April, 3-5, 2006 J.L. Halbwachs : VIM0 1

The VIMO software : What are VIMOs ?

VIMO: “Variability-Induced Mover with Orbit”= astrometric binary with a component with variable

brightness.

Parameters:

•5 or 6 for the single-star astrometric solution: , , , , , [].

•4 Thiele-Innes parameters: A, B, F, G.

•3 non-linear “normal binary” terms: P, e, T0

•1 VIMO term: g = 10 -0.4m2 (1 + q) / q a0 = a1(1 - g.10 0.4mT)

April, 3-5, 2006 J.L. Halbwachs : VIM0 2

The VIMO strategy: search of g (1)g = 10 -0.4m2 (1 + q) / q a0 = a1(1 - g.10 0.4mT)

• Minimum g : comes from the minimum variation of a0 for a VIM effect; a0 > c X , a1 = a q / (1+q)

gmin = c X 10 -0.4 mmin / [ aMax (1 – 10 -0.4m) ] with m = mmin-mMax >0

• Maximum g : m2 > mT gMax = 10 -0.4 mmin (1 + qmin) / qmin

assuming aMax = 50 mas, X = 40 as, c = 1, m = 1 mag, qmin= 0.5, g.10 0.4mmin ranges from 0.00133 up to 3.

For security, aMax = 100 mas, c=1/4 and qmin=0.01 are assumed, leading to gMax/gmin=6 105 when m=1 mag.

April, 3-5, 2006 J.L. Halbwachs : VIM0 3

The VIMO strategy: search of g (2)Fortunately, the interval of g may be restricted

assuming fixed P (ie 100 d), and e (ie 0) and trial values of g, all the other parameters are derived (linear system).

When the star is a VIMO, 2(g) is usually varying for g around its actual value

new interval : g0/10 – 10 g0 (100 trial values) Having g, we adapt the method

of Pourbaix & Jancart (2003)

April, 3-5, 2006 J.L. Halbwachs : VIM0 4

The VIMO strategy: search of PAssuming e =0 and using trial P and g, we have a system of linear equations. P is then generally corresponding to the minimum of 2(P,g) (Pourbaix & Jancart, DMS-PJ-01, 2003).

Otherwise, up to 5 minima of 2(P,g;e=0) are tested.

If an acceptable P is still not found, other eccentricities are tried: e=0.7 (6 trial T0) and e=0.9 (8 trial T0)

Time-consuming !!

April, 3-5, 2006 J.L. Halbwachs : VIM0 5

The VIMO strategy: the solutionFor a minimum of 2(P, e=0),

preliminary values of e and T0 are still obtained trying e=0.1 and e=0.5 in place of e=0;

having starting values for P ( log P ), g ( log g), e ( -log(-log e)), T0 and for the “linear” parameters, a solution is calculated using the Levenberg-Marquardt method of 2 minimisation.

Other minima of 2 are tried until a solution with GOF < 3 is obtained ( risk to keep a local minimum when the orbit is small !)

April, 3-5, 2006 J.L. Halbwachs : VIM0 6

VIMO simulation: hypothesesVIMO with large orbits :

• a = 50 mas, X = 40 as,

• f(log P ;10days < P < 10 years) = Cst,

• f(e < 0.9) = Cst,

• f(q ; 0.1 < q < 1.2) = Cst ( q = M2/M1, where “1” refers to the variable)

• m2-m1 = 6.6 log q + m ; f(m) = N (0,1)

• photometric variations a0 varying a0 2.5 X

April, 3-5, 2006 J.L. Halbwachs : VIM0 7

VIMO simulation: resultsSolutions for 93 systems/100

<(Psolution– Pactual)/IP> = -0.08 <(esolution– eactual)/Ie> = 0.16

E/I = 1.03 E/I = 1.11

April, 3-5, 2006 J.L. Halbwachs : VIM0 8

The Goodness-Of-Fit problem• F2 (Hipparcos, vol 1, 112) : inadequate since the model is not

linear.

• Degrees of freedom derived by Fourier transform (Pourbaix DMS-DP-02, 2005) : e and T0 exchanged with (many) linear terms.

With large VIMO orbits :

<F2> = 0.12, = 1.4

<GOFF> = 2.2, = 1.7

both N (0,1)

April, 3-5, 2006 J.L. Halbwachs : VIM0 9

Still to be done ..- Adaptation of the software to small orbits (trying other minima of 2).

- To compute faster ! current rate : 10 stars/hour (PC). The number of trial values could be reduced.

- False candidates must be discarded : acceleration+VIM solutions must be searched (3 VIMacc parameters).

April, 3-5, 2006 J.L. Halbwachs : VIM0 10

Searching VIMO in Hipparcos ..Hipparcos published archive:

• Stochastic solutions.

• Epoch Photometry Annex: H-mag for each transit

• Intermediate Astrometric Data: Average abscissae for 1 orbit of the satellite (10 h) !!!

no short period variable, no variable with fast variation (WW UMa, BY Dra, flare stars ..)

no nearby stars

no astrometric orbit with P 3 years,

.. but an opportunity to see the difficulties with real data

April, 3-5, 2006 J.L. Halbwachs : VIM0 11

Is HIP 88848 a VIMO (1) ?

single star model:

2 =280, F2=17.8 (NDAC)

1st VIMO solution :

2 = 26.5, F2=-0.17

P = (1595 1574) d

g = (4.3 2.1) 10-4

A SB+AB (Feckel,..,Jancart & Pourbaix 2005) : P = 2092 days

April, 3-5, 2006 J.L. Halbwachs : VIM0 12

HIP 88848 assuming SB elementsWith NDAC data:

2 = 26.4, F2=-0.18, GOFF=1.02

P = (21473855) d

g = (3.72.7) 10-4

a1 = 91 mas (instead of

48..)

New VIMO solution starting from the SB elements:

NOT a VIMO !!!

April, 3-5, 2006 J.L. Halbwachs : VIM0 13

Searching g in stochastic stars

“Stochastic stars” = single stars generated with X much larger than assumed in the VIMO calculation.

April, 3-5, 2006 J.L. Halbwachs : VIM0 14

ConclusionA prototype software

• providing reliable solutions when variability induces important astrometric effects,

• but inept results otherwise ...

The software must still be adapted to small orbits

VIMO solutions

• must be searched only when no “normal” solution can be found

• must be kept only when the estimated errors are reasonable

April, 3-5, 2006 J.L. Halbwachs : VIM0 15

Searching VIMO in Hipparcos ..