Appendix A: Partial derivatives for the Levenberg-Marquardt algorithm

As it was mentioned in Sect. 3, the following eight parameters can be adjusted in our non-linear Levenberg-Marquardt code: c₀, P, $A\equiv a_{12}\sin{i'}/c$, e', $\omega'$, l'₀, i', $D\equiv\Omega'-\Omega$. Here we list the analytical form of the partial derivatives of the Fourier-coefficients with respect to these parameters. The a_k, b_k symbols refer to the coefficients of the geometrical light-time effect, while a^*_k, b^*_k denote the dynamical terms. (We note that in the following: $A_k\equiv a_k+a^*_k$, $B_k\equiv b_k+b^*_k$.)

$$\frac{\partial{A_k}}{\partial A}=\frac{a_k}{A}+3\frac{M_{12}}{M_{123}+M_{12}}\frac{a^*_k}{A},$$ (A.1)

$$\frac{\partial{B_k}}{\partial A}=\frac{b_k}{A}+3\frac{M_{12}}{M_{123}+M_{12}}\frac{b^*_k}{A},$$ (A.2)

$$\frac{\partial{A_1}}{\partial{e'}}=-A\frac{e'}{2}\left[\frac{3}{2}\cos\left(l'_0+\omega'\right)+\cos\omega'\cos{l'_0}\right]+\frac{a^*_1}{e'},$$ (A.3)

$$\frac{\partial{B_1}}{\partial{e'}}=-A\frac{e'}{2}\left[\frac{3}{2}\sin\left(l'_0+\omega'\right)+\cos\omega'\sin{l'_0}\right]+\frac{b^*_1}{e'},$$ (A.4)

$$\frac{\partial{A_2}}{\partial{e'}}=\frac{a_2}{e'},$$ (A.5)

$$\frac{\partial{B_2}}{\partial{e'}}=\frac{b_2}{e'},$$ (A.6)

$$\frac{\partial{A_3}}{\partial{e'}}=\frac{2a_3+a^*_3}{e'},$$ (A.7)

$$\frac{\partial{B_3}}{\partial{e'}}=\frac{2b_3+b^*_3}{e'},$$ (A.8)

$$\frac{\partial a_1}{\partial\omega'}=A\left[\left(A.1-\frac{3e'^2}{8}\right)\sin\left(l'_0+\omega'\right)-\frac{e'^2}{4}\sin\omega'\cos{l'_0}\right],$$ (A.9)

$$\frac{\partial b_1}{\partial\omega'}=-A\left[\left(A.1-\frac{3e'^2}... ...right)\cos\left(l'_0+\omega'\right)+\frac{e'^2}{4}\sin\omega'\sin{l'_0}\right],$$ (A.10)

$$\frac{\partial a^*_1}{\partial\omega'}=-2A^*e'\left[{\cal{C}}\cos\left(l'_0+2\omega'\right)+{\cal{S}}\sin\left(l'_0+2\omega'\right)\right],$$ (A.11)

$$\frac{\partial b^*_1}{\partial\omega'}=2A^*e'\left[{\cal{S}}\cos\left(l'_0+2\omega'\right)-{\cal{C}}\sin\left(l'_0+2\omega'\right)\right],$$ (A.12)

$$\frac{\partial A_2}{\partial\omega'}=-\left(b_2+2b^*_2\right),$$ (A.13)

$$\frac{\partial B_2}{\partial\omega'}=a_2+2a^*_2,$$ (A.14)

$$\frac{\partial A_3}{\partial\omega'}=-\left(b_3+2b^*_3\right),$$ (A.15)

$$\frac{\partial B_3}{\partial\omega'}=a_3+2a^*_3,$$ (A.16)

$$\frac{\partial A_k}{\partial l'_0}=-k\left(b_k+b^*_k\right),$$ (A.17)

$$\frac{\partial B_k}{\partial l'_0}=k\left(a_k+a^*_k\right),$$ (A.18)

$$\frac{\partial A_1}{\partial i'}=-3\frac{M_{12}}{M_{123}+2M_{12}}a^*_1\cot{i'}$$

$$\qquad\quad +A^*e'\sin2i_{\rm m}\big\{\cos{u'_{\rm m}}\left[6\cos{l'_0}+\cos\left(l'_0+2\omega'\right)\right]$$

$$\qquad\quad +\sin{u'_{\rm m}}\sin\left(l'_0+2\omega'\right)\big\},$$ (A.19)

$$\frac{\partial B_1}{\partial i'}=-3\frac{M_{12}}{M_{123}+2M_{12}}b^*_1\cot{i'}$$

$$\qquad\quad +A^*e'\sin2i_{\rm m}\big\{\cos{u'_{\rm m}}\left[6\sin{l'_0}+\sin\left(l'_0+2\omega'\right)\right]$$

$$\qquad\quad -\sin{u'_{\rm m}}\cos\left(l'_0+2\omega'\right)\big\},$$ (A.20)

$$\frac{\partial A_2}{\partial i'}=-3\frac{M_{12}}{M_{123}+2M_{12}}a^*_2\cot{i'}$$

$$\qquad\quad -A^*\sin2i_{\rm m}\big[\cos{u'_{\rm m}}\cos2\left(l'_0+\omega'\right)$$

$$\qquad\quad +\sin{u'_{\rm m}}\sin2\left(l'_0+\omega'\right)\big],$$ (A.21)

$$\frac{\partial B_2}{\partial i'}=-3\frac{M_{12}}{M_{123}+2M_{12}}b^*_2\cot{i'}$$

$$\qquad\quad -A^*\sin2i_{\rm m}\big[\cos{u'_{\rm m}}\sin2\left(l'_0+\omega'\right)$$

$$\qquad\quad -\sin{u'_{\rm m}}\cos2\left(l'_0+\omega'\right)\big],$$ (A.22)

$$\frac{\partial A_3}{\partial i'}=-3\frac{M_{12}}{M_{123}+2M_{12}}a^*_3\cot{i'}$$

$$\qquad\quad -A^*\frac{7e'}{3}\sin2i_{\rm m}\big[\cos{u'_{\rm m}}\cos\left(A.3l'_0+2\omega'\right)$$

$$\qquad\quad +\sin{u'_{\rm m}}\sin\left(3l'_0+2\omega'\right)\big],$$ (A.23)

$$\frac{\partial B_3}{\partial i'}=-3\frac{M_{12}}{M_{123}+2M_{12}}b^*_3\cot{i'}$$

$$\qquad\quad -A^*\frac{7e'}{3}\sin2i_{\rm m}\big[\cos{u'_{\rm m}}\sin\left(3l'_0+2\omega'\right)$$

$$\qquad\quad -\sin{u'_{\rm m}}\cos\left(3l'_0+2\omega'\right)\big],$$ (A.24)

$$\frac{\partial A_1}{\partial D}=2A^*e'\big\{I_1\left[6\cos{l'_0}-\cos\left(l'_0+2\omega'\right)\right]-I_2\sin\left(l'_0+2\omega'\right)$$

$$\qquad\quad -\cos{i'}\left[{\cal{C}}\cos\left(l'_0+2\omega'\right)+{\cal{S}}\sin\left(l'_0+2\omega'\right)\right]\big\},$$ (A.25)

$$\frac{\partial B_1}{\partial D}=2A^*e'\big\{I_1\left[6\sin{l'_0}-\sin\left(l'_0+2\omega'\right)\right]+I_2\cos\left(l'_0+2\omega'\right)$$

$$\qquad\quad -\cos{i'}\left[{\cal{C}}\sin\left(l'_0+2\omega'\right)-{\cal{S}}\cos\left(l'_0+2\omega'\right)\right]\big\},$$ (A.26)

$$\frac{\partial A_2}{\partial D}=2b^*_2\cos{i'}+2A^*\left[I_1\cos2\left(l'_0+\omega'\right)+I_2\sin2\left(l'_0+\omega'\right)\right],$$ (A.27)

$$\frac{\partial B_2}{\partial D}=-2a^*_2\cos{i'}+2A^*\left[I_1\sin2\left(l'_0+\omega'\right)-I_2\cos2\left(l'_0+\omega'\right)\right],$$ (A.28)

$$\frac{\partial A_3}{\partial D}=2b^*_3\cos{i'}+A^*\frac{14e'}{3}\big[I_1\cos\left(3l'_0+2\omega'\right)$$

$$\qquad\quad +I_2\sin\left(3l'_0+2\omega'\right)\big],$$ (A.29)

$$\frac{\partial B_3}{\partial D}=-2a^*_3\cos{i'}+A^*\frac{14e'}{3}\big[I_1\sin\left(3l'_0+2\omega'\right)$$

$$\qquad\quad -I_2\cos\left(3l'_0+2\omega'\right)\big],$$ (A.30)

where

$$I_1=-I\sin{i'}\sin{i_{\rm m}}\sin{u'_{\rm m}},$$ (A.31)

$$I_2=I\sin{i'}\sin{i_{\rm m}}\cos{u'_{\rm m}}.$$ (A.32)

Acknowledgements

We thank Dr. László Szabados and Mr. Szilárd Csizmadia for their comments on the manuscript, as well as Mrs. Rita Borkovits-József and Mr. Imre Barna Bíró for the linguistic corrections. This research has made use of NASA's Astrophysics Data System Bibliographic Services.