# V1412 Aql: a simulation

###### Michel Bonnardeau

24 Aug 2009

### Abstract

*A Monte Carlo simulation which may be useful to search for eclipses
is proposed. *

### Introduction

The white dwarf G24-9 (or V1412 Aql) was observed to be unexpectedly
dim on 2 occasions, in 1985 and 1988. This is interpreted as eclipses
due to a dark, substellar companion (Zuckerman & Becklin (1988)).

Feb 2009, the AAVSO lauched a campaign to
observed this object to detect its eclipses (AAVSO Special Notice #148).

I propose here a Monte Carlo simulation which may be useful to speed
up the discovery of eclipses.

### Observations

The 2 observed eclipses are:

19851007.11 (Landolt (1985)), heliocentric correction 180.56s,
t85=2446345.612 HJD;

19880715.3 (Carilli & Conner (1988)), heliocentric correction 441.74s,
t88=2,447,357.805 HJD.

The AAVSO has 271 negative (i.e. no eclipse) observations on 23 Aug 2009.

### Simulation

The orbital period is P=(t88-t85)/n where n is an integer.

G29-4 is a white dwarf, so it has a small size (about that of Earth),
then the eclipse duration tau is given mostly by the diameter D of the
occulting body:

with M the mass of the system, G the gravitational constant (taking for
the inclination i=90° and for the eccentricity e=0).

The computer simulation is a Monte Carlo one where a large number of
random sets of n, M, D are used to derived ephemeris. The ephemeris that
are retained are those that give the 2 observed eclipses and that do not
give eclipses for the negative AAVSO observations.

The algorithm works the following way:

1,000,000 random sets of n, M, D are generated, with n between 1 and 100,
M between 0.1 and 2 solar masses, D between 0.08 and 5 jovian diameters;
for each set the period P0=(t88-t85)/n and the eclipse duration tau is
computed;
the ephemeris is HJD(E)=T+P*E with T an random number between t85-tau and t85+tau,
and P a random number between P0-tau/n and P0+tau/n;
the ephemeris than do not give the eclipses at t85 and t88 are rejected;
the ephemeris that give an eclipse for one of the negative AAVSO observations are
rejected.
~~About 1/3 of the random sets give acceptable solutions~~. The spectra
of n, M and D solutions are:

*The "probability" is actually the number of acceptable solutions from
the simulation.*

The probability for the eclipse duration:

The probability for future eclipses may be computed:

Close-up:

*The peak around 75.7 (1 Sep 5hTU) comes from solutions with values
of n that are multiple of 8. If indeed the period is given by n=8 or n=16,
etc., a time-series around this date will pick up an eclipse. If not,
this will eliminate all these solutions. *

The peak around 82.4 (7 Sep 21hTU) comes from values of n that are multiple
of 19. Again a time-serie will pick up an eclipse if n=19, 38, etc. or
will eliminate these solutions.

*The peak around 87.2 (12 Sep 17hTU) comes from values of n that are
multiple of 11, and the one around 93.8 (19 Sep 7hTU) from values multiple
of 14.*

*This large peak between 117 and 118.5 (12 Oct 12hTU and 14 Oct 0hTU)
comes from values of n that are multiple of 3.*

### References

Carilli C., Conner S. (1988) IAU Circ. 4648.

Landolt A.U. (1985) IAU Circ. 4125.

Zuckerman B., Becklin E. (1988) IAU Circ. 4652.