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 ..