A&A 398, 10911102 (2003)
DOI: 10.1051/00046361:20021688
On the detectability of long period perturbations in close
hierarchical triple stellar systems
T. Borkovits^{1} 
B. Érdi^{2} 
E. ForgácsDajka^{2} 
T. Kovács^{2,}^{}
1  Baja Astronomical Observatory of BácsKiskun County,
PO Box 766, 6500 Baja, Szegedi út, Hungary
2 
Department of Astronomy, Eötvös Loránd University,
1117 Budapest, Pázmány P. sétány 1/A, Hungary
Received 26 September 2002 / Accepted 12 November 2002
Abstract
We study the possibility of the detection of the low amplitude long (P') period perturbative
effect of a distant third companion on the motion of a close binary. We give a new, more accurate analytical formula
for this kind of perturbation affecting the moments of the times of minima in eclipsing binaries. The accuracy of
this formula is tested by numerical integrations carried out for several
initial configurations. We also describe a numerical method based on a nonlinear LevenbergMarquardt
algorithm which makes it possible to separate this dynamical effect from the
pure geometrical lighttime effect in the eclipsing OC diagram. The
capabilities of this new method are demonstrated by the analysis of numerically
simulated OCs for test systems having physical parameters very similar to
Algol and IU Aur. The results show that the above mentioned
effect would be detectable in these systems nowadays, observing almost each
minima events in a 12 yearlong interval.
Key words: methods: analytical  methods: numerical  celestial mechanics  binaries: close 
binaries: eclipsing
Several close binary stars have third, distant companion. Due to the presence
of this further component, the motion of the binary no longer will be purely
Keplerian, but different types of periodic and nonperiodic (secular)
perturbations would occur. According to the classification of Brown (1936)
the periodic perturbations can be divided into the following three groups:
 Short period perturbations. The typical period is equal
to the orbital period P of the close binary, while the amplitude has the
order (P/P')^{2} (where P' denotes the period of the wide orbit);
 Long period perturbations. This group has a typical period of P', and magnitude of the
order (P/P');
 Apsenode terms. In this group the typical period is about P'^{2}/P, and the order of the
amplitude reaches unity.
(We have to note that this classification differs from what is used in the classical planetary perturbation theories.
There the first two of our groups called together as "short period'' perturbations, while the "apsenode terms'' are
called as "long period'' ones. In the stellar threebody problem this latter classification was used by Harrington 1968, 1969.)
These effects can be most easily detected in those triple systems, where the close binary happens to be an eclipsing one.
This follows from different reasons. First, the usual orbital periods of the eclipsing binaries are several days,
so in favourable cases even the apsenode terms appear in a timescale of some decades or centuries, which
for nowadays almost can be covered at least for a few systems. Furthermore, the variation of the orbital elements may
produce very spectacular effects in the characteristics of the eclipses. Here we refer to the
variable eclipsedepth at some eclipsing binaries in a timescale of decades. The wellknown examples are SS Lac
(Torres 2001, and further references therein), V907 Sco (Lacy et al. 1999),
SV Gem (Guilbault et al. 2001)^{},
where the eclipses disappeared in the last decades, furthermore, the yeteclipsing binary IU Aur
also shows fast inclination variations (Drechsel et al. 1994). Another important effect in eccentric binaries
is the precession of the line of the apsides caused by the third star. Nevertheless, this phenomenon is not so easily
observable in triple systems, since the main sources of the apsidal motion in the known cases are the tidal forces
arising from the close proximity of the members of such binaries. In the cases of some systems with abnormally slow
apsidal motion the superposition of the tidegenerated and the third bodyforced apsidal motion
might explain the discrepancies between the theory and the observations (see e.g. Khodykin & Vedeneyev 1997; Kozyreva et al. 1999
for the binary AS Cam).
A further main advantage of dealing with eclipsing binaries is that all of the aforementioned phenomena (as well as further ones) affect the occurrence of eclipse events too.
Apart from several other physical mechanisms which can modify the observable eclipsing minima times (e.g. mass flow in/from the system,
tidal forces etc.), the effect of the third body on the eclipsing OC diagram can be divided into a geometrical and a dynamical part.
The geometrical contribution is the wellknown lighttime effect. This reflects the motion of the eclipsing pair around the centre of
mass of the triple system. If its quasisinusoidal pattern can be separated from the other distortions of the OC curve, some of
the orbital elements of the wide orbit (P', e', ,
,
,
where a_{12} denotes the semimajor axis of the
orbit of the eclipsing pair around the common centre of mass of the triple system) can be determined. (Perhaps the easiest
way of this calculation was introduced by Kopal 1978.)
The dynamical contributions arise from the different perturbations. The typical amplitudes of these terms are listed in Söderhjelm (1975).
During an apsenode cycle the magnitude of the OC variations can reach even the order of days. Nevertheless, on a timescale
which is significantly shorter than the apsenode period this variation can be manifested e.g. as a parabolic pattern in the
OC curve, and its nature very easily can be misinterpreted (for an extended discussion see Borkovits et al. 2002).
Up to this moment we mainly concentrated only for the largest amplitude apsenode terms. Nevertheless, the OC diagram might
give a unique possibility to detect some kinds of long periodic perturbations. Of course, the OC curve reflects the
long, and even the short periodic variations of the orbital elements in the same way as in the case of the apsenode timescale
perturbations, but the amplitude of such variations usually much smaller than the limit of observability. The only exception
(at least in some cases) arises from the direct (long period) perturbation in the mean motion of the close binary, which due to
its cumulative effect on the OC diagram may exceed the limit of detectability.
In this paper we concentrate on this long period contribution of the OC diagram. In Sect. 2 we give an analytical formula for
this term which is valid for arbitrary value of the mutual inclination, although only for nearly circular close orbits. We also
compare our result with the outputs of direct numerical integrations. In Sect. 3 we present a numerical process to
separate the dynamical and geometrical term of the OC, and we examine whether and how this dynamical term can be
used to determine the real mutual spatial orientation of the orbits. We illustrate our results with numerically
generated OC data. Finally in Sect. 4 we consider the chance of the separability in real triple systems.
By the use of the theory of Harrington (1968, 1969), based on the von Zeipel averaging method of the canonical equations,
Söderhjelm (1975, 1982) derived analytical formulae for the long period perturbations in the standard Delaunay variables.
Although these formulae are exact up to second order in the (a/a') ratio, their practical use is limited, at least in their original
forms. Mayer (1990) gave a simple, useful form with the assumptions that the elements of the wide orbit are constant, the close
orbit is circular, and the relative orientation of the two planes is invariant. Nevertheless, as it will be shown,
the expression of Mayer (1990) maybe somewhat inaccurate. More explanation will come later.
In the following we present a corrected new formula, which is valid for the same assumptions.
Such a solution could be get easily from the original formulae of Söderhjelm as it was done by Mayer. Despite
this fact we follow a different way. Instead of the perturbing potential we depart from the perturbing force, and we calculate directly
the perturbations in the eclipsing period in the function of the true anomaly of the eclipsing pair along the wide orbit. As it will be shown
this method is more effective and faster for this particular problem, than the usual methods, furthermore, this automatically helps
to avoid that kind of inaccuracies which occured in the former solution.

Figure 1:
The spatial orientation of the orbital planes. See text for details. 
Open with DEXTER 
Using the masspoint approximation, the perturbing force acting upon the close binary is:

(1) 
where G denotes the gravitational constant, m_{3} is the mass of the tertiary, while
stands for
the position vector between the ith component of the binary and the third star.
The above expression, as it is wellknown, can be expanded into a series of Lagrangian polynomials of the following form:



(2) 
where M_{12} means the total mass of the binary,
and
denote the absolute value of the first two Jacobian positionvectors,
(e.g.
means the separation of the members of the eclipsing pair, while
is the distance between the centre of mass of the binary and
the distant third companion), while
stands for the direction cosine between
and
.
Let us define
a Cartesian coordinate system whose origin is at the centre of mass of the binary, and the three axes are parallel
with the vectors
,
,
,
respectively. The direction
cosines between the vector
and the axes (as it can be seen in Fig. 1) are as follows:
where w and w' denote the true longitude of the secondary and the tertiary measured from the intersection of the orbits,
and
is the mutual inclination.
According to these the three (radial, transversal and normal) components of the perturbing force in the first order of the
ratio
are
where

(10) 
(As it can be seen, the radial force component is divided into two parts. The first one contains terms depending on w, and very similar to
the
transversal component, while
does not depend on the revolution of the eclipsing binary.)
In what follows we will refer the orbital elements to a plane perpendicular to the line of sight of the observer,
and going accross the centre of mass of the binary. We will call it as the plane of the sky. It is clear that the
distance of this plane from the observer varies in time, according to the

(11) 
function, where

(12) 
furthermore v' denotes the true anomaly of the outer body,
is the argument of the periastron of the binary's
orbit around the centre of mass of the triple system, and M_{123} stands for the total mass of the triple.
As it is wellknown this motion is the source of the lighttime effect detected in several triple systems.
Using the above mentioned true anomaly, v', of the outer body, and the true longitude, u, of
the secondary measured from the plane of the sky, the forcecomponents have the following forms:
where the phase angles are
(In the above expressions
refers to the true longitude of the intersection of the two orbits
measured along the inner orbit, while
has the same meaning for the outer one.)
Using the expansions (13)(16) the analytical form of the OC can
be calculated very easily. To do that we depart from the wellknown fact, that at the moment of a midminimum

(19) 
where k is an integer. (Strictly speaking the above equation is valid exactly only if the eccentricity
of the eclipsing binary is zero, or the visible inclination is 90
,
nevertheless, in our treatment
the first condition is practically fulfilled.)
Let us define the so called instantaneous period of the binary in the following way:

(20) 
where as it is wellknown (see e.g. Milani et al. 1987, Chap. 3.2):

(21) 
Here we note an important fact. The first term on the rhs. is independent from the plane of reference, while
the second one has different values using different reference planes. (Please keep in your mind that in the
above expression, as well as in the following ones
means the special angular momentum of the binary, e.g. the length of vector ,
and not the velocity of light.)
Furthermore, let us denote by
the elapsed time between the ith and the (i+1)th eclipsing
(let's say primary) minima. (Hereafter we refer to
as eclipsing period.) Then it can be seen easily that

(22) 
where

(23) 
According to this the eclipsing period is the average of the instantaneous period during a revolution.
Consequently the occurrence of the Nth primary minimum after a t_{0} epoch can be determined as

(24) 
Consequently, if the
(or )
versus
(or u) dependence is known, the theoretical form of the
OC curve can be calculated formally by an integration. (Of course, the OC curve is not a continuous
function, only at the integer values of the independent variable has physical meaning.)
To get this relation we rewrite
as

(25) 
where n_{0} stands for the mean motion of the unperturbed twobody revolution in a fixed (let's say t =t_{0}) moment, and

(26) 
while

(27) 
As far as the perturbations are small
in the mean motion the relation between the instantaneous period
and the Keplerian period P_{0} of the unperturbed
motion can be written as

(28) 
(In what follows we omit the "0'' subscripts from the initial values of the quantities referring to the unperturbed motion.)
First we calculate the effect of the
component on the instantaneous period. It can be easily seen, that

(29) 
The second derivative of the
part of the true longitude has the following form (cf. Milani et al. 1987, Chap. 3.2):

(30) 
where

(31) 
and v denotes the true anomaly of the secondary component. Here we have to note an important fact. For the first sight
the presence of the second term in Eq. (30) contradicts our previous assumption, that the orbit of the binary is circular.
In fact it is not true.
Although the eccentricity is close to zero, it cannot be permanently exactly zero in perturbed systems. Even if at some moment the
close orbit was circular in the next moment due to the perturbing forces it would not be that. (For the possible
astrophysical importance of this small nonzero eccentricity especially for semidetached systems see Eggleton et al. 1998.)
So in nearly circular systems the eccentricity oscillates between zero and a small value (typically some tenhundredthousandths,
see e.g. our numerical integrations for the system IM Aur in Borkovits et al. 2002). In that case e approximately has
the same magnitude as ,
and so it can be shown that the two terms on the rhs of (30) may have the
same order. Furthermore, since both
and
have the same order of magnitude as the small variation
in the
radius, the denominators in (30) can be replaced by a constant average
distance, which is the Keplerian semimajor axis a. On the other hand, we note that this is important only in the case
of the
term of Eq. (30), as in the first expression the multiplicator
in the amplitude of
cancels the denominator.
Let us define the expessions



(32) 



(33) 
respectively. Then

(34) 
For the evaluation of the first term we have to express the true anomaly of the third component by .
This can be done in two steps. First, we can change from the true anomaly to the mean anomaly by the use of the
expansions of Cayley (1861), and after that we approximate the mean anomaly l' by the following formula:

(35) 
where l'_{0} is the mean anomaly at the epoch t_{0}. Now using the expressions



(36) 



(37) 
the evaluation of (32) is trivial, and we get that

(38) 
Our next task is the calculation of (33). The dependence of the integrand on
can be written as
where the further integrands can be evaluated with good approximation as

(40) 
and

(41) 
(see e.g. Milani et al. 1987, Chap. 3.2).
Performing the integrations we obtain that

(42) 
where
contains the constant terms and those depending upon only u (via some
trigonometric functions). These terms will not give any contribution to the values of the OC, since they have
the same values in every minima, which fact is the direct consequence of (19).
Finally we give the
function.
As it is well known

(43) 
Since
has itself the same order as the components of
,
the other quantities in (43) can be treated as constants.
Consequently, our approximation for
is the following:
Let us turn back to the expression (28) of .
It can be seen easily that all
of the above calculated perturbative terms
have the order of (P/P')^{2}, which is in the order of 10^{2}10^{4} even for the closest hierarchical
systems. So, our expansion is verified.
A further integration of (28) gives the analytical form of the effect of the long period perturbations on the
OC curve. We keep only the terms which depend also on v'. (The constant terms will give a linear contribution
to the OC, and so they will build up into the observed eclipsing period, while terms which contain pure trigonometric
functions of u will disappear.) First let's treat the terms which depend on only the true anomaly v' of the tertiary.
For these the integration can be carried out directly with respect to v', using the expression (cf. e.g. Roy 1988, p. 292)

(45) 
Consequently, the amplitude of these integrated expressions is multiplied by
.
On the other hand, according to (36) and (37) terms which contain both v' and uafter the integration will have the same order of magnitude than before. Consequently, the terms which depend on purely
the orbital motion of the tertiary will be dominant. Keeping only these terms we obtain that
(Kepler's third law has been used for the transformation of the amplitude.)
For a comparison of our result with the formula of Mayer (1990) we enclose here his solution:
where

(48) 
(We used our notations instead of the original ones, furthermore, some obvious misprints were corrected here.)
The fundamental difference between (47) and (46) manifests in
the phase of the trigonometric terms. The phasing would be identical if
in Mayer's paper would
be measured from the intersection of the two orbital planes. Nevertheless, he used the same notation for
the argument of the periastron in the lighttime contribution, where
evidently has to be measured from
the plane of the sky. However, the two meanings of the
would be identical only if the observational
and the dynamical system of references were the same, or if the two orbital planes intersected each other
in the plane of the sky. As it is wellknown the calculation of the perturbational
problems is usually carried out in the dynamical frame of reference, where the fundamental plane is the invariable
plane of the system. In the case of the hierarchical triple stellar systems the net angular momentum of the system
mainly concentrates in the wide orbit (see e.g. Eq. (26) of Söderhjelm 1975), consequently the plane of
the wide orbit is very close to the invariable plane, and in the immovable wide orbit approximation
(which was used by Mayer 1990) the two planes become identical.
The other discrepancies also arise from the same problem. If the plane of reference is the plane of the wide
orbit,
,
consequently the terms multiplied by
will disappear.

Figure 2:
The long period dinamical contribution of OCs calculated by numerical integration,
furthermore, with the analytical formulae presented in this paper, and in
Mayer (1990). Upper panels: low mutual inclinations. Middle panels: medium mutual inclinations.
Lower panels: high mutual inclinations. (For the exact input parameters see Tables 2, 3.) 
Open with DEXTER 
In order to illustrate the accuracy of our result, and to compare it with the formula of
Mayer (1990) we carried out several numerical integrations with different initial conditions.
The description of the integrator can be found in Borkovits et al. (2002). The only alteration applied here is, that
the sampling of the Jacobian coordinates and velocities is done after the integration step closest to
the center of an eclipse, and not to the vicinity of the periastron. Only masspoint
approximation was applied. As initial parameters the physical properties and orbital elements of two
wellknown close triple systems were chosen (see Tables 13).
As it can well be seen in Fig. 2, in the exact coplanar case (upper left panel), as well as in the case,
when the two orbital planes intersect each other on the plane of the sky (upper right panel) both Mayer's and our
results give similarly accurate approximations, while in the other cases the differences are significant.
Table 1:
The initial parameters of the close systems. (The masses are given in solar mass, the period in days, and the angular
orbital elements in degrees.)
The nonarbitrary parameters are taken from Söderhjelm (1980), Lestrade et al. (1993) for Algol,
and from Drechsel et al. (1994) for IU Aur.
System 
m_{1} 
m_{2} 
P 
e 
i 

u 
"Algol AB'' 
3.7 
0.8 
2.8673 
0.0 
82.3 
52 
60 
"IU Aur AB'' 
21.3 
14.4 
1.811474 
0.0 
88.0 
60 
90 
Table 2:
The fixed initial parameters of the wide systems. The mass function f(m_{3}) is calculated from the amplitude of the OC curve, and is given in solar mass. The period P' is given in days, while the periastron passage
in HJD2 400 000.
System 
f(m_{3}) 
P' 
e' 

"Algol ABC'' 
0.125 
679.9 
0.23 
50 000.0 
"IU Aur ABC'' 
1.89 
294 
0.54^{a} 
50 000.0 
^{a} In the (last) run I10 e'=0.24 was chosen. (See text for details.)
In this section first we show how the presence of the dynamical term can influence the usual
method of lighttime solutions, and then, we give a numerical method to separate
the two terms, which can improve the accuracy of the lighttime solution, and, furthermore,
may give additional information about the spatial orientation of the triple system.
A usual way of calculation of the lighttime 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 Fourierexpansion
of the lighttime curve, where the fundamental frequency is the period ratio, e.g. .
Consequently, if the harmonic coefficients of the OC were determined by some numerical methods (typically by
weighted leastsquares 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 lighttime effect the mathematical form of the OC curve is:

(49) 
where N is the cycle number, and

(50) 
while



(51) 



(52) 
where

(53) 
furthermore,



(54) 



(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 nonzero coefficients are as follows:



(56) 



(57) 



(58) 



(59) 



(60) 



(61) 
Using the expansions of Cayley (1861) Eq. (46) also can be easily expanded into trigonometric series of the
mean anomaly l', as
where
We omitted terms which are multiplied by
,
as in eclipsing systems the observable inclination of the
binary is usually close to
,
consequently these terms give only a minor contribution. (E.g. even for
,
.)
A comparison between (62) and (49) reveals that in this case the coefficients are
as follows:



(66) 



(67) 



(68) 



(69) 



(70) 



(71) 
where the amplitude A^{*} is

(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 OC 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
lighttime 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 noncoplanar, but circular outer orbit, the amplitude of the lighttime 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 Fouriercoefficients) may give
an error of several 10 percents. Finally, if the outer orbit is significantly eccentric, both the massfunction
(via the amplitude), and the outer eccentricity is affected.
The separation of the dynamical terms from the lighttime curve may give two important advantages. The first
is the determination of more accurate orbital elements, mainly the "projected'' thirdbody 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 lighttime orbit  depend on the third body mass (m_{3}), the observable inclination of the wide orbit
(i'), and the mutual inclination (). It is evident that the dependence on m_{3} appears in the
amplitude A^{*}, while the effect of the inclinations manifests through the following equations of spherical triangles:
In order to make this separation we developed a computer code which is based on a nonlinear
LevenbergMarquardt algorithm (see Press et al. 1989, Chap. 14.4). In the present state the code adjusts the following eight parameters: c_{0} (a zeropoint correction), P, A (the amplitude
of the lighttime terms), e', ,
l'_{0}, i', and instead of the mutual inclination,
.
(Here we note, that although P is an adjusted parameter, de fundamental frequency
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 lighttime solution from the OC curve in the usual way. These orbital elements are used as input parameters for the LevenbergMarquardt
method. For the remaining two parameters (i', D) several initial trialvalues are applied automatically.
(Using the massfunction, the mass m_{3} 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.
The results of our numerical fitting can be found in Tables 4, 5.
In the case of the "Algollike'' system we deduced the following conclusions:
 In most cases more accurate orbital elements were gained for the elements e', ,
than
it was obtained from the simple lighttime (L) fit. In these runs the
value also improved with respect to the corresponding Lfits;
 Significant exceptions arise in fits A3, A6, A12, e.g. at those runs, where
.
In these cases
,
while
,
which is also zero for run A3, while
and
for runs A6 and A12, respectively.
Consequently, in these
cases only the term, and the coefficients a^{*}_{1}, b^{*}_{1} have significant contributions.
On the other hand, in run A15, where
was zero a good fit was also found. In this case
already the
term is the dominant. (Here we note, that in this integration
the mutual inclination was
,
which is very close to that "critical'' value where the
term disappears.) It is interesting, that despite the weaker fits in these cases, the
,
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 m_{3}, 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 lighttime 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 apsenode time scale
of the largest amplitude perturbations is significantly shorter, and these disturbances may manifest on the OC
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 OC 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.
In this paper we examined the possibility of detecting certain kind of perturbations, which are manifesting
themselves on a very short time scale (even during a yearly observing window of the eclipsing variable),
but usually omitted due to their low amplitude. This question naturally has different aspects. These are as follows:
 There are only a few known systems where the amplitude of the dynamical term in the OC exceeds
significantly the present observational accuracy. We can easily estimate the maximumvalue
of the P'/P ratio which is necessary to fulfill this condition. Supposing that the three components have
equal masses, and the third body revolves in a circular orbit, according to Eq. (72) the amplitude may exceed the
10^{3} day limit, if P'/P<40P, at least when the two orbits are perpendicular to each other. If the outer orbit
has significant eccentricity (say e'=0.50), or the system consists of stars with different masses, then this limit
in the function of the mutual inclination may even grow up to
.
From that point of view
the two systems for which the numerical integrations were carried out in this paper are placed near the
limit of the detectability, as for Algol
,
while for IU Aur the same value is
.
The three presently known closest eclipsing triple systems are Tau (
),
DM Per (
), and VV Ori (
)
(see e.g. the catalogue of Chambliss 1992). In the cases of the latter two binaries unfortunately only a very
few times of minima observations can be found in the literature, although we could expect the largest
effects at these stars. On the other hand, we have to note that as the amplitude of the light timeeffect decreases with P'^{2/3},
at these stars already the detection of the pure geometrical effect is also a challenge. In the most interesting case
of Tau, the amplitude of the perturbative terms may be larger with an order of magnitude than
that of the lighttime terms. Furthermore, at this system due to the large amount of proximity our initial assumptions
may loose their validity.
 The second aspect is mainly a technical question. It refers to the observing strategy. This was written already
in the conclusion of Borkovits et al. (2002), but, for the sake of the completeness, we repeat it. In order to have any
chance for the detection of this phenomenon frequent and accurate timings are necessary. It is desirable to
cover a few revolutions of the distant object as densely as possible. The shorter the time interval of such coverage,
the smaller the apsenode time scale or secular variations in the orbital elements which could modify the results.
 Finally, the mathematical modelling of perturbations are necessary in order to extract all information
from the observations.
In this paper we concentrated mainly on this third item. We calculated a new analytical formula
which gives the long period perturbations of the times of minima in eclipsing binaries.
We found that this formula is very similar to the earlier expression of Mayer (1990), nevertheless, some
errors are corrected. Using this expression we developed a numerical method to separate this dynamical effect from
the pure geometrical light time effect. We tested the capabilities of our model by the analysis of numerically simulated
OC curves. In the case of the test runs for outer orbits with moderate eccentricity, significantly better
solutions were found than in the larger eccentricity cases.
Naturally the next step would be the application of the method for observed OC diagrams of real systems. Unfortunately,
up to now there are not any OC diagrams with sufficient accuracy for the possible target systems. This is why
we plan to observe some of the few such systems in the near future to collect as many new times of minima as possible.
As it was mentioned in Sect. 3, the following eight parameters can be adjusted in our
nonlinear LevenbergMarquardt code: c_{0}, P,
,
e', ,
l'_{0}, i',
.
Here we list the analytical form of the partial derivatives of the Fouriercoefficients with respect to these parameters. The a_{k}, b_{k} symbols refer to the
coefficients of the geometrical lighttime effect, while a^{*}_{k}, b^{*}_{k} denote the dynamical terms.
(We note that in the following:
,
.)



(A.1) 



(A.2) 



(A.3) 



(A.4) 



(A.5) 



(A.6) 



(A.7) 



(A.8) 



(A.9) 



(A.10) 



(A.11) 



(A.12) 



(A.13) 



(A.14) 



(A.15) 



(A.16) 



(A.17) 



(A.18) 











(A.19) 











(A.20) 











(A.21) 











(A.22) 











(A.23) 











(A.24) 











(A.25) 







(A.26) 







(A.27) 







(A.28) 







(A.29) 







(A.30) 
where



(A.31) 



(A.32) 
Acknowledgements
We thank Dr. László Szabados and Mr. Szilárd Csizmadia for their comments on the manuscript,
as well as Mrs. Rita BorkovitsJózsef and
Mr. Imre Barna Bíró for the linguistic corrections. This research has made use of NASA's
Astrophysics Data System Bibliographic Services.

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