1. 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, T 0 1 VIMO term: g = 10 -0.4m 2  (1 + q) / q  a 0 = a 1 (1 - g.10 0.4m T )

2. April, 3-5, 2006 J.L. Halbwachs : VIM0 2 The VIMO strategy: search of g (1) g = 10 -0.4m 2  (1 + q) / q  a 0 = a 1 (1 - g.10 0.4m T ) Minimum g : comes from the minimum variation of a 0 for a VIM effect;  a 0 > c   X, a 1 = a  q / (1+q)  g min = c   X 10 -0.4 m min / [ a Max (1 – 10 -0.4  m ) ] with  m = m min -m Max >0 Maximum g : m 2 > m T  g Max = 10 -0.4 m min (1 + q min ) / q min  assuming a Max = 50 mas,  X = 40  as, c  = 1,  m = 1 mag, q min = 0.5, g.10 0.4m min ranges from 0.00133 up to 3. For security, a Max = 100 mas, c  =1/4 and q min =0.01 are assumed, leading to g Max /g min =6 10 5 when  m=1 mag.

3. 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 : g 0 /10 – 10 g 0 (100 trial values) Having  g, we adapt the method of Pourbaix & Jancart (2003)

4. April, 3-5, 2006 J.L. Halbwachs : VIM0 4 The VIMO strategy: search of P Assuming 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 T 0 ) and e=0.9 (8 trial T 0 )  Time-consuming !!

5. April, 3-5, 2006 J.L. Halbwachs : VIM0 5 The VIMO strategy: the solution For a minimum of  2 (P, e=0), preliminary values of e and T 0 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)), T 0 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 !)

6. April, 3-5, 2006 J.L. Halbwachs : VIM0 6 VIMO simulation: hypotheses VIMO 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 = M 2 / M 1, where “1” refers to the variable) m 2 -m 1 = 6.6 log q +  m ; f(  m ) = N (0,1) photometric variations  a 0 varying   a 0  2.5  X

7. April, 3-5, 2006 J.L. Halbwachs : VIM0 7 VIMO simulation: results Solutions for 93 systems/100 = -0.08 = 0.16  E/I = 1.03  E/I = 1.11

8. 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 T 0 exchanged with (many) linear terms. With large VIMO orbits : = 0.12,  = 1.4 = 2.2,  = 1.7 both  N (0,1)

9. 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 VIM acc parameters).

10. 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

11. 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

12. April, 3-5, 2006 J.L. Halbwachs : VIM0 12 HIP 88848 assuming SB elements With NDAC data:  2 = 26.4, F2=-0.18, GOF F =1.02 P = (2147  3855) d g = (3.7  2.7) 10 -4 a 1 = 91 mas (instead of 48..) New VIMO solution starting from the SB elements: NOT a VIMO !!!

13. 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.

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

15. April, 3-5, 2006 J.L. Halbwachs : VIM0 15 Searching VIMO in Hipparcos..