3 Separation of the dynamical term from the O-C

In this section first we show how the presence of the dynamical term can influence the usual method of light-time solutions, and then, we give a numerical method to separate the two terms, which can improve the accuracy of the light-time solution, and, furthermore, may give additional information about the spatial orientation of the triple system.

3.1 The effect of the dynamical term on the light-time solution

A usual way of calculation of the light-time solution is based on the fact that there are some very simple relations (at least in the first and second order in e') between the orbital elements of the wide orbit and the first two or three pairs of coefficients of the Fourier-expansion of the light-time curve, where the fundamental frequency is the period ratio, e.g. $2\pi P/P'$ . Consequently, if the harmonic coefficients of the O-C were determined by some numerical methods (typically by weighted least-squares fit), then the orbital elements could be calculated in a very simple way.

For the sake of completeness we describe here the most important formulae after Kopal (1978, Chap. V). In the case of the pure light-time effect the mathematical form of the O-C curve is:

$\begin{displaymath}% {\rm O}{-}{\rm C}=\sum_{k=1}^\infty\left[a_k\sin(k\nu N)+b_k\cos(k\nu N)\right]-\sum_{k=1}^\infty b_k, \end{displaymath}$

(49)

where N is the cycle number, and

$\begin{displaymath}% \nu=2\pi\frac{P}{P'}, \end{displaymath}$

(50)

while

		$\displaystyle a_k=A\left[g_k\left(e'\right)\cos\omega'\cos{kl'_0}-h_k\left(e'\right)\sin\omega'\sin{kl'_0}\right],$	(51)
		$\displaystyle b_k=A\left[g_k\left(e'\right)\cos\omega'\sin{kl'_0}+h_k\left(e'\right)\sin\omega'\cos{kl'_0}\right],$	(52)

where

$\begin{displaymath}% A=\frac{a_{12}\sin{i'}}{c}, \end{displaymath}$

(53)

furthermore,

		$\displaystyle g_k\left(e'\right)=2\sqrt{1-e'^2}\frac{J_k\left(ke'\right)}{ke'},$	(54)
		$\displaystyle h_k\left(e'\right)=\frac{2}{k}\frac{{\rm d}J_k\left(ke'\right)}{{\rm d}ke'},$	(55)

and in the latter expressions J_k represents the Besselian function of the kth order. (We note, that in (53) c stands for the velocity of light.) Considering a quadratic approximation in the outer eccentricity the non-zero coefficients are as follows:

		$\displaystyle a_1=A\left[\left(1-\frac{3e'^2}{8}\right)\cos\left(l'_0+\omega'\right)-\frac{e'^2}{4}\cos\omega'\cos{l'_0}\right],$	(56)
		$\displaystyle b_1=A\left[\left(1-\frac{3e'^2}{8}\right)\sin\left(l'_0+\omega'\right)-\frac{e'^2}{4}\cos\omega'\sin{l'_0}\right],$	(57)
		$\displaystyle a_2=A\frac{e'}{2}\cos\left(2l'_0+\omega'\right),$	(58)
		$\displaystyle b_2=A\frac{e'}{2}\sin\left(2l'_0+\omega'\right),$	(59)
		$\displaystyle a_3=A\frac{3e'^2}{8}\cos\left(3l'_0+\omega'\right),$	(60)
		$\displaystyle b_3=A\frac{3e'^2}{8}\sin\left(3l'_0+\omega'\right).$	(61)

Using the expansions of Cayley (1861) Eq. (46) also can be easily expanded into trigonometric series of the mean anomaly l', as

$\displaystyle % {\rm O}{-}{\rm C}$	$\textstyle \approx$	$\displaystyle \frac{3}{8\pi}\frac{m_3}{M_{123}}\frac{P^2}{P'}\biggl\{{\cal{C}}\cos2\left(l'+\omega'\right)+{\cal{S}}\sin2\left(l'+\omega'\right)$
		$\displaystyle +e'\biggl[{\cal{M}}\sin{l'}-{\cal{C}}\cos\left(l'+2\omega'\right)-{\cal{S}}\sin\left(l'+2\omega'\right)$
		$\displaystyle +\frac{7}{3}{\cal{C}}\cos\left(3l'+2\omega'\right)+\frac{7}{3}{\cal{S}}\sin\left(3l'+2\omega'\right)\biggl]\biggl\}+{\cal{O}}\left(e'^2\right),$	(62)

where

		$\displaystyle {\cal{C}}=-\left(1-I^2\right)\sin2u'_{\rm m},$	(63)
		$\displaystyle {\cal{S}}=\left(1-I^2\right)\cos2u'_{\rm m},$	(64)
		$\displaystyle {\cal{M}}=6I^2-2.$	(65)

We omitted terms which are multiplied by $0.5\cot{i}$ , as in eclipsing systems the observable inclination of the binary is usually close to $90\hbox{$^\circ$ }$ , consequently these terms give only a minor contribution. (E.g. even for $i=80\hbox{$^\circ$ }$ , $0.5\cot{i}<0.09$ .) A comparison between (62) and (49) reveals that in this case the coefficients are as follows:

		$\displaystyle a^_1=A^e'\left[{\cal{M}}\cos{l'_0}-{\cal{S}}\cos\left(l'_0+2\omega'\right)+{\cal{C}}\sin\left(l'_0+2\omega'\right)\right],$	(66)
		$\displaystyle b^_1=A^e'\left[{\cal{M}}\sin{l'_0}-{\cal{S}}\sin\left(l'_0+2\omega'\right)-{\cal{C}}\cos\left(l'_0+2\omega'\right)\right],$	(67)
		$\displaystyle a_2^=A^\left[{\cal{S}}\cos2\left(l'_0+\omega'\right)-{\cal{C}}\sin2\left(l'_0+\omega'\right)\right],$	(68)
		$\displaystyle b_2^=A^\left[{\cal{S}}\sin2\left(l'_0+\omega'\right)+{\cal{C}}\cos2\left(l'_0+\omega'\right)\right],$	(69)
		$\displaystyle a_3^=A^\frac{7e'}{3}\left[{\cal{S}}\cos\left(3l'_0+2\omega'\right)-{\cal{C}}\sin\left(3l'_0+2\omega'\right)\right],$	(70)
		$\displaystyle b_3^=A^\frac{7e'}{3}\left[{\cal{S}}\sin\left(3l'_0+2\omega'\right)+{\cal{C}}\cos\left(3l'_0+2\omega'\right)\right],$	(71)

where the amplitude A^* is

$\begin{displaymath}% A^*=\frac{3}{8\pi}\frac{m_3}{M_{123}}\frac{P^2}{P'}\cdot \end{displaymath}$

(72)

(It will be seen in the next section, that in the case of systems interesting for us, for moderate outer eccentricities, the order of A^* is about Ae'. This is the reason why the quadratic terms were held in (56)-(61).) It is evident that a numerical modelling of the O-C curve in the form (49) yields to coefficients which are the sums of the corresponding (56)-(61) and (66)-(71) terms.

Now some qualitative remarks can be easily done about the effect of the dynamical terms on a usual light-time solution. First, if the third star revolves in a circular orbit, which is coplanar with the orbit of the binary, the dynamical terms diminish, e.g. the geometrical terms are unaffected in that case. Furthermore, for a non-coplanar, but circular outer orbit, the amplitude of the light-time solution remains invariant, at least as far as the quadratic term is not counted. On the other hand, in the case of this configuration the usual determination of the outer eccentricity (from the second Fourier-coefficients) may give an error of several 10 percents. Finally, if the outer orbit is significantly eccentric, both the mass-function (via the amplitude), and the outer eccentricity is affected.

Table 4: The results of the parameter search for different runs in the "Algol''-like system. The "L''-rows list the results of the simple light-time solutions. Quantities marked with ^* are calculated for the input values of i', which are also listed in the "L''-rows in parenthesis. (For the other input parameters see Tables 2, 3.)

Run No. e' $\omega'$ $\tau '$ a₁₂ i' D $i_{\rm m}$ m₃ $\chi^2$

$\hbox{$^\circ$ }$ MHJD $10^6~{\rm km}$ $\hbox{$^\circ$ }$ $\hbox{$^\circ$ }$ $\hbox{$^\circ$ }$ $M_\odot$ 10^-7

A1 L 0.22 101 50 071 128^* (82) (90) 2.0^* 848

1 0.23 57 49 993 140 65 63 62 2.2 678

2 0.23 57 49 993 130 99 247 113 2.0 719

3 0.21 148 50 157 226 35 14 49 4.2 837

A2 L 0.22 77 50 034 131^* (82) (45) 2.0^* 359

1 0.22 59 49 999 128 91 219 140 2.0 216

2 0.22 59 49 999 128 89 320 40 2.0 217

A3 L 0.25 58 50 005 136^* (82) (0) 2.1^* 266

1 0.16 54 49 995 175 131 158 141 3.0 400

2 0.21 73 50 027 141 69 185 151 2.2 637

A4 L 0.22 78 50 036 131^* (82) (315) 2.0^* 397

1 0.22 59 49 998 135 72 41 41 2.1 205

2 0.22 58 49 998 129 92 222 138 2.0 216

3 0.22 58 49 998 129 88 318 42 2.0 216

A5 L 0.25 80 50 039 131^* (82) (135) 2.0^* 514

1 0.23 56 49 995 143 64 44 46 2.3 365

2 0.23 56 49 994 130 84 227 132 2.0 411

A6 L 0.24 59 50 006 135^* (82) (0) 2.1^* 237

1 0.21 64 50 011 134 76 194 155 2.1 493

2 0.21 66 50 014 134 103 347 25 2.1 554

3 0.21 67 50 015 148 61 12 24 2.4 564

A7 L 0.17 76 50 031 130^* (82) (225) 2.0^* 448

1 0.23 59 49 999 128 88 42 43 2.0 258

2 0.23 60 50 000 135 109 221 139 2.1 266

A8 L 0.17 103 50 077 128^* (82) (270) 2.0^* 907

1 0.23 58 49 995 144 63 298 62 2.3 607

2 0.23 57 49 995 129 84 68 68 2.0 645

3 0.17 168 50 197 235 147 162 129 4.5 722

4 0.17 167 50 197 187 42 199 122 3.2 729

Run No. e' $\omega'$ $\tau '$ a₁₂ i' D $i_{\rm m}$ m₃ $\chi^2$

$\hbox{$^\circ$ }$ MHJD $10^6~{\rm km}$ $\hbox{$^\circ$ }$ $\hbox{$^\circ$ }$ $\hbox{$^\circ$ }$ $M_\odot$ 10^-7

A9 L 0.36 166 50 030 139^* (82) (270) 2.2^* 679

1 0.23 144 49 990 131 87 68 67 2.0 796

2 0.23 144 49 989 151 61 299 61 2.4 799

3 0.30 186 50 065 153 115 324 49 2.4 1084

4 0.29 186 50 066 185 131 146 135 3.1 1169

A10 L 0.22 106 50 081 145^* (60) (90) 2.3^* 968

1 0.23 57 49 994 142 64 68 67 2.2 778

2 0.23 57 49 994 133 107 253 108 2.1 811

3 0.23 155 50 170 231 34 15 50 4.4 958

A11 L 0.16 72 50 025 149^* (60) (45) 2.4^* 408

1 0.05 147 50 165 212 37 20 48 3.9 238

2 0.22 62 50 004 128 95 39 41 2.0 276

3 0.22 63 50 004 139 66 323 39 2.2 302

A12 L 0.26 53 49 993 154^* (60) (0) 2.5^* 280

1 0.17 40 49 968 177 47 25 41 3.0 265

2 0.17 39 49 967 149 119 333 45 2.4 289

3 0.23 71 50 023 144 65 180 147 2.3 635

4 0.23 73 50 025 166 52 359 31 2.7 744

A13 L 0.24 133 50 129 252^* (30) (90) 5.0^* 1820

1 0.23 58 49 996 199 40 84 81 3.5 1256

2 0.34 174 50 204 206 37 198 118 3.7 2332

3 0.25 59 49 997 543 167 222 108 20.1 9377

A14 L 0.07 355 49 876 252^* (30) (45) 5.0^* 1474

1 0.22 60 50 002 186 43 133 111 3.2 601

2 0.23 61 50 003 229 213 42 121 4.3 685

3 0.21 270 49 713 201 322 33 114 3.6 1325

4 0.21 268 49 712 250 150 151 123 4.9 1330

A15 L 0.35 31 49 950 264^* (30) (0) 5.3^* 953

1 0.23 64 50 011 202 321 356 122 3.6 724

2 0.32 6 49 903 151 60 221 126 2.4 1307

3.2 The numerical method of the separation

The separation of the dynamical terms from the light-time curve may give two important advantages. The first is the determination of more accurate orbital elements, mainly the "projected'' third-body mass. The second one is the possibility to determine the relative spatial orientation of the two orbital planes. This latter arises from the fact, that the dynamical terms - through the orbital elements which determine the light-time orbit - depend on the third body mass (m₃), the observable inclination of the wide orbit (i'), and the mutual inclination ( $i_{\rm m}$ ). It is evident that the dependence on m₃ appears in the amplitude A^*, while the effect of the inclinations manifests through the following equations of spherical triangles:

		$\displaystyle \sin{i_{\rm m}}\cos{u'_{\rm m}}=-\cos{i}\sin{i'}+\sin{i}\cos{i'}\cos\left(\Omega'-\Omega\right),$	(73)
		$\displaystyle \sin{i_{\rm m}}\sin{u'_{\rm m}}=\sin{i}\sin\left(\Omega'-\Omega\right),$	(74)
		$\displaystyle \cos{i_{\rm m}}=\cos{i}\cos{i'}+\sin{i}\sin{i'}\cos\left(\Omega'-\Omega\right).$	(75)

In order to make this separation we developed a computer code which is based on a non-linear Levenberg-Marquardt algorithm (see Press et al. 1989, Chap. 14.4). In the present state the code adjusts the following eight parameters: c₀ (a zero-point correction), P, A (the amplitude of the light-time terms), e', $\omega'$ , l'₀, i', and instead of the mutual inclination, $D\equiv\Omega'-\Omega$ . (Here we note, that although P is an adjusted parameter, de fundamental frequency $\nu$ is constant during the iteration.) The partial derivatives of the coefficients (56)-(71) are listed in the Appendix.

The method of the computation is the following. As a first step we determine the light-time solution from the O-C curve in the usual way. These orbital elements are used as input parameters for the Levenberg-Marquardt method. For the remaining two parameters (i', D) several initial trial-values are applied automatically. (Using the mass-function, the mass m₃ and the amplitude A^* are calculated in each iteration step.) If a solution is convergent, the program saves the final parameters, and takes the following set of the angular (i', D) parameters.

Table 5: The results of the parameter search for different runs in the "IU Aur''-like system. The "L''-rows list the results of the simple light-time solutions. Quantities marked with ^* are calculated for the input values of i', which are also listed in the "L''-rows in parenthesis. (For the other input parameters see Tables 2, 3.)

Run No. e' $\omega'$ $\tau '$ a₁₂ i' D $i_{\rm m}$ m₃ $\chi^2$

$\hbox{$^\circ$ }$ MHJD $10^6~{\rm km}$ $\hbox{$^\circ$ }$ $\hbox{$^\circ$ }$ $\hbox{$^\circ$ }$ $M_\odot$ 10^-7

I1 L 0.40 6 50 000 150^* (88) (90) 16.0^* 2806

1 0.09 18 50 010 411 160 359 72 71.3 2289

2 0.47 4 49 999 143 89 318 42 15.1 2694

3 0.47 4 49 999 144 85 42 42 15.2 2705

I1S L 0.40 6 50 000 150^* (88) (90) 16.0^* 2691

1 0.09 15 50 007 402 159 358 71 69.3 2187

2 0.47 5 49 999 143 88 319 41 15.1 2641

3 0.47 5 49 999 143 86 40 41 15.2 2665

I2 L 0.49 7 50 003 184^* (88) (45) 21.0^* 2418

1 0.53 7 50 001 159 88 205 154 17.1 3560

2 0.53 7 50 001 160 83 25 26 17.2 3639

I3 L 0.55 6 50 002 220^* (88) (0) 26.7^* 3506

1 0.63 10 50 004 172 94 201 159 19.6 6624

2 0.63 11 50 004 172 90 159 159 19.4 6840

I4 L 0.46 73 50 004 169^* (88) (90) 18.8^* 3157

1 0.46 71 49 999 165 86 322 38 18.1 2619

2 0.46 70 49 999 165 89 38 38 18.0 2623

I5 L 0.50 65 50 006 183^* (88) (45) 20.8^* 2823

1 0.43 76 50 009 183 108 339 29 20.5 3286

2 0.43 77 50 010 189 67 20 29 21.6 3469

Run No. e' $\omega'$ $\tau '$ a₁₂ i' D $i_{\rm m}$ m₃ $\chi^2$

$\hbox{$^\circ$ }$ MHJD $10^6~{\rm km}$ $\hbox{$^\circ$ }$ $\hbox{$^\circ$ }$ $\hbox{$^\circ$ }$ $M_\odot$ 10^-7

I6 L 0.54 55 50 005 203^* (88) (0) 23.9^* 3043

1 0.39 54 50 002 260 132 155 134 34.2 3501

2 0.39 54 50 002 247 129 35 47 31.7 3536

3 0.48 82 40 020 235 55 180 143 29.3 5968

4 0.49 84 50 021 247 51 359 37 31.5 6275

I7 L 0.43 78 50 006 233^* (45) (90) 28.9^* 3544

1 0.47 67 49 998 167 77 314 46 18.4 3224

2 0.47 67 49 998 164 83 134 133 18.0 3239

I8 L 0.43 62 50 000 249^* (45) (45) 31.7^* 2540

1 0.47 80 50 009 187 66 160 148 21.3 3657

2 0.47 81 50 009 194 62 341 32 22.4 3809

I9 L 0.52 53 50 000 262^* (45) (0) 34.2^* 3092

1 0.33 42 49 990 303 216 332 123 42.1 3579

2 0.49 81 50 015 224 128 6 41 27.1 5511

3 0.49 82 50 016 236 48 354 40 29.2 5815

I10 L 0.23 75 50 009 184^* (88) (45) 20.9^* 1243

1 0.22 56 49 997 190 73 42 44 22.1 1341

2 0.22 56 49 997 188 102 337 44 21.6 1343

3 0.15 106 50 038 271 138 345 51 36.6 1409

The results of our numerical fitting can be found in Tables 4, 5. In the case of the "Algol-like'' system we deduced the following conclusions:

In most cases more accurate orbital elements were gained for the elements e', $\omega'$ , $\tau '$ than it was obtained from the simple light-time (L) fit. In these runs the $\chi^2$ value also improved with respect to the corresponding L-fits;
Significant exceptions arise in fits A3, A6, A12, e.g. at those runs, where $D_{{\rm initial}}=0\hbox{$^\circ$ },180\hbox{$^\circ$ }$ . In these cases ${\cal{C}}\equiv0$ , while ${\cal{S}}=\sin^2i_{\rm m}$ , which is also zero for run A3, while ${\cal{S}}\approx0.07$ and ${\cal{S}}\approx0.14$ for runs A6 and A12, respectively. Consequently, in these cases only the ${\cal{M}}$ -term, and the coefficients a^*₁, b^*₁ have significant contributions. On the other hand, in run A15, where $D_{{\rm initial}}$ was zero a good fit was also found. In this case already the ${\cal{S}}\approx0.62$ term is the dominant. (Here we note, that in this integration the mutual inclination was $52\hbox{$.\!\!^\circ$ }3$ , which is very close to that "critical'' value where the ${\cal{M}}$ -term disappears.) It is interesting, that despite the weaker fits in these cases, the $D\approx0\hbox{$^\circ$ }$ , $180\hbox{$^\circ$ }$ values were reproduced well;
The determination of the visible inclination of the outer orbit is less accurate. This is of course not so surprising. We note that the dependence of the tertiary mass on the visible inclination (i') is very weak in the high inclination region. A comparison of the upper or the middle panels in Fig. 2 shows this clearly. Nevertheless, for the last three runs (in the low visible inclination region), where a smaller variation in the visible inclination i' results already significant variation in m₃, really a smaller inclination, and consequently larger third body mass was found;
Finally, we can conclude that the better fits were reached when the mutual inclination had a medium value.

Considering the "I'' runs with the parameters of the IU Aur system our results are less satisfactory (see Table 5). Although the light-time parameters were improved in more than the half of these runs, the accuracy is far from that achieved in the "A'' runs. What could be the reason? There are three significant differences between the configurations of the two systems. The first is the large outer eccentricity in the IU Aur system, while the other two are the larger masses of the stars, and the smaller P'/P ratio, e.g. the closeness of this triple. The effects of these properties for the behaviour of our solution are very complex. A purely mathematical effect is that due to the larger eccentricity our expression gives a weaker approximation. On the other hand, we note, as perhaps the most important physical effect, that because the latter system is more compact, the apse-node time scale of the largest amplitude perturbations is significantly shorter, and these disturbances may manifest on the O-C diagram within years. (E.g. according to Drechsel et al. 1994 the period of the nodal regression is about 300 years in the IU Aur system.)

In order to study these two above mentioned phenomena we carried out the following two tests. In the last run (I10) we changed the outer eccentricity for the value e'=0.24, while the other parameters had the same values as in run I5. A significant improvement was found in most of the parameters (although a false result also occurred). Furthermore, we calculated the O-C solution of run I1 for a shorter (about half of the original) time interval. (These results are listed in the "I1S'' rows of Table 5.) We did not find significant discrepancy with respect to the original "I1'' solutions. We can conclude, that the larger eccentricity plays a more important role in our weaker results in this case.

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