The impact of a merger episode in the galactic disc white dwarf population

Santiago Torres,1 Enrique Garcıa-Berro,2,3P Andreas Burkert4 and Jordi Isern3,5

1Departament de Telecomunicacio i Arquitectura de Computadors, EUP de Mataro, Universitat Politecnica de Catalunya, Av. Puig i Cadafalch 101,

08303 Mataro, Spain2Departament de Fısica Aplicada, Universitat Politecnica de Catalunya, Jordi Girona Salgado s/n, Modul B-4, Campus Nord, 08034 Barcelona, Spain3Institut d’Estudis Espacials de Catalunya (CSIC/UPC), Edifici Nexus, Gran Capita 2-4, 08034 Barcelona, Spain4Max-Planck-Institut fur Astronomie, Koenigstuhl 17, 69117 Heidelberg, Germany5Institut de Ciencies de l’Espai, C.S.I.C., Edifici Nexus, Gran Capita 2-4, 08034 Barcelona, Spain

Accepted 2001 August 3. Received 2001 July 25; in original form 2000 September 25

A B S T R A C T

In this paper we analyse the consequences in the white dwarf population of a hypothetical

merger episode in our Galactic disc. We have studied several different merging scenarios with

our Monte Carlo simulator. For each one of these scenarios we have derived the main

characteristics of the resulting white dwarf population and we have compared them with the

available observational data, namely the white dwarf luminosity function and the kinematic

properties of the white dwarf population. Our results indicate that very recent (less than

,6 Gyr ago) and massive (,16 per cent of the mass of our Galaxy) merger episodes are quite

unlikely in view of the available kinematical properties of the disc white dwarf population.

Smaller merger episodes (of the order of ,4 per cent of the mass of our Galaxy) are, however,

compatible with our current knowledge of those kinematical properties. Finally, we prove that

the white dwarf luminosity function is quite insensitive to such a merger episode.

Key words: stars: luminosity function, mass function – white dwarfs – Galaxy: stellar

content.

1 I N T R O D U C T I O N

Current cosmological models predict that large galaxies like the

Milky Way formed through the hierarchical merging of smaller

subunits which, owing to their large densities, remained

gravitationally bound until they had spiralled into the central

regions of their parent galaxies where they were tidally disrupted.

As a result, a substantial amount of substructure should still exist

within virialized dark matter haloes (Klypin et al. 1999; Moore

et al. 1999). The satellites of the Milky Way and the recently

detected Sagittarius galaxy (Ibata, Gilmore & Irwin 1994)

represent visible examples of the underlying substructure of the

Galaxy, which is still growing in mass by accretion and merging.

An important constraint on the merging frequency of satellites

was presented by Ostriker (1990) and Toth & Ostriker (1992) who

argued that in a high-density cold dark matter universe about 80 per

cent of all dark matter haloes should have experienced a merger

during the last 5 Gyr that increased their mass by 10 per cent or

more. If the merging satellites were able to sink into the disc region

before being disrupted, one would have to expect a substantial

amount of disc heating, leading to an increase in the thickness and

vertical velocity dispersion of discs. Toth & Ostriker (1992)

concluded that the observed thinness and coldness of the Galactic

disc indicates that no more than 4 per cent of its present mass

within the solar circle could have been accreted during the last 5

billion years. More recently, Velazquez & White (1999) found that

disc heating depends on the orbital parameters of the satellite. In

general, spiral galaxies could accrete quite massive satellites,

particularly if their orbits are retrograde, without a substantial

increase in the disc scaleheight.

The past accretion and merging history of the Galactic disc can

be investigated through its old stellar population. As the stellar disc

evolves practically dissipationless, the kinematical properties of

the disc stars as function of their age contain information about the

past merging events of the Milky Way. White dwarfs are ideal

candidates in order to probe the disc evolution in the solar

neighbourhood, because they have very long evolutionary time-

scales and, at the same time, their evolution is relatively well

understood – see Salaris et al. (2000) and references therein – at

least for moderately low luminosities, say logðL=L(Þ * 24:5,

because the observed blue turn for hydrogen rich white dwarfs

(Hodgkin et al. 2000; Hansen 1999) occurs at smaller luminosities.

The white dwarf luminosity function has for example been used

frequently to determine the age of the disc (Winget et al. 1987;

Garcıa-Berro et al. 1988; Hernanz et al. 1994) and the past history

of its star formation rate (Noh & Scalo 1990; Dıaz–Pinto et al.

1994; Isern et al. 1995a,b; Garcıa-Berro et al. 1999). In addition,PE-mail: garcia@fa.upc.es

Mon. Not. R. Astron. Soc. 328, 492–500 (2001)

q 2001 RAS

the observed proper motions of white dwarfs provide information

about the kinematical evolution of the solar neighbourhood.

In this paper we analyse the white dwarf luminosity function and

the kinematical properties of the local white dwarf population with

special focus on past merger episodes in the galactic disc. We adopt

a Monte Carlo approach, which is described in Section 2, and we

compare our results with a handful of selected white dwarfs, which

is described in Section 3. Using this model, Section 4 demonstrates

that the white dwarf population does not contain a signature of a

major merger episode in the galactic disc during the past 6 Gyr.

Discussions and conclusions follow in Section 5.

2 T H E M O N T E C A R L O S I M U L AT I O N S

A detailed description of our Monte Carlo simulator can be found

in Garcıa-Berro et al. (1999) and, therefore, we will only explain

here the changes introduced for the simulations described in this

paper. In summary, these changes are the following. For the sake

of simplicity we have adopted a constant disc scaleheight of

Hp ¼ 150 pc, which is an average of the exponentially decreasing

scaleheight law used in Garcıa-Berro et al. (1999), and which

reproduces reasonably well the observed kinematical properties of

the white dwarf population. Additionally, and also for the sake of

simplicity, we have adopted a constant star formation rate per unit

volume. It should be mentioned however that our results are not

very sensitive to the exact time dependence of the star formation

rate. Finally, for the three components of the velocity dispersions

(sU,sV,sW) we have taken constant values of 40 km s21,

30 km s21 and 20 km s21, which are also in good agreement with

the available observational data (see Table 3 later). The cooling

sequences adopted for the calculations reported here are the fully

evolutionary sequences of Salaris et al. (2000), which incorporate

the latest advances in the physics of white dwarf atmospheres

(Saumon & Jacobson 1999), and which use the best prescriptions

available for the equation of state of the degenerate core.

Given a disc age (which we adopt to be 9.8 Gyr in order to fit the

observed cut-off of the white dwarf luminosity function), our

Monte Carlo simulator produces as a result a synthetic population

of potentially observable white dwarfs (hereinafter the ‘original’

sample). In order to build the white dwarf luminosity function

using the 1/Vmax method (Schmidt 1968) a smaller sample of white

dwarfs (hereinafter the ‘restricted’ sample) must be culled from the

original sample and in order to do this a set of restrictions in visual

magnitude and proper motion must be adopted. We have chosen

the following restrictions: m $ 0:16 arcsec yr21 and

mV # 18:5 mag, as in Oswalt et al. (1996) and in Garcıa-Berro

et al. (1999). Besides, the 1/Vmax method requires that all the

objects belonging to the restricted sample must have known

parallaxes. This, in turn, means that all the white dwarfs belonging

to this sample are within a sphere of radius of roughly 200 pc

centred on the location of the Sun. In addition, all stars with

tangential velocities greater than 250 km s21 have also been

discarded, because these would be probably classified as halo white

dwarfs (Torres, Garcıa-Berro & Isern 1998). Finally, all white

dwarfs brighter than mV # 13 mag were automatically included in

the restricted sample, regardless of their proper motion, because

the bright portion of the white dwarf luminosity function is

generally assumed to be complete (Liebert, Dahn & Monet 1988).

In summary, all the white dwarfs of the restricted sample have

known absolute magnitudes, distances and proper motions and

hence tangential velocities.

Each one of the Monte Carlo simulations discussed in Section 4

consists of an ensemble of 50 independent realizations of the

synthetic white dwarf population, for which the average of any

observational quantity along with its corresponding standard

deviation were computed. Here the standard deviation means the

ensemble mean of the sample dispersions for a typical sample size

of ,250 objects (see Section 3 below). In this way we can assess

the expected fluctuations for a typical sample. We have carried out

a first set of five different simulations, which can be described as

follows: in the first one (t0v0), which we consider to be our

reference simulation, no merger episode was assumed. For the

remaining four simulations we assumed that all the stars that were

born during the first 3 (6) Gyr of the life of our Galaxy have

suffered a kinematical ‘kick’ which increases the modulus of their

heliocentric velocity by a factor of 3 (6). We will refer to these

simulations as t3v3, t3v6, t6v3 and t6v6, respectively.

Of course, with this procedure we are assuming that the time-

scale for the sedimentation of the gas is short compared with the

evolutionary time-scale of the Galaxy. In addition, we assume that

the merger leads to an isotropic heating of the relevant disc region

by a certain constant factor. Note that this is just a first simple

approach. It is beyond the scope of this paper to study the change of

the velocity distribution function of disc stars that are affected by a

satellite infall. Additional simulations are, however, planned to

study this interesting question in greater detail. In fact this kick

should be interpreted as fast heating of the disc by whatever

process. This would, of course, apply to merging processes that

happened on a dynamical time-scale or to mergers that happened

fast, leading to wave patterns in the disc with adiabatic changes.

Finally, it is worth mentioning at this point that we have ignored

disc heating by other processes than mergers, such as scattering

with large- (spiral modes, bar modes) and small-scale perturbers.

However it should be mentioned as well that these last processes

cannot easily increase the velocity dispersions to an amount

comparable to that of a merger process, because they are more

efficient if the dispersion is much lower than 20 km s21 – see, for

instance, Wielen (1977).

No increase in the disc scaleheight was assumed for this set of

simulations. These, of course, are extreme cases and even if they

are unrealistic they maximize the effects of a merging episode in

the white dwarf luminosity function, as will be shown in Section 4.

However, the natural effect of such a kick is to increase the

scaleheight, whence the distribution of white dwarfs is back in

equilibrium. Thus this set of simulations cannot be taken as a firm

upper limit because it overestimates the effect of a merger episode.

As we shall see below, the effects of an increase of the scaleheight

are taken properly into account in several other sets of simulations

which are described later on.

Since the kinematical kick amounts, at least, to a factor of 3 in

the velocity dispersion, for the average velocity of disc white

dwarfs this translates into an increment in the velocity of roughly

80 km s21 and, therefore, for a typical infall velocity of the satellite

of 200 km s21 this means that the mass of the satellite should be

roughly 16 per cent of the mass of the Milky Way, assuming that

the efficiency of the merging process is 100 per cent. In order to

simulate a less efficient or less massive merging scenario we have

performed a second set of simulations in which we have assumed

that only one out of four white dwarfs in our simulated samples

have suffered a kinematical kick of the same strength and at the

same times as the previous set of simulations. Thus, these

simulations correspond to a mass of the satellite of ,4 per cent of

the mass of our Galaxy or, equivalently, to an efficiency of ,25 per

cent. Both sets of simulations bracket the fiducial 10 per cent

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increment in the mass of Galaxy (Ostriker 1990). We will refer to

these simulations as s3v3, s3v6, s6v3 and s6v6, respectively.

Again, we have not considered for this set of simulations any

increase in the disc scaleheight. As the scaleheight scales as the

square of the velocity dispersion perpendicular to the Galactic

plane, for both the first and the second set of simulations a large

increase in the resulting scaleheight is expected. However for the

second set of simulations, in which only one out of four white

dwarfs is affected by a kinematical kick, it is difficult to estimate ‘a

priori’ which would be the final scaleheight of the resulting white

dwarf population. Thus, this second set of simulations provides us

with a reasonable estimate of the inflation factor that must be used

for subsequent simulations and, consequently, we have used the

same procedure for the first set of simulations.

Additionally we have performed two more sets of simulations in

which we vary the adopted scaleheight, one for the case in which a

strong impact is adopted (r3v3 to r6v6) and the other one for the

case in which a less massive impact is assumed (h3v3 to h6v6).

Finally we have also computed a fifth set of simulations in which

we vary the initial velocity dispersions (l3v3 to l6v6). These sets

of simulations, for the sake of clarity, will be described in greater

detail later in Section 4.

3 T H E O B S E RVAT I O N A L S A M P L E

In order to compare the results of our Monte Carlo simulations with

an observational sample, we have used the catalogue of

spectroscopically identified white dwarfs of McCook & Sion

(1999). This catalog contains information of 2 249 white dwarfs, of

which only 250 have determinations of their parallax and proper

motions, to verify the restrictions in proper motion and magnitude

used to obtain the luminosity function explained in Section 2 and,

thus, allow a reasonable comparison with the kinematical

properties of the restricted sample. We emphasize here that we

are only considering white dwarfs with known proper motion and

parallax. It is worth noticing at this point that the radial velocities

of cool white dwarfs are difficult to measure and, thus, in practice

we are left with only two thirds of the kinematical information.

However our Monte Carlo simulations can easily reproduce this

observational biass and, consequently, it is still possible to obtain

useful information. Regarding the completeness of the observa-

tional sample it should be mentioned that the bright portion of the

white dwarf luminosity function [say white dwarfs with

logðL=L(Þ * 23:0� is complete (Fleming, Liebert & Green

1986) to a distance of about 200 pc, whereas the faint branch it

is generally assumed that it is fairly complete, although there is still

some debate about the degree of completeness, the current

estimates ranging from about 80 per cent (Bergeron, Leggett &

Ruiz 2001) to 60 per cent out to 20 pc (Holberg, Oswalt & Sion

2001).

4 R E S U LT S A N D D I S C U S S I O N

Our results are shown in Figs 1–6 and Tables 1–4. In Fig. 1 we

show the white dwarf luminosity function obtained for each one of

the first five cases described in the previous section (each panel is

clearly labelled for the sake of clarity). The observational

luminosity function of Oswalt et al. (1996) is shown for

comparison in the upper left panel of Fig. 1, with its corresponding

error bars, and it is reproduced in each one of the other panels

without the error bars in order to facilitate the comparison.

Following the usual procedure, the theoretical white dwarf

Figure 1. Panel showing the first set of simulations of the white dwarf luminosity function compared with the observational luminosity function of Oswalt et al.

(1996). For the sake of clarity, the observational luminosity function is also shown in each subpanel as a dotted line.

494 S. Torres et al.

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luminosity functions have been normalized to the point with

logðL=L(Þ ¼ 23:5 of the observational luminosity function. It is

important to mention here that in all five cases the same age of the

disc has been adopted. As can be seen in Fig. 1, the agreement

between the simulated samples and the observational data is good,

no matter at which time the kinematical kick happened and what its

actual value was. In particular, the position of the simulated cut-off

does not depend on the kinematical kick. Consequently, the age of

the galactic disc obtained by fitting the theoretical models to the

observed cut-off of the white dwarf luminosity function is

completely insensitive to the existence of a merger episode and,

therefore, from this point of view, the white dwarf luminosity

function provides a robust estimate of the age of the galactic disc.

Moreover, this figure clearly proves that the white dwarf

luminosity function is almost insensitive to a merger episode

because there is not any special signature in the shape of the white

dwarf luminosity function. There are several reasons for this

behaviour.

First, the kinematical kick increases considerably the tangential

velocities of a sizeable fraction of white dwarfs in the simulated

samples. As a result some white dwarfs acquire velocities in excess

of 250 km s21 and thus, following the standard observational

procedure, should automatically be classified as halo members.

Consequently, these white dwarfs are not taken into account in the

calculation of the resulting disc white dwarf luminosity function.

The situation is clearly shown in Table 1, where we display for the

same number of white dwarf progenitors the total number of

objects belonging to the restricted sample – second column – if we

withdraw the condition that the white dwarfs of the restricted

sample should have tangential velocities smaller than 250 km s21

and the number of objects with velocities smaller than this limit –

third column – for a typical realization of each one of the cases

studied in the first set of simulations. This fact is important by itself

because we do not see too many white dwarfs with such high

tangential velocities in the solar neighbourhood (Torres et al.

1998).

Secondly, we are drawing white dwarfs with relatively large

proper motions and since the results of our Monte Carlo simulator

are normalized to the observational density of white dwarfs, the

intrinsic differences introduced by the kinematical kicks are

effectively erased. However, it should be mentioned here that for

the same number of simulated stars in the original sample, the total

number of white dwarfs that belong to the restricted sample (the

sample used to build the white dwarf luminosity function) differs

considerably from one simulation to another, being larger the

stronger the kinematical kick is, as clearly seen in Table 1. Since

the observational white dwarf luminosity function is built using

approximately 250 objects, and our results are conveniently

normalized to the observed value, we have carried out our

simulations in such a way that the total size of the restricted sample

Figure 2. Normalized tangential velocity distributions of the restricted sample for the first set of simulations.

Table 1. Number of objects belonging tothe restricted sample when the same numberof white dwarf progenitors is adopted –NWD – and the number of these objects thathave tangential velocities smaller than250 km s21 – N0WD.

Model NWD N0WD

t0v0 116 112t3v3 173 162t3v6 211 155t6v3 251 227t6v6 333 197

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is always the same. This procedure allows meaningful compari-

sons. Note, however, that several realizations (see Section 2) were

performed for each Monte Carlo simulation.

Finally, the simulations in which a merger episode occurs

produce, as expected, a larger fraction of white dwarfs with large

proper motions. As the white dwarf luminosity function is built

using the 1/Vmax method, and as the contribution of any object to

the corresponding bin of the white dwarf luminosity function is

inversely proportional to the volume where it could be found, it

turns out that the contribution of these high proper motion white

dwarfs is very small owing to the fact that they should be

potentially found in much larger volumes. All these three reasons

conspire in such a way that the observational effects of a merger

episode in the white dwarf luminosity function should be very

modest. We stress at this point that this is, perhaps, one of our

major findings.

Now we turn our attention to the derived kinematical properties

of the white dwarf population. The tangential velocity distributions

of the restricted sample for each one of the different simulations of

the first set (t0v0 to t6v6) are shown in Fig. 2. As in the previous

figure, each panel is clearly labelled and the observational data is

shown in the upper left panel of Fig. 2. All the distributions have

been normalized to unit area and, therefore, are frequency

distributions. The expected standard deviations for each of the bins

of the theoretical distributions are also shown in order to compare

our results properly with the observational sample. The error bars

of the observational distribution have been computed using the

following procedure. First we assume that the distribution of errors

in proper motion and parallax are Gaussian, and we assume a

dispersion of 0.1 arcsec yr21 for the proper motion and of

0.01 arcsec for the parallax, which correspond to the typical values

quoted in the catalogue of McCook & Sion (1999). We then assign

a new value of proper motion and parallax for each one of the white

dwarfs in the observational sample according to these distributions

of errors. After this we build again the observational tangential

velocity distribution. This is repeated several times and the

standard deviation is computed for each bin. This procedure allows

us to estimate the influence in the tangential velocity distribution of

the observational errors. On the other hand we would like also to

assess the influence of the completeness. To this regard we have

used the following simple procedure. We randomly eliminate from

the observational sample one out of three white dwarfs (see

Section 3), and we recompute the tangential velocity distribution.

This is again repeated several times and, in this way, the standard

deviation can also be computed. The total error is the square root of

both errors added in quadrature. The result is shown in the upper

left panel of Fig. 2.

As it can be seen in this figure the kinematical properties of the

white population are extremely sensitive to a putative merger

episode. For instance, Fig. 2 clearly shows that the stronger and

more recent the merger episode is, the larger is the fraction of white

dwarfs with tangential velocities in excess of say 100 km s21 and,

correspondingly, the less pronounced is the peak at moderate

(,40 km s21) velocities, as we should expect. In fact, none of the

simulations but the t0v0, in which no merger episode was

assumed to occur, agree with the observational distribution and,

hence, must be discarded. One should note as well the absence of

low tangential velocity objects, as one should expect given that the

objects belonging to this sample have been selected on the basis of

moderately high proper motion.

Now we will discuss the second set of simulations, in which only

25 per cent of the white dwarfs of the simulated samples have

suffered a kinematical kick. Such a situation could occur, for

instance, if only parts of the disc were affected by the infalling

satellite. From the previous discussion it is easy to understand that

the luminosity functions of this second set of simulations are in

good agreement with the observational data and are insensitive to

the gross properties of the assumed merger episode. Therefore, we

will only focus on the kinematical properties of the simulated

samples. We expect to see smaller differences with respect to the

observational case, and this is indeed the case.

In Fig. 3 we show as solid lines the frequency distributions of the

tangential velocity for the simulations s3v3, s3v6, s6v3 and

s6v6. Also shown in each panel is the observational distribution

(dotted histogram). As it can be seen there, the maximum of the

distribution is not significantly affected by the strength of the

kinematical kick or the time at which it occurred. However,

the formation of extended tails in the tangential velocity

distribution is quite apparent, those tails being more extended the

more recent and stronger the kinematical kicks are. Therefore, this

Figure 3. Normalized tangential velocity distributions of the restricted sample for the second set of simulations.

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set of simulations also shows that a relatively inefficient and recent

merging process can also be apparently discarded on the basis of

the comparison with the observational distribution.

Additionally, and in order to make more quantitative estimates,

in the upper section of Table 2 we show a Z 2 statistic test of the

compatibility of the observational tangential velocity distribution

and both sets of Monte Carlo simulated samples. This statistic test

(Lucy 2000) improves the x 2 statistics and it is especially designed

for meagre data sets. We have used as statistical weights the error

bars of both the observational distribution and those of the Monte

Carlo simulated samples added in quadrature. Additionally, the Z 2

test already takes into account the frequency of each of the bins of

the observational distribution. As can be seen in the top section of

table 2, the only simulation that is acceptable at the 2s confidence

level is the t0v0, in which no merger episode was assumed to

occur. At the 1s confidence level, however, the s3v3, the s3v6and the s6v3 simulations are compatible with the observational

data. Thus, clearly strong and efficient merger episodes can be

discarded whereas only old and inefficient merging processes are

compatible with the available observational data but to a lower

probability.

A more sophisticated comparison can also be done. We proceed

as follows. First, for each observed datum and each model, we

compute (from the distribution of the corresponding simulated

datum among the ensemble of samples) the probability P, that a

value less than the actual observed one occurs. After computing P,

for each bin we compute the Kolmogorov–Smirnov probability

that the distribution of P, follows indeed a uniform distribution,

and thus determine whether or not the observed numbers are

consistent with the model. We have used the same statistical

weights that we used previously for the Z 2 test. The results are

shown in Table 3. As can be seen in the top section of this table, the

only models consistent with the observational data are again the

t0v0 and the s3v3, but not the s3v6 and the s6v3. Thus, again

only modest and old merger episodes seem to be compatible with

the observational distribution of tangential velocities.

There is still another way of obtaining valuable additional

information. In order to do that in Table 4 we show the average

values of the heliocentric components of the tangential velocity

and of its corresponding velocity dispersions for each one of the

ensembles of Monte Carlo simulations studied so far and we

compare them with the observational data. The typical standard

deviations of the average values are of the order of 4 km s21 for the

velocities and of about 8 km s21 for the velocity dispersions. We

see that for the simulations in which a massive satellite is involved

the value of kUl, and kVl are moderately sensitive to the

characteristics of the merger process, the range of variation being

1.5, and 1.6, respectively. However, the range of variation of their

corresponding velocity dispersions, ksUl and ksVl, is somehow

larger, 2.3 and 2.6, as one should expect. In contrast, the value of

kWl and of the velocity dispersion perpendicular to the Galactic

plane, ksWl, vary more significantly by factors of 3.3 and 3.8,

respectively. In fact, the value of ksWl is the most sensitive to a

merger episode, which clearly shows the heating effect of such a

putative episode. As a matter of fact, by visual inspection of Table 4

one can already discard all the simulations in which a massive

merger episode was assumed to occur, because all of them double,

at least, the value of the W component of the tangential velocity

dispersion.

The situation is less clear for the simulations in which a small

satellite is involved, because in this case the average values of the

Table 3. Kolmogorov–Smirnov test of the compatibility of the observa-tional tangential velocity distribution and the Monte Carlo simulatedsamples. See text for details.

Model P Model P Model P

t0v0 1.0t3v3 2.7� 10213 s3v3 1.0 r3v3 6.6� 10215

t3v6 1.1� 10217 s3v6 3.9� 1024 r3v6 1.5� 10219

t6v3 4.1� 10221 s6v3 2.0� 1026 r6v3 2.8� 1029

t6v6 1.5� 1027 s6v6 1.2� 10210 r6v6 1.0� 10219

h3v3 1.0 l3v3 1.0h3v6 2.1� 1025 l3v6 7.8� 1025

h6v3 1.6� 1028 l6v3 1.0h6v6 3.5� 10211 l6v6 6.7� 10211

Table 4. Average values of the heliocentric velocitiesand velocity dispersions (both in km s21) for the severalsets of simulations described in the text and for theobservational one. Typical errors are of the order of4 km s21 for the velocities and about 8 km s21 for thevelocity dispersions.

Model kUl kVl kWl ksUl ksVl ksWl

Obs. 9 222 26 40 31 22t0v0 11 214 26 37 29 20t3v3 13 218 210 59 48 39t3v6 13 217 212 67 57 52t6v3 16 222 215 73 60 51t6v6 17 221 220 88 76 75

s3v3 10 215 26 45 36 27s3v6 10 215 26 48 40 34s6v3 12 218 29 54 44 36s6v6 13 217 29 60 51 48

r3v3 13 219 210 59 48 39r3v6 16 219 211 64 54 54r6v3 17 224 215 73 61 52r6v6 15 223 217 89 76 79

h3v3 11 216 25 45 36 27h3v6 12 216 25 48 39 34h6v3 14 218 28 53 43 35h6v6 11 218 28 59 49 46

l3v3 17 210 27 22 18 13l3v6 25 214 217 40 34 28l6v3 22 213 210 27 22 17l6v6 34 220 224 51 42 37

Table 2. Z 2 statistic test of the compatibility of theobservational tangential velocity distribution and theMonte Carlo simulated samples.

Model P Model P Model P

t0v0 0.989t3v3 0.630 s3v3 0.939 r3v3 0.530t3v6 0.538 s3v6 0.893 r3v6 0.423t6v3 0.221 s6v3 0.746 r6v3 0.180t6v6 0.108 s6v6 0.665 r6v6 0.040

h3v3 0.908 l3v3 0.999h3v6 0.758 l3v6 0.878h6v3 0.694 l6v3 0.998h6v6 0.589 l6v6 0.691

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components of the tangential velocity are consistent with the

observational data. Note, however, that the actual values of the

dispersions of the three components of the tangential velocity are

still quite sensitive to the properties of the merger. Particularly, the

s3v6, the s6v3 and s6v6 simulations can be discarded on

the basis of their velocity dispersions as the W component of the

dispersion of tangential velocities is at least 1.5 times larger than

the observational one. Hence, recent merger episodes, even if they

correspond to a small mass of the infalling satellite, can also be

rejected on the basis of the available observational data of the

kinematics of the white dwarf population.

Since the scaleheight, Hp, is proportional to the square of the

velocity dispersion perpendicular to the galactic plane we have run

a third set of simulations in which we have adopted a self-

consistent scaleheight, derived from sW obtained in the first and

second set of simulations. These simulations are referred to as

r3v3, r3v6, r6v3, r6v6 and h3v3, h3v6, h6v3, h6v6,

respectively. The results obtained for this set of simulations are

shown in Figs 4 and 5 and in the third and fourth sections of Table

4. The compatibility of these simulations with the observational

sample is again assessed in Tables 2 and 3. Once more we see that

only modest and old merger episodes like that of the h3v3simulation can be favourably compared with the observational

distribution. In fact all these simulations are not compatible with

the observational data at the 2s level (Table 2) and do not pass the

Kolmogorov–Smirnov test (Table 3). Moreover, it should be noted

that the h3v3 simulation has average values of the components of

the tangential velocity and of their corresponding dispersions

relatively close to their observational values. Thus, for these cases

we conclude that only old merger episodes involving a satellite

with relatively small mass are compatible with the observational

data.

Finally, it would be also quite interesting to see the results of a

new set of calculations, starting with a low-velocity dispersion of

the order of 8 km s21 and assuming a merger that heated the white

dwarf population to 22 km s21 in the z component. Of course one

Figure 5. Normalized tangential velocity distributions of the restricted sample for the fourth set of simulations.

Figure 4. Normalized tangential velocity distributions of the restricted sample for the third set of simulations.

498 S. Torres et al.

q 2001 RAS, MNRAS 328, 492–500

should then also reduce the initial velocity dispersions for the U

and V components correspondingly and assume a similar heating

for them. Therefore, we have run a fifth set of simulations in which

we have varied the initial velocity dispersions, decreasing the value

of each one of the three components to 1/3 of the original value

adopted for all the previous simulations and assuming that the

infalling satellite was small. The adopted scaleheight is the original

one. These simulations are referred as l3v3, l3v6, l6v3 and

l6v6. The results obtained for this set of simulations are shown in

Fig. 6 and in the bottom sections of Tables 2, 3 and 4. An analysis

of Fig. 6 reveals that only small merger episodes like that of the

l3v3 and l6v3 simulations can reproduce the observational

distribution, their corresponding likelihood (see Table 2) being

very high. Both distributions also pass the Kolmogorov–Smirnov

test (Table 3). Thus, again we conclude that only small and old

merger episodes are consistent with the observational data

5 C O N C L U S I O N S

We have used a Monte Carlo simulation to test the effects of a

hypothetical merger episode in our Galactic disc. In order to mimic

the effects of such a merging process we have artificially increased

the modulus of the velocity of the white dwarf simulated samples.

Our simulations encompass a broad range of times at which the

merger episode occurred and large enough kinematical kicks.

Additionally we have explored the possibility of both small (or

inefficient episodes) and large masses of the accreted satellite (4

and 16 per cent of the mass of the Milky Way, respectively). Our

major findings can be summarized as follows.

We have found that the white dwarf luminosity function remains

virtually unaffected by such a hypothetical merging episode, even

in the case of the largest, most recent and massive impacts. We

have traced back the reasons for this, and we have found that a

combination of three factors produces this behaviour. (i) The

observational procedure of discarding white dwarfs with velocities

in excess of 250 km s21 when computing the disc white dwarf

luminosity function discards a sizeable fraction of disc white

dwarfs for the stronger merger episodes. (ii) Since we are forced to

normalize to the observational white dwarf number density and any

realistic Monte Carlo simulation must contain a similar number of

objects in the final sample, the computational procedure effectively

smears out the fact that those simulations that include a merger

produce a considerably larger number of white dwarfs with high

proper motions than those simulations that do not include such a

merger. (iii) White dwarfs with large proper motions, like those

produced in the simulations that incorporate a strong merger

episode, contribute little to the luminosity function, as they could

be potentially found in much larger observational volumes.

Additionally we have found that the position of the observed drop-

off of the white dwarf luminosity function remains unaltered by the

presence of the merging episode. Thus, the age of the Galactic disc

obtained by fitting the theoretical models to the observed cut-off is

insensitive to a merger episode and, therefore, this method provides

a robust estimate of the age of the Galactic disc.

In contrast, we have found that the kinematical properties of the

white dwarf population are very sensitive to the gross properties of

the merger episode. In particular we have found that the only

simulations that seem capable of reproducing the observational

distribution of tangential velocities are those that involve a modest

impact, with a small mass of the accreted satellite, and are

sufficiently old. Kinematical kicks in excess of a factor of 3 can be

clearly discarded for a massive satellite (of the order of 14 per cent

of the mass of the Galaxy), whereas for a smaller satellite (of the

order of 4 per cent of the mass of the Galaxy) larger kinematical

kicks could be accommodated provided that the merging episode

happened at the very first stages of the formation of our Galactic

disc. Thus, although the situation is not yet clear, and we cannot

totally discard merger episodes that are sufficiently old, we would

like to emphasize at this point that future astrometric missions, like

GAIA, which are expected to collect high-quality astrometric and

photometric data for the local population of white dwarfs (Figueras

et al. 1999) will, undoubtedly, allow us to distinguish clearly the

effects of past putative merger episodes.

AC K N OW L E D G M E N T S

This work has been supported by the DGES grants PB98–1183–

C03–02 and ESP98–1348, by the MCYT grant AYA2000-1785

and by the CIRIT. One of us, EGB, also acknowledges the support

received from Sun MicroSystems under the Academic Equipment

Figure 6. Normalized tangential velocity distributions of the restricted sample for the fifth set of simulations.

Impact of a merger episode 499

q 2001 RAS, MNRAS 328, 492–500

Grant AEG–7824–990325–SP. We also acknowledge the com-

ments of an anonymous referee which greatly improved the

manuscript.

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