# V1412 Aql: a simulation

###### Michel Bonnardeau

First version 24 Aug 2009

Updated 2 Sep 2009 (more observations)

Updated and corrected 3 Sep 2009 (more observations; uncertainties on
the ToM of the 2 eclipses taken into account at the suggestion of Wolfgang
Renz; correction of the number of solutions)

Updated 5 Sep 2009 (more observations; simulation with the period down to 1 d;
inclination)

Corrected 10 Sep 2009 (uncertainties
on the ToM of the 2 eclipses)

Updated 15 Sep 2009 (more observations)

Updated 20 Sep 2009 (more observations, slightly different ranges for
the simulation)

### 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
faint on 2 occasions, in 1985 and 1988. This is interpreted as eclipses
due to a dark, substellar companion (Zuckerman & Becklin (1988)).

This white dwarf has a V magnitude of 15.77 and an absolute one of Mv=13.73
(Greenstein (1984)); it is then at a distance of 25.6 pc.

From its color B-V=0.33 and Reed (1998), the white dwarf effective temperature
may be estimated as T=7248°K (neglecting the interstellar extinction
for this close object).

From Stefan's law (assuming black bodies), the white dwarf radius may
be estimated as:

with Mvo=4.83 the Sun absolute V magnitude, Ro=700,000km the Sun radius,
To=5780°K the Sun effective temperature, and BC(T) the bolometric correction
given by Reed (1998):

R= 7098 km.

Feb 2009, the AAVSO lauched a campaign to
observe this object so as 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=2,446,345.612
HJD. The uncertainty is taken as et85=0.005 HJD;

19880715.3 (Carilli & Conner (1988)), heliocentric correction 441.74s,
t88=2,447,357.805 HJD. The uncertainty is taken as et88=0.05 HJD.

The AAVSO has 902 negative (i.e. no eclipse) observations on 15 Sep 2009.

Arne Henden obtained 14 time-series showing no eclipses, from 30 Oct to
8 Dec 2008, total duration 34.1 h. And also 7 short time-series, in 2002
and 2008, total duration 2.6h.

Robert Fridrich obtained a time-series on 1 sep 2009, duration 0.3 h.

Christian Knigge also obtained 329 negative measurements in 2009.

### Simulation

The orbital period is P=(t88-t85)/n=(1012.193 days)/n where n is an
integer (assuming it is constant, i.e. no mass transfer, no third body).

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 observations.

The algorithm works the following way:

10,000,000 random sets of n, M, D are generated, with n between 1 and
1000, M between 0.1 and 1.6 solar masses, D between 0.15 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 observations
or for one of the 14+7+1 time-series are rejected.
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:

and the period:

The probability for future eclipses may be computed:

However, the uncertainties et85 and et88 introduce an uncertainty on their
timing of (where t is the current time):

dt = et85+(et85+et88)*(t-t85)/(t88-t85)

dt = 0.48 day

Close-up:

*The small peaks correspond to n between 10 and 30. There is an uncertainty
of dt=0.48 on their position (see an illustration below for 7-15 Oct 2009).*

*Thick red line: the probability with P0=(t88-t85)/n;
*

blue line: P0=(t88+et88-t85+et85);

green line: P0=(t88-et88-t85-et85).

The large peak around 118 (12 Oct 2009) +/-dt=0.48 comes from values
of n that are multiple of 3.

The red peak is smaller than the other two because of time-series around
peaks from the "red" prediction.

Other (than n=3) large periods:

n=1 and n=2: 455 THJD (15 Sep 2010)

n=4: 202 THJD (5 Jan 2010)

n=5: 253 THJD (25 Feb 2010)

n=7: 166 THJD (30 Nov 2009)

n=13: 143 THJD (8 Nov 2009)

### 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 (or time-series)
and by having some luck.
Periods below 1 day are now eliminated by the time-series.
The minimum inclination of the orbit (so as to have an eclipse) may be estimated as:

*Red line: for D=1 and M=1.5; Blue line: for D=3 and M=1.*

Long periods require an inclination of almost exactly 90° which would
be a somewhat improbable coincidence.

### References

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

Greenstein J.L. (1984) ApJ __276__ 602.

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

Reed B.C. (1998) J. of the Royal Astronomical Society of Canada __92__ 36.

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