designed for the study of ultracool dwarfs and quasars. It will provide a band to allow M, L, T and Y dwarfs to be distinguished both from each other and from hotter main sequence stars. Additionally, SDSS (Adelman-McCarthy 2006) photometry can be used to remove quasars from the sample (Hewett et al. 2006). The final survey will cover an area around the Northern Galactic Pole as well as a small strip in the south Galactic cap. Both areas are scanned by the SDSS (Adelman-McCarthy 2006) allowing additional optical photometry to be utilized. The full seven-year survey will go to depths of Y= 20.5, J= 20.0, H= 18.8 and K= 18.4 (detection in K was not required in the survey selection mechanism as many cooler objects have their luminosity severely reduced in this band due to methane and water absorption) with an additional second scan in J to allow proper motion measurements. Simulations of the full seven-year LAS area were carried out as described in Section 2. Fig. 4 shows the simulated proper motion histograms for objects of different spectral types. Notice that later spectral types' histograms peak at higher proper motions. This is simply a selection effect as cooler objects will only be observable nearby, where they will typically have large proper motions.3

The expected numbers of detected objects of different spectral types are shown in Table 1. Fig. 5 shows the effect on the (J−H)_MKO colour4 histogram of altering the IMF. The sharp drop at (J−H)_MKO= 0.4 is due to objects bluer than this divide being mid T dwarfs and objects redder than this being (on the whole) the much more easily detectable early L dwarfs. It is clear that for more steeply declining IMFs fewer T dwarfs are observed. Note that the lognormal and flat (α= 0) IMFs produce very similar results.

The effects of different birthrates are shown in Fig. 6. Here, birthrates which produced more objects earlier in the Galaxy's history have fewer T dwarfs due to the cooling of brown dwarfs with time. The higher numbers of early L dwarfs in simulations with birthrates which were higher in the past is due to our normalization and the way we have modelled scaleheight evolution. As the age of a population increases it becomes more spread out and hence the density in the Galactic plane drops. Since we are normalizing in the local region (the Galactic plane) the number of objects here is kept constant. Hence as the population spreads the total number of objects increases. So if a particular class of object has enough mass for stable hydrogen burning, as early L dwarfs do, the number of detectable objects is increased by a birthrate which was historically higher. Distributions with very long scale times (either increasing or decreasing) would be difficult to distinguish from a constant birthrate. Note that in all cases the numbers of early T dwarfs are much smaller than those for late T dwarfs. This appears counterintuitive as early T dwarfs are brighter and hence more detectable. However, as the atmospheric chemistry of early T dwarfs changes quickly with effective temperature the spectral types T0–T4 cover a very small region of an effective temperature distribution (barely 100 K, see Figs 2 and 3). Hence even though late T dwarfs are more difficult to detect than early T dwarfs, their larger temperature spread (and hence higher numbers) lead to a greater number of detections.

4 THE ULTRA DEEP SURVEY

The UDS is a deep pencil-beam survey, covering 0.77 deg², primarily designed for extragalactic studies. For this purpose it uses the J, H and K passbands down to depths of J= 25, H= 24 and K= 23. This presents a problem when trying to detect ultracool dwarfs. T and Y dwarfs have similar near infrared colours to main sequence stars due to methane absorption around two microns. Hence with only J, H and K photometry detection can be difficult. Luckily for the study of ultracool dwarfs the area covered by the UDS is also covered by the Subaru/XMM–Newton Deep Survey which provides optical photometry. This will make it easier to distinguish ultracool dwarfs from other objects. Of course, many of the individual ultracool dwarfs found in the UDS will be too faint for spectroscopic follow-up observations. However, they will still contribute to the observed sample. Note that this survey is so deep that the Galactic disc scalelength had to be taken into account along with the scaleheight; we used a value of 3.5 kpc (de Vaucouleurs & Pence 1978). The results are shown in Table 2, where we see that a few tens of T dwarfs, a few hundred L dwarfs and a handful of Y dwarfs will be detected. While this sample will not be as useful as that of the LAS for studying the IMF, birthrate and spectroscopic properties of these objects, it may provide valuable data on their distribution within the Galaxy.

5 CONSTRAINING THE IMF AND BIRTHRATE

In the previous two sections, we have shown that it is possible to predict the numbers of objects detected in UKIDSS surveys using different birthrates and IMFs. Clearly, we can attempt also the reverse, and use the observed numbers to constrain the underlying birthrate and IMF. For example, suppose that we assume that the IMF (in the range 0.1–0.003 M_⊙) and birthrate have the following functional forms:

 

where a positive value of β implies a declining birthrate. We now simulate a grid of J−H distributions with values of α ranging from −2.0 to +2.0 and β ranging from −0.2 to +0.2. We then take another simulated distribution with known α and β. A multiplication factor γ is then used to vary the total number of objects for each simulated (J−H) distribution. The values of χ² for a range of values of γ are calculated for each value of α and β. Hence, we get a value of χ² for each value of γ and for each (J−H) histogram (and hence for each value of α and β). This distribution is then marginalized over γ (i.e. the probability distribution is integrated over γ) such that

 

This procedure produces a probability distribution over α and β. Marginalizing this further will produce distributions solely over α or β.

In order to study the constraints that could be set in more detail we first simulate a coarse grid in the ranges −2.0 < α < 2.0 and −0.2 < β < 0.2. Once we identify the area of maximum probability in this coarse grid we simulate a second, finer grid centred around this region. This has 10 times the resolution in α and five times the resolution in β. The results for such finer grids are shown in Fig. 7(α= 0, β= 0), Fig. 8(α=−1, β= 0) and Fig. 9(α= 0, β=−0.1). The first thing that becomes apparent is that the noise on these finer grids means that there is not a smooth probability distribution. However, it is obvious that there is a degeneracy between α and β. In order to glean information on the typical errors expected, the probability distribution was marginalized over α to produce a distribution in β and vice versa. The mean value of each parameter along with their s.d. values can then be calculated. These results are shown in Table 3. It is clear that the calculated values are in agreement with the values of the parameters used to generate the J−H histograms.

5.1 Constraining the IMF and birthrate using existing data

To allow some real results in advance of UKIDSS data being available, we simulated the results of an existing ultracool dwarf survey, viz. the J-band LF of Cruz et al. (2003), covering the L dwarf regime. We simulated a grid in the same manner as for the UKIDSS LAS with the appropriate cuts and limiting magnitudes quoted in Cruz et al. We excluded the bin centred on M_J= 10.75 as, although our maximum mass of 0.1 M_⊙ equates to a stable main sequence absolute J magnitude of 10.2 (Baraffe et al. 2003), the scatter into this bin from brighter bins caused by photometric errors would not be modelled correctly. We also excluded the two faintest bins from the probability analysis as they are described as being incomplete (however we did simulate them to correctly model scatter). Finally after going through the probability analysis we excluded a secondary peak which appeared at the edge of our grid as it was at a very high value of α excluded by other studies. The resulting probability surface is shown in Fig. 10. The measured values of the parameters were; α= 0.95 ± 1.17 (implying a mass function rising at lower masses) and β=−0.134 ± 0.173. The question remains over what range of masses is this result valid. If we take 0.1 M_⊙ as a maximum mass then our minimum mass will by given by the mass of an object with M_J= 14.0 at an age of 10 Gyr, the maximum calculated in the Baraffe et al. (2003) models – we find this to be m= 0.072 M_⊙. Hence the value of α covers the range 0.072 < m < 0.1 M_⊙. This result (albeit with a large error) is consistent with both Kroupa (2001), who measured a value in the range 0.08 < m < 0.5 M_⊙ of α= 0.3 ± 0.5, and with the Chabrier (2001) lognormal IMF peaking at 0.75 M_⊙. It differs by just over 1σ from the Allen et al. (2005) value of −0.7 ± 0.6. The value of β is consistent with a constant birthrate.

6 DISCUSSION

The series of simulations presented here will have two main uses. In the short term they can be used as a method for predicting the results of the UKIDSS surveys, yielding more accurate values for the expected number of extremely cool objects. This shows that – given current models and reasonable assumptions of the IMF and birthrate – tens of Y dwarfs should be detected. The second use will be using LAS data in conjunction with these simulations to constrain underlying distributions. Our simulations show that for a sample size with a typical local density in the range 0.09 < m < 0.1 M_⊙ the exponent of a power-law IMF can be constrained with an error of approximately 0.06 while the birthrate parameter β can be constrained to an error of 0.016.

This method could prove very useful in determining both the IMF and birthrate. However, binarity has not been taken into account in these simulations. The level of unresolved binarity will provide another parameter to characterize the results of the LAS. Furthermore, the simulations do not take contamination of the sample into account. Photometric errors will scatter objects such as hotter stars and white dwarfs across colour–colour diagrams so that they have colours similar to ultracool dwarfs. This contamination will have to be quantified in order for accurate comparisons to be made with the simulations. The quasar locus crosses the ultracool dwarf locus on a (J−H) versus (H−K) plot (e.g. Leggett et al. 2005), and such objects may also cause contamination of the ultracool dwarf sample. However, Hewett et al. (2006) have produced a method to separate quasars from ultracool dwarfs using SDSS (Adelman-McCarthy 2006) photometry, and this should minimize quasar contamination.

7 CONCLUSIONS

Clearly the techniques outlined here provide useful tools for both predicting the results of UKIDSS surveys and using those results to constrain the IMF (to an error in α of 0.06) and birthrate (to an error in β of 0.015). We have demonstrated that using an existing small dataset (55 objects) we can utilize this technique to produce loose constraints of α and β that are consistent with other studies: using the LF of Cruz et al. (2003), we have found values of α= 0.95 ± 1.17 in the range 0.072 < m < 0.1 M_⊙ and β=−0.134 ± 0.173. These techniques are also complimentary to those used by Pinfield et al. (2006) to examine the IMF by empirical spectroscopic methods. While our techniques have the discussed limitations they will provide useful information to constrain the IMF and birthrates.

The authors would like to thank Sandy Leggett and Isabelle Baraffe for providing unpublished data, and Steve Warren, David Bacon, Thomas Kitching and Andy Taylor for helpful discussions. Thanks are also due to Sandy Leggett for refereeing this paper and for making many useful suggestions that have resulted in a great improvement over the original manuscript.

REFERENCES

Author notes