T. Yokoyama - M. R. Mana - C. do Nascimento - M. T. Santos - N. Callegari Jr
Universidade Estadual Paulista, IGCE-DEMAC Caixa Postal 178,CEP 13.500-970 Rio Claro (São Paulo), Brasil
Received 28 April 2004 / Accepted 8 September 2004
Abstract
A probable capture of Phobos into an interesting resonance was
presented in our previous work.
With a simple model, considering Mars in a Keplerian and circular
orbit, it was shown that once captured in the resonance, the inclination of the satellite
reaches very high values. Here, the integrations are extended
to much longer times and escape situations are analyzed.
These escapes are due to the interaction of new additional resonances, which
appear as the inclination starts to increase reaching some specific
values. Compared to classical capture in mean motion resonances, we see
some interesting differences in this problem.
We also include the effect of Mars' eccentricity
in the process of the capture.
The role played by this eccentricity becomes important, particularly when
Phobos encounters a double resonance at
.
Planetary
perturbations acting on Mars and variation of its equator are
also included. In general, some possible scenarios of the future of Phobos are presented.
Key words: planets and satellites: general - solar system: general
In Yokoyama (2002), hereafter PYO, we showed that in the future, Phobos will face some interesting resonance because its semi major axis a is continuously decreasing due to the action of tides. This means that the frequencies associated with the longitude of the node () and the argument of the pericenter (w) are also varying, so that in the neighbourhood of a specific value of the semi major axis, these frequencies can satisfy resonant relations like , where is the mean motion of the Sun, k_{i} are integers, and dots mean time derivatives. This kind of commensurability is rather different to usual mean motion resonances, since only one mean longitude (the Sun) is involved. Moreover, this mean motion is not affected by the tidal effects. The current semimajor axis (a), eccentricity (e) and inclination (I) of Phobos are: , and , where inclination is referred to Mars' equator and is the equatorial radius of Mars. In our previous work we showed that Phobos can be captured in the resonance in the neighbourhood of . Even in the absence of tides, i.e., for the ordinary restricted three body problem, the effect of this resonance is clearly visible. Indeed, taking and integrating the Cartesian Eqs. (1) as given in PYO (Mars is in a Keplerian orbit), we get Fig. 1. The obliquity of the equator is artificially fixed to (Panels A, B). For this high value of , the inclination varies up to almost while for (Panels C, D), the maximum inclination is . If however the semi major axis a is slightly changed to a=2.145, the angle does not librate any more and the variation of the inclination is negligible.
Figure 1: Left panels: variation of the inclination for resonant semi major axes. Right panels: libration of the resonant argument: . Initial conditions for the satellite: Top: , e=0.015, , , , Bottom: a=2.1498, e=0.015, , , . Mars is assumed in a Keplerian orbit with initial conditions: a=1.5236 AU, e=0.0933, , , , . | |
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Now, let us suppose a case when the semi major axis of Phobos decreases secularly due to tides and passes through the resonant value ( ). According to Szeto (1983) the time rate variation of Phobos' semimajor axis is of the order of 10^{-7} cm s^{-1}. The current obliquity of Mars is , but this value is chaotically varying (Ward 1979; Touma & Wisdom 1993; Laskar & Robutel 1993) and over a long time scale it can sweep a large interval. In particular, in a frequency analysis over 45 Myr, Laskar & Robutel (1993) found a chaotic zone of which can range from to . If at the moment when , the current value of is larger than , then capture usually occurs and inclination grows secularly to high values, much greater than the present value. As mentioned in PYO, the reason for this large increase is due to the continuous outward displacement of the libration center (Fig. 2 in PYO) as Phobos' semi major axis decreases. In PYO we always considered Mars in a circular orbit and integrations were not long enough, so that no escape or possible rupture of the resonance were presented. This time we want to explore some escape situations. This problem was studied by many authors: Beaugé & Ferraz-Mello (1994), Gomes (1995a,b, 1997a,b, 1998), Liou & Zook (1997), Hamilton (1994), Jancart et al. (2003), etc. In particular, Gomes has presented a deep study of the phenomenon of resonance lock. He also deduced some interesting formulae that govern the evolution of the eccentricity, inclination and semi major axis not only in the restricted three body problem but in the mutual case as well. Gomes considers different kind of dissipative forces; however in all cases the resonance is mainly due to the combination of two mean longitudes. Therefore, for the restricted three body problem, in general, once a capture takes place, the semi major axis of the particle must stop its secular variation and remains librating around some resonant value. Depending on the dissipative force, the amplitude of the libration of the semi major can either decrease to a stationary value (stable lock) or increase beyond some tolerable limit (unstable lock), so that the escape is unavoidable and capture is only temporary (Gomes 1995b). A noteworthy point in the present problem is the resonant angle: instead of two mean motions, the resonance comes from Phobos' longitude of node and the mean motion of the Sun. Therefore, in opposition to mean motion resonances, in our problem, the satellite remains captured, even if its semi major axis is decreasing (Sect. 3). In PYO we list other similar resonances, e.g., the evection: . Although in our problem we are mostly interested in the capture of inclination, it is instructive to comment briefly on some points related to the dynamics of the evection resonance. This commensurability might have been important in the past of the Earth-Moon system (Touma & Wisdom 1998). Capture in this case was possible only for the increasing semi major axis of the Moon. The duration of this regime is interesting, since unlike our problem, the increase of the eccentricity generates some consequences for the evolution of the semi major axis: in the Touma & Wisdom problem, the eccentricity grows rapidly, and after some time, its value is high enough that the Moon's semi major axis stops increasing. This occurs because, when the orbit becomes very eccentric, at the pericenter, the angular motion of the Moon becomes faster than the angular rotation of the Earth. This is not the situation in our problem, that is, the capture in inclination does not dampen the decay of Phobos' semi major axis. In the Earth-Moon problem the dynamics of the evection is very complex, but as pointed out by Touma & Wisdom (1998), capture in this resonance is always temporary (for tidal dissipative forces). Compared to problems involving two mean motion resonance, or evection resonances, we show that the dynamics of our resonance presents some clear differences in the evolution during the capture.
This work is organized in the following way: Sect. 2 shows some situations of capture and escape. To explain these escapes and to put the problem in a workable form, we still keep Mars' equator as the reference plane. The dynamics of the problem when Mars' eccentricity is taken into account is described in Sect. 3. In Sect. 4 we include the planetary perturbations acting on Mars and also the precessional equations of Mars' equator. Section 5 is devoted to conclusions.
As mentioned before, many authors have studied the phenomenon of capture and escape in resonances involving mean motion commensurability. Very roughly speaking, due to the dissipative force, the semimajor axis of the perturbed body varies secularly so that at some moment a particular orbital resonance involving the mean motions can be satisfied. Then, provided that certain conditions are fulfilled (see for instance, Henrard 1982; Peale 1989; Malhotra 1990; Gomes 1995a, 1997a, etc.) a capture takes place. Usually, in this case, the ratio of semi major axes, or the semi major axis of the perturbed body (restricted three body problem), stops varying secularly. Also, under the capture, a specific angle connected to the current resonance starts to librate, whereas the eccentricity or inclination begins to vary secularly. Beaugé & Ferraz-Mello (1994) showed that in the case of drag forces, the eccentricity varies secularly during some time, but after that, it has the tendency to stabilize around a specific value which they called universal eccentricity. In spite of this stabilization, the capture may not be permanent, that is, if the temporal variation of the libration angle increases in amplitude, an escape from the resonance will appear and this rupture is essentially due to the unstable character of the resonance lock. Therefore, according to Gomes (1995b), a resonance lock is stable or unstable and this stability depends on the particular dissipative force involved. A detailed study yielding analytical criteria about the permanency in resonance trapping is given in his paper.
Similarly to universal eccentricity, the concept of universal inclination was introduced by Gomes (1997b) considering only one specific resonant librating angle.
However, it seems that the existence of such universal values is not completely clear, at least for arbitrary dissipative forces or when it is not possible to define one specific resonant angle in the libration regime. Actually, Gomes (1997b) says: although the concept of a universal inclination is presented here, I could not get any example where this feature showed in an unequivocal way, with the inclination coming to a value equal to the universal inclination.
In this section, we discuss the escapes in our problem. Even
though an analytical proof is not given, we show through numerical
experiments that all the escapes occur, without any previous
stabilization around some possible universal value. We put the
problem into the simplest workable form, so that we still keep a
simplified model taking Mars equator as the reference plane. The
Sun is in a Keplerian circular orbit, with
its current inclination with respect to Mars' equator.
Therefore, we consider Phobos disturbed by the Sun and the
oblateness of Mars. The coordinate system is fixed at the center
of the planet. Besides the mean motion of the Sun (),
the principal frequencies are:
where (
)
are respectively: semi major
axis, eccentricity, inclination, mean motion, argument of the
pericenter and longitude of the node. J_{2} is the oblateness
coefficient, k^{2}, is the gravitational constant and
is
the equatorial radius of the planet. The complete disturbing
function of the problem is given in Eqs. (4) to (5) in PYO. We
integrate the averaged equations of the motion and the tidal
effect on the Phobos orbit is computed through the simplest model
for the semi major axis:
Figure 2: Left panels: evolution of the resonant angle ( top), eccentricity ( center), inclination ( bottom) taking the Radau integrator with LL=12. Initial conditions: , e=0.01, , , , .Right panels: similar but with , e=0.02, , , , . As the semi major axis and inclination evolve, new resonances can appear. Whenever a new resonance is encountered the dynamics of the libration of is clearly affected and either an escape or a different mode of libration occurs. The passage through the new resonances provokes some jumps in the eccentricity. | |
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where k_{2} is the second order Love number of Mars, while Q_{2200} is the dissipation function (Szeto 1983; Yoder 1981).
Figure 2 shows two different capture/escapes in resonance. The panels on the left correspond to the variation of the resonant angle, eccentricity and inclination with respect to the semi major axis of a single orbit starting with , e=0.01, , , , . Similarly, on the right, the initial conditions are: , e=0.02, , , , . In these integrations, the precision parameter LL in the Radau (Everhart 1985) integrator is fixed to be LL=12. Panel A shows the evolution of the resonant angle . Once captured (at ), libration takes place and lasts until , when we have escape. In panel B, Phobos is captured in the same way as before, but it does not escape when . Instead, changes its libration mode, and remains librating in this new mode until . Again, as before, it acquires a new libration mode and it continues until when it escapes definitively. Connected to each of these mentioned semi major axes, the eccentricity undergoes some clear jumps, although the inclination seems to ignore such effects. Starting from different initial conditions, the qualitative results are similar, i.e., at some specific values of the semi major axis, changes its libration mode. In some cases, this change causes the escape from resonance, so that inclination stops its secular variation. To understand these dynamics, we simply calculate the values of the semi major axis where different resonances can occur. Of course, in the beginning, for the current I of Phobos and , only is possible. Then capture takes place at , however, after that, since I is increasing, we note that other resonant combination can be fulfilled. Considering each of the arguments of the cosines given in PYO (Eq. (5)), the following additional resonances can occur together with :
In the present situation, due to the interaction of these two resonances, a chaotic motion appears, but it is temporary since the semi major axis is varying and the double resonance occurs for specific semi major axes given in the above equations. This disturbs the original capture, so that the occurrence of an escape or some change in the libration mode, shown in Fig. 2, seems to be quite natural. However, so far, we are not able to predict those encounters that provoke escapes.
Figure 3: A) circles are solutions of Eq. (2), while the continuous curve is obtained from direct integration of an orbit starting at , e=0.01, , , , LL=12, . B) circles are solution of Eq. (3), while the continuous curve results from numerical integration of an orbit starting at: , e=0.01, , , , LL=10. In both cases, analytical results match very well the numerical integration. ( C)-D)): even considering only one resonant term ( in , both inclination and eccentricity undergo large variations. Initial conditions: a=1.98, e=0.01, , , , LL=10. ( E)-F)): similar to ( C)-D)), but for resonance. This time inclination decreases. Initial conditions: , e=0.01, , , , LL=10. | |
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Assuming only one specific resonant librating angle, Gomes (1997b) was able to find very interesting separate expressions for the evolution of the eccentricity and inclination. They are differential equations that give the variation of these orbital elements in the case of capture. Through these relations, the concepts of universal inclination and eccentricity can be derived.
Here we can also obtain very simple relations giving the evolution
of I and e during the capture. On the other hand, the
existence of the universal inclination in this
problem seems to be not so clear. Jancart, Lemaître &
Letocart (2003) working with general dissipation
forces could not find a universal value for the inclination.
Again, the problem analyzed by these authors involved mean motion
resonances. In the present problem, the evolution of the capture
may present some particular features, since the resonance
relation and the dissipative force are different. In order to see
that, we make some simplified assumptions: instead of considering
the whole effect of the tidal dissipative force, we assume that
the tidal effect is simply given through Eq. (1), that
is, only semi major axis variation is taken. Therefore, if Phobos
is to be trapped in a specific resonance, the resonant relations
like
must remain valid all during the
capture. For instance, in the
resonance, the following relation between eccentricity and
inclinations should hold:
In the present problem, if the initial eccentricity is small, usually, its variation is also very small, so that Eq. (2) is an analytical relation that gives the inclination variation versus semi major axis. Unlike problems involving orbital resonances (Gomes 1997b), in the only one resonant angle is in the libration regime. Therefore the theoretical variation of the inclination given by Eq. (2) is always valid until an escape occurs.
In Fig. 3, panel A, the continuous curve comes from direct numerical integration. The circles are solutions of Eq. (2) but we interrupted at because the numerical integration shows an escape at this point. The agreement between the continuous and the circled curves is excellent.
The situation is similar for the evection resonance:
Therefore, provided that eccentricity (or inclination) does not vary greatly, Eq. (2) (or Eq. (3)) gives the variation of inclination (or eccentricity) with respect to a.
Note, however, that if the initial inclination is not small, we have some cases where even in an eccentricity type resonance, the inclination suffers large variation during the evolution of the capture. Although Eq. (3) remains valid, both inclination and eccentricity can undergo large variation. In Fig. 3C-E we show the significant increase and decrease of the inclination in the and resonances respectively. Panel D shows the variation of the eccentricity, and it is the corresponding pair of panel C. Analogously, panel F is the corresponding pair of panel E. In order to decouple and to get separate equations for eccentricity and inclination, we need an extra relation, involving a,e,I. In this sense, we consider, for instance, the action: , where p is the canonical momentum associated with the resonant librating angle q. Interesting interpretations and geometric idea of this quantity are given by several authors (Henrard 1982; Henrard & Lemaître 1986; Peale 1989; Gomes 1995a,b, etc.). Let be the gradient operator while J_{1} and J_{2} are the two first integrals of the motion. If and are not colinear vectors in the (p, q) phase space, then we say that J_{1} and J_{2} are independent integrals. With this reasoning, although and Eq. (3) are not integrals of the motion, separated equations for I and e could be obtained (provided that the previous relations are independent). Usually, an explicit analytical expression for is not possible, and in that case we have to resort to approximate numerical methods (Henrard & Lemaître 1986).
In the two previous sections we considered Mars in a circular orbit.
The current inclination of Phobos is
.
Under
normal conditions this value should remain almost unchanged, so
that capture of Phobos in the
commensurability is almost certain (Yokoyama 2002),
provided that Mars' obliquity is not less than
.
However before reaching this resonance, two previous
commensurabilities are encountered:
(N), and
(E). They appear almost
simultaneously at
so that we called them
double resonance. The former, for a decreasing semi major axis, is
favorable for capture while the latter (evection), provides
capture only if a is increasing. Since they appear
simultaneously, capture in (N) resonance, for a decreasing
semi major axis, is not certain. Indeed, some numerical
experiments in our previous work (PYO), showed that for the
circular case, capture in the
resonance
usually does not occur. However, a salient fact is noted: some
slight jumps in the inclination of Phobos were observed during the
passage through these resonances. In the circular case, these
jumps are not significant, and therefore, capture in the next
resonance,
,
should not be affected. But
if the height of these jumps becomes large, the post passage
inclination may exceed some threshold value so that the capture in
the next
resonance may not be certain.
So, in this section we show the effect of Mars' eccentricity. It
can play an important role mainly in the presence of the double
resonance. Besides the resonances given in PYO, the following
additional combinations (of order )
will appear for
:
Figure 4: Evolution of the log of the frequencies versus semi major axis. At the point where the fundamental frequency goes to zero we have the corresponding semi major of the separatrix. Since these separatrices are close, they can overlap and we expect the onset of chaos. Each integration was extended to 1 996 050 years, generating 2^{16} outputs, sampled at every 30 years. Phobos initial conditions: e=0.051, , , . Mars initial conditions: e=0.0, , , , , . | |
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We examine the dynamics when the semi major of Phobos varies, passing through these two resonant values. This time, Mars' eccentricity is considered (up to fourth order terms) and also all cosines of the disturbing function given in Eq. (5) in PYO.
Figure 5: Due to the action of the two simultaneous resonances in the neighbourhood of , the motion is very irregular (chaotic). Depending on the input values, the inclination and eccentricity are strongly excited during the passage through the double resonance. The final stabilized values are not predictable. On the left, the system leaves the double resonance with inclination and eccentricity 0.08. On the right, these values are smaller than those used in the input data. Initial conditions ( left panels): a=2.625, e=0.015, , , , ( right panels): a=2.630, e=0.015, , , . | |
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Figure 5 shows the situation when Phobos encounters (E) and (N) resonances. As expected, it is clear that the motion is quite irregular in the vicinity of . Because of this, in all integrations of this figure, we again fixed high precision in the Radau integrator (LL=12). In Fig. 5A the inclination goes through jumps during the passage through the critical value of the semi major axis, but after that it stabilizes at . Also the eccentricity is excited to (Fig. 5C). This significant increase in the inclination is important since unless some extra damping occurs, the next resonance ( ) will be faced with an inclination so large that capture in this resonance may not be possible. Indeed, certain capture is possible only when the initial action (see PYO, Eq. (9)), far from the resonance, is less than the area inside of the separatrix first formed, during the evolution of the semi major axis (Henrard 1982; Peale 1989). This means that we cannot guarantee capture for an arbitrary initial inclination. For instance, several numerical experiments show that for capture depends on the phase, while for we always can get capture no matter the initial values for w or .
Figure 5B shows another interesting situation: as before, the motion is also very complicated due to the double resonance and the inclination is excited to high values. However, at the end, it decreases back to about . The final eccentricity is also smaller than the initial value before entering the critical region (Fig. 5D). Since the motion is very irregular it seems that the precision parameter LL=12 in the Radau integrator is important. In this sense, a curious case appears if LL is relaxed to LL=9: we have a capture in the and inclination increases to high values similar to those shown in Fig. 2. After a lot of numerical experiments, we prefer to say that this capture is improbable and that the percentage of initial conditions leading to capture is very small. Moreover, tests showed that the general dynamics is very sensitive to the starting point and to the precision parameter LL used in the integrator. In particular, the initial conditions we used to get the mentioned capture were: , e=0.015, , , , , with LL=9.
We believe that the inclusion of Mars' eccentricity enhances the effect of the double resonance. The final value of the inclination when the semi major axis crosses the region seems to be completely uncertain.
Up to now, the orbit of Mars was considered Keplerian with a fixed
equator. In a more realistic model, we should take into account
the planetary perturbations which will disturb not only Mars'
orbit but also its equator. However, an integration starting from
present data until Phobos semi major axis decreases to
or
,
needs an enormous and probably a
prohibitive computational effort (Hamilton 2002).
Without a reliable tidal model, and corresponding reliable
constants, we cannot be confident of the final result of this long
time integration. Usually tidal models demand knowledge of
important physical constants, but in general, the accuracy of
these values is always questionable: the dissipation function Q,
Love's number ,
mass satellite (Phobos) and also rigidity and
dissipation in Phobos. For the precession of Mars' equator, the
problem is the same and the important constants involved are: the
moment of inertia and spin rate. In spite of all these
difficulties we think it is interesting to have a rough
qualitative idea of the main effects when planetary perturbations
are included. To avoid the problem of computational time, we
artificially change the initial semi major of Phobos, and start
our integration in the neighbourhood of
and
not
,
which is the current value of Phobos' semi major
axis. In order to consider the perturbation of the planets on
Mars, we use Laskar's secular theory (Laskar 1988). We
also take the variation of Mars equator through the equations
(Woolard 1953):
K_{0} | = | ||
K_{1} | = |
The Laskar secular theory (Laskar 1988) is given
through the series:
Figure 6: A typical situation of capture in the resonance including planetary perturbation on Mars and variation of its equator. In panel A, Phobos' inclination with respect to the ecliptic of 2000 is shown, while in panel B, the same inclination is shown with respect to the moving Mars equator. Panel C, shows the behaviour of . Initially these two nodes are almost synchronous and after the capture, they start to diverge ( more details are given in the text). Panel D, exhibits the variation of the obliquity. Initial conditions a=2.152, e=0.015, , , , | |
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The equations to be integrated are the same as before, but this time the orbit of Mars is given through series 6 and we also add Eq. (5). As usual, the initial conditions are chosen such that Phobos' inclination with respect to Mars' equator is and . Figure 6 shows a case of a capture in the resonance, when planetary perturbations on Mars and motion of its equator are considered.
Panel A shows the variation of the inclination of Phobos with respect to the ecliptic plane. A fast oscillation around the initial inclination ( ) is noticed. The amplitude of this oscillation starts to increase as the capture occurs. In fact, this inclination may reach zero or values greater than (Gomes 1997b), so that regularized variables are needed in these cases. Panel B shows the corresponding inclination referred to the equatorial plane. This time, once capture occurs, instead of the fast oscillations of the previous case, we have the well-known slow and secular increasing behaviour of the inclination. Panel C shows the difference . Its behaviour is very similar to the inclination in panel A. Finally, the variation of the obliquity () of Mars is shown in panel D. A few tests have shown that sometimes capture fails if the initial inclination with respect to the equator is , while we always found successful capture when we took it as . Therefore we see that even considering planetary perturbations and the variation of the equator, capture still occurs and the general path of the variation of the inclination with respect to the equator is quite similar to the cases without these two effects.
Considering the simplest tidal model for Phobos, we analyzed some situations of escape from the resonance. The escapes are due to new resonances that appear as the inclination reaches some specific values: , , , , , , . However, we cannot predict the inclinations for which escapes will occur. After several numerical simulations, we conclude that the existence of a universal inclination is not clear in this problem, since no stabilization of the inclination was found. Some analytic equations that give the path followed by the inclination during the capture are derived for each resonance. We also show that for , the role played by the eccentricity of Mars is decisive, since Phobos' inclination can be excited to about , so that capture in is not possible any longer. Finally, including planetary perturbations on Mars and precession equations for the motion of its equator, we showed that capture of Phobos is still possible and probably, the general features of the dynamics should be similar to the simplified model.
Acknowledgements
Part of this work was supported by FAPESP and FUNDUNESP. An anonymous referee is gratefully thanked for very important comments.