# V1412 Aql: a simulation

###### Michel Bonnardeau

First version 24 Aug 2009

Updated 2 Sep 2009 (more observations)

### 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 600 negative (i.e. no eclipse) observations on 1 Sep 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 30% 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.*

Most solutions with n=8 and multiples are eliminated owing to the time-series
of 1 Sep 2009 of AAVSO observers Stephano Padovan (observer code PSD)
and Jim Roe (ROE).

The probability for the eclipse duration:

The probability for future eclipses may be computed:

Close-up:

*
The peak around 82.4 (7 Sep 21hTU) comes from values of n that are multiple
of 19. 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.

The small peak around 84.2 (9 Sept 16hTU) comes from n values multiple of 30.

*The bump at 90.8 (16 Sep 7hTU) comes from n values multiple of 25 and
the peak around 93.8 (19 Sep 7hTU) from values multiple
of 14.*

*The peak around 98 (23 Sep 12hTU) comes from n values multiple of 17
and the one around 103.2 (28 Sep 16hTU) comes from values multiple of 23.*

*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.*

### Discussion

The peaks correspond to possible long periods and they can be checked
by time-series.
Outside the peaks, the probabilities are not nil, and correspond to
short periods. They can be checked by single measurements and by having
some luck.
Very short periods (a few hours) are now eliminated by the time-series.
### References

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

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

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