Issue 
A&A
Volume 555, July 2013



Article Number  A133  
Number of page(s)  8  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201220510  
Published online  12 July 2013 
The quest for companions to postcommon envelope binaries
IV. The 2:1 meanmotion resonance of the planets orbiting NN Serpentis^{⋆}
Institut für Astrophysik, GeorgAugustUniversität, FriedrichHundPlatz 1, 37077 Göttingen, Germany
email: k.beuermann@tonline.de
Received: 6 October 2012
Accepted: 9 May 2013
We present 69 new mideclipse times of the young postcommon envelope binary (PCEB) NN Ser, which was previously suggested to possess two circumbinary planets. We have interpreted the observed eclipsetime variations in terms of the lighttravel time effect caused by two planets, exhaustively covering the multidimensional parameter space by fits in the two binary and ten orbital parameters. We supplemented the fits by stability calculations for all models with an acceptable χ^{2}. An island of secularly stable 2:1 resonant solutions exists, which coincides with the global χ^{2} minimum. Our bestfit stable solution yields current orbital periods P_{o} = 15.47 yr and P_{i} = 7.65 yr and eccentricities e_{o} = 0.14 and e_{i} = 0.22 for the outer and inner planets, respectively. The companions qualify as giant planets, with masses of 7.0 M_{Jup} and 1.7 M_{Jup} for the case of orbits coplanar with that of the binary. The twoplanet model that starts from the present system parameters has a lifetime greater than 10^{8} yr, which significantly exceeds the age of NN Ser of 10^{6} yr as a PCEB. The resonance is characterized by libration of the resonant variable Θ_{1} and circulation of , the difference between the arguments of periapse of the two planets. No stable nonresonant solutions were found, and the possibility of a 5:2 resonance suggested previously by us is now excluded at the 99.3% confidence level.
Key words: binaries: close / binaries: eclipsing / white dwarfs / stars: individual: NN Serpentis / planets and satellites: detection / planets and satellites: dynamical evolution and stability
Table 1 is available in electronic form at http://www.aanda.org
© ESO, 2013
1. Introduction
Planets orbiting postcommon envelope binaries (PCEB) are a recent discovery, and only a few PCEB are known or suspected to harbor planetary systems. These planets are detected by the lighttravel time (LTT) effect, which measures the variations in the observed mideclipse times caused by the motion of the binary about the common center of mass. The derived orbital periods significantly exceed those of planets orbiting single stars, because the LTT method preferably selects long orbital periods and the radialvelocity and transit methods short ones. Since eclipsetime variations can also be brought about by other mechanisms, it is necessary to prove the strict periodicity of the LTT signal to confirm its planetary origin. The orbital periods on the order of a decade represent a substantial challenge, however. For a system of circumbinary planets, one additionally has to demonstrate the secular stability of a particular solution.
The discovery of two planets orbiting the dG/dM binary Kepler 47 with semimajor axes less than 1 AU (Orosz et al. 2012) has convincingly demonstrated that close binaries can possess planetary systems, but also raised questions about their orbital coevolution. The evolutionary history of planets orbiting PCEB may be complex: they either formed before the commonenvelope (CE) event and had to survive the loss of a substantial amount of matter from the evolving binary and the passage through the ejected shell, or they were assembled later from CEmaterial. Even if the planets existed before the CE, their masses may have increased in the CE by accretion, making a distinction between first and secondgeneration origins difficult. In both cases the CE may have significantly affected the planetary orbits, and we expect that the dynamical age of the system equals the age of the PCEB, which is the cooling age of the white dwarf. The hot white dwarf in NN Ser has an age of only 10^{6} yr (Parsons et al. 2010a), so the planets in NN Ser are dynamically young. Migration of planets is expected to occur in the CE, but our poor knowledge of the CE structure complicates predictions about the dynamical state of PCEB planetary systems.
Only a handful of PCEB have been suggested to contain more than one circumbinary companion. Of these, NN Ser (Beuermann et al. 2010; Horner et al. 2012) and HW Vir (Beuermann et al. 2012) have passed the test of secular stability. A final conclusion on HU Aqr (Goździewski et al. 2012; Hinse et al. 2012) is pending, and the eclipsetime variations in QS Vir are presently not understood (Parsons et al. 2010b). For a few other contenders, the data are still insufficient for theoretical modeling. In the case of NN Ser, a period ratio of the two proposed planets near either 2:1 or 5:2 was found (Beuermann et al. 2010, henceforth Paper I), with both models being stable for more than 10^{6} yr. Horner et al. (2012) confirm the dynamical stability of the proposed orbits, but suggest that further observations are vital in order to better constrain the system’s true architecture. In this paper, we report further eclipsetime observations of NN Ser and present the results of extensive stability calculations, which show that a twoplanet system that starts from our best fit will be secularly stable and stay tightly locked in the 2:1 meanmotion resonance for more than 10^{8} yr.
2. Observations
We obtained 69 additional mideclipse times of the 16.8 mag binary with the MONET/North telescope at the University of Texas’ McDonald Observatory via the MONET browserbased remoteobserving interface. The photometric data were taken with an Apogee ALTA E47+ 1k × 1k CCD camera mostly in white light with 10 s exposures separated by 3 s readout. In most cases photometry was performed relative to a comparison star 2.0 arcmin SSW of NN Ser, but in some nights absolute photometry yielded lower uncertainties. The eclipse light curves were analyzed with the white dwarf represented by a uniform disk occulted by the secondary star (see Backhaus et al. 2012). A sevenparameter fit was made to each individual eclipse profile of the relative or absolute source flux. The free parameters of the fit were the mideclipse time T_{c}, the fluxes outside and inside eclipse, the full width at half maximum (FWHM) of the profile, the ingress/egress time, and the two parameters a_{1} and a_{2} of a multiplicative function f = 1 + a_{1}(t − T_{c}) + a_{2}(t − T_{c})^{2}, which allowed us to model photometric variations outside of the eclipse. Such variations can be caused by effects intrinsic to the source, as the illumination of the secondary star, or by observational effects, as colordependent atmospheric extinction. The formal 1σ error of T_{c} was calculated from the measurement errors of the fluxes in the individual CCD images, employing standard error propagation. The distribution of the measured FWHM of all eclipses is Gaussian with a mean of 569.20 s and a standard deviation of 0.79 s. In Cols. 1−3 of Table 1 we list the cycle numbers, the observed mideclipse times T_{c} in UTC, and the 1σ errors of the 69 new eclipses. Reanalysis of earlier MONET/N data led to small corrections, and we also include the seven mideclipse times published already in Paper I. The errors of T_{c} vary between 0.17 and 0.83 s, depending on the quality of the individual light curves. The mean timing error is 0.38 s. All mideclipse times were converted from Julian days (UTC) to barycentric dynamical time (TDB) and corrected for the light travel time to the solar system barycenter, using the tool provided by Eastman et al. (2010)^{1}. These corrected times are given as barycentric Julian days in TDB in Col. 4 of Table 1. Together with the errors in Col. 3, they represent the new data subjected to the LTT fit in the Sect. 4.1.
Fig. 1 Residuals of the mideclipse times of NN Ser since 2007 relative to the linear ephemeris of Eq. (1), with the bestfitting model of Fig. 5 and Table 2 shown as solid curve. See text for further explanation. 
Figure 1 shows the O − C_{lin} residuals relative to the linear ephemeris of the binary quoted in Eq. (1) and derived in Sect. 4.1 below. The data points to the left of the dashed line are from Paper I and those to the right from this work. The residuals of the individual MONET eclipse times are displayed as crosses with error bars. Overplotted are the weighted mean O − C values for 19 groups of timings typically collected in the dark periods of individual months (cyanblue filled circles). Plotting these mean values avoids cluttering up the graphs on the expanded ordinate scales in Figs. 5 and 6. All fits presented in this paper were made to the total set of 121 individual mideclipse times, 52 from Paper I and 69 from this work. The solid curve in Fig. 1 represents the bestfit twoplanet LTT model derived in Sect. 4.1. The residuals O − C_{ell} of the individual timings relative to this fit are included in Col. 5 of Table 1. The new feature that allows us to derive a significantly improved orbital solution within the framework of the twoplanet LTT model is the detection of the upturn in O − C_{lin} that commences near JD 2 455 650.
3. General approach
In this paper, we adopt a purely planetary explanation of the eclipsetime variations in NN Ser and represent the set of mideclipse times by the sum of the linear ephemeris of the binary and the LTT effect of two planets. A model that only involves a single planet was already excluded in Paper I and is not discussed again, given the very poor fit with a reduced . Specifically, we describe the motion of the center of mass of the binary in barycentric coordinates by the superposition of two Kepler ellipses, which reflect the motion of the planets (Irwin 1952, Eq. (1); Kopal 1959, Eqs. (8)−(94); Beuermann et al. 2012, Eq. (2); Paper I, Eq. (1)^{2}). We justify the Keplerian model at the end of this section.
The central binary was treated as a single object with the combined mass of the binary components M_{bin} = 0.646 M_{⊙}, (Parsons et al. 2010a). This approach is justified, because the gravitational force exerted by the central binary on either of the planets varies only by a fraction of 10^{7} or less over the binary period. Hence, the NN Ser system was treated as a triple consisting of the central object and two planets. We assumed coplanar planetary orbits viewed edgeon, which coincide practically with the orbital plane of the binary with an inclination i_{bin} = 89.6° ± 0.2° (Parsons et al. 2010a). In spite of the high inclination, transit events, which would present proof of the planetary hypothesis, are unfortunately extremely improbable. Accurate eclipsetime measurements over a sufficiently large number of orbits could, in principle, provide information on the inclinations of the gravitationally interacting orbits, but this is presently not feasible given the long periods of the planets in NN Ser. The measured orbital periods and amplitudes of the LTT effect yield minimum masses, with the true masses scaling as 1/sin i_{k}, where i_{k} is the unknown inclination of planet k. The semimajor axes depend on the total system mass, implying a minor dependence on i_{k}.
Even with the new data, the leastsquares fits of the LTT model permit a wide range of model parameters. As in Paper I, we required, therefore, that an acceptable model provides a good fit to the data and fulfills the side condition of secular stability. Formally, a minimum lifetime of only 10^{6} yr is required, the cooling age of the white dwarf and the age of NN Ser as a PCEB, but most models in the vicinity of the best fit reached more than 10^{8} yr, suggesting that a truly secularly stable solution exists.
We performed a large number of leastsquares fits in search of the global χ^{2} minimum, using the LevenbergMarquardt minimization algorithm implemented in mpfit of IDL (Marquardt 1963; Markwardt 2009). The time evolution of all models that achieved a χ^{2} below a preset limit (Sect. 4.1) was followed numerically until they became either unstable or reached a lifetime of 10^{8} yr. Most models that fit the data developed an instability within a few hundred years or less and fewer than 200 survived for 10^{8} yr, allowing us to calculate the evolution of all models with an acceptable overall CPUtime requirement. We used the hybrid symplectic integrator in the mercury6 package (Chambers 1999), which allows one to evolve planetary systems very efficiently with high precision over long times. The model treats the central object and the planets as point masses and angular momentum and energy are conserved to a high degree of accuracy. Time steps of 0.1 yr were used, which is not adequate for the treatment of close encounters, but such incidents do not occur in the successful models (see Beuermann et al. 2012 for more details).
Fig. 2 Temporal variation of the mideclipse times of NN Ser as predicted by our bestfitting dynamical twoplanet model of Table 2 relative to the underlying linear ephemeris of the binary and folded over the orbital period of the outer planet. The first two orbits, covering a time interval of 30 yr (black curves), and the next 29 orbits (30–500 yr, red curves) are displayed. 
These dynamical model calculations provide us with information on the complex time variations of the orbital parameters, but also allow us to test the validity of the Keplerian assumption in fitting the data. Figure 2 shows the calculated variation of the mideclipse times for the first 500 yr of our bestfit longlived model, starting from the Keplerian fit. The LTT effect is displayed folded over the 15.47yr orbital period of the outer planet, which contributes 87% of the signal. The first two orbits (30 yr) agree closely and are indistinguishable from the Keplerian fit, but in the long run the signal changes due to the dynamical evolution. The orbital phase interval covered in Fig. 2 is the same as in Fig. 5 (top panel), where the data and the Keplerian fit are displayed. We conclude that in this special case fitting the data by a Keplerian model is justified. Data trains that extend over more orbits or involve more massive companions will require a dynamical model.
Fig. 3 Results of fitting the Keplerian twoplanet model to the eclipsetime data of NN Ser. Left: χ^{2} distribution in the e_{o}, e_{i} plane. The + sign indicates the minimum χ^{2} and the contour lines refer to the increments Δ χ^{2} indicated in the legend. Center: maximum lifetime τ of models with χ^{2} at the 99.9% confidence level (Δ χ^{2} = + 13.8 for two degrees of freedom). For ease of comparison, the χ^{2} contours of the left panel are included. Right: same for models with χ^{2} at the 68.3% confidence level (Δ χ^{2} = + 2.3). 
Fig. 4 Histograms of the orbital periods P_{o} and P_{i} of the outer and inner planets of NN Ser and the period ratio P_{o}/P_{i} for solutions with a lifetime exceeding 10^{6} yr and a χ^{2} corresponding to the 99.9% confidence level (yellow) or the 68.3% confidence level (green) for two degrees of freedom. 
4. Results
Our leastsquares fit to the 121 mideclipse times involves twelve free parameters, two for the linear ephemeris of the binary, the epoch T_{bin} and period P_{bin}, and five orbital elements for each planet. With all parameters free, the number of degrees of freedom of the fit is therefore 109. The five parameters for planet k are the orbital period P_{k}, the eccentricity e_{k}, the argument of periapse ω_{bin,k} measured from the ascending node in the plane of the sky, the time T_{k} of periapse passage, and the amplitude of the eclipse time variation, K_{bin,k} = a_{bin,k} sin i_{k}/c, with a_{bin,k} the semimajor axis of the orbit of the center of mass of the binary about the common center of mass of the system, i_{k} the inclination of the planet’s orbit against the plane of the sky, and c the speed of light. The arguments of periapse of the center of mass of the binary (with the index “bin”) and that of the planet differ by π. For clarity, we use the indices k = i or k = o for the inner and outer planets, respectively^{3}.
We fitted a total of 101 618 models to the data, adopting three lines of approach. Run 1 with 81 552 models is a grid search in the e_{o}, e_{i} plane. In Run 2 with 10 598 models, we started again from the e_{o}, e_{i} grid values, but subsequently optimized the eccentricities along with the other parameters. Run 3 with 9468 models was performed to study the properties of selected solutions, in particular, models in the vicinity of the best fit. It also includes models with a circular outer orbit, as advocated in Paper I, but dismissed now and not discussed further in this paper.
4.1. Grid search in the e_{o}, e_{i} −plane
The grid search in the e_{o}, e_{i} plane covers eccentricities from zero to 0.40 in steps of 0.01, with an extension to e_{o} = 0.70 in steps of 0.05. The ten other parameters were optimized, starting from values chosen randomly within conservative limits. Figure 3 (left panel) shows the resulting χ^{2} distribution for e_{o} < 0.3. For each grid point, the figure displays the best χ^{2} value of typically 50 model fits. The minimum is attained at e_{o},e_{i} = 0.15,0.22 with . In the extension of the grid to e_{o} = 0.70, no model fit with χ^{2} < 150 was found. The contour lines refer to increments Δχ^{2} of +1.0, +2.3, +4.6, and +13.8, corresponding to confidence levels of 40%, 68.3%, 90.0%, and 99.9% for two degrees of freedom. Color coding displays the lowest χ^{2} as dark red, fading into white at the 99.9% level. The 68.3% contour encloses a substantial fraction of the e_{o}, e_{i} plane, indicating that the data permit fits with a wide range of eccentricities and appropriate adjustment of the remaining parameters. Notably, those describing the inner planet are not well defined by the data alone, and the independent stability information is required for further selection.
We calculated the temporal evolution of all models with χ^{2} < 110 (99.93% confidence level), starting from the fit parameters and requiring that the lifetime τ exceeds the age of NN Ser of 10^{6} yr. We found that an island of secularly stable models is located close to the minimum χ^{2}, establishing an internal consistency between the fits to the data and the results of the stability calculations. The requirement of τ > 10^{6} yr imposes severe constraints on the orbital parameters of the successful models and, not surprisingly, the minimum χ^{2} of the stable models is slightly inferior to that of the unconstrained fits. The best stable model has , eccentricities e_{o},e_{i} = 0.14,0.21, and a lifetime τ = 5 × 10^{6} yr. The center panel of Fig. 3 shows the lifetime distribution in the e_{o}, e_{i} plane, with the peak lifetime in each bin displayed. The detailed structure of the lifetime distribution is complex. In particular, the appearance of secularly stable solutions outside the main island of stability is reminiscent of a skerry landscape. These solutions fit the data less well and disappear if the χ^{2} limit is reduced or, to stay in the picture, the sea level is raised. This is shown in the righthand panel of Fig. 3, where only solutions with τ > 10^{6} yr and χ^{2} < 97.9 are retained (68.3% confidence). The efficiency of the lifetime selection is demonstrated by the reduced size of the island of stability, which covers only a fraction of the χ^{2}space delineated by the 1σ contour level in the lefthand panel (solid curve). Its size decreases a bit further if the eccentricities are also optimized as done in Run 2.
We combined the results of Runs 1 and 2 to investigate the spread of the fit parameters for the solutions that pass the lifetime criterion, irrespective of their position in the e_{o}, e_{i} plane. Figure 4 shows the distributions of the orbital periods P_{o} and P_{i} and the period ratio P_{o}/P_{i} for model fits with χ^{2} < 109.4 and χ^{2} < 97.9, corresponding to the selections in the center and righthand panels of Fig. 3, respectively. For the more lenient χ^{2} limit, 169 of 173 solutions yield nearly identical periods and a period ratio of 2.022 with a standard deviation σ = 0.025. The remaining four solutions interestingly have a period ratio of 2.512 with σ = 0.026. The requirement of longterm stability implies that resonant solutions are selected, preferably the 2:1 case, but also 5:2 for a few fits. Other period ratios do not occur and no nonresonant secularly stable model was found in the entire parameter space. The stability island, together with all longlived outliers (dark red) in the center panel of Fig. 3, represents the 2:1 case, while the four 5:2 solutions lie in the light red region around e_{o},e_{o} = 0.12,0.14 on a χ^{2} ridge. The most longlived of the four has τ = 6.7 × 10^{6} yr with χ^{2} = 106.0 and the bestfitting τ = 1.1 × 10^{6} yr with χ^{2} = 105.6, clearly inferior to the island solutions. We exclude the 5:2 resonant solution at the 99.3% confidence level. Truly longterm stable solutions that provide good fits to the data exist only in the 2:1 meanmotion resonance explored in more detail in the next section.
Fig. 5 Best fit of the Keplerian twoplanet model to the eclipsetime variations of NN Ser. The lower panels display the residuals from the fit for two different time intervals. Data points with errors larger than 1 s are omitted. 
Reducing the χ^{2} limit to the 1σ level of 97.9 leaves us with 56 solutions with τ > 10^{6} yr, 44 from Run 1 and 12 from Run 2 (Fig. 3, righthand panel, and Fig. 4, green histograms). In the multidimensional parameter space, these solutions lie close together and all parameters have narrow quasiGaussian distributions with welldefined mean values, as shown for the periods and the period ratio in Fig. 4. Some of the parameters are strongly correlated. We quote in Table 2 the mean values and the standard deviations of the respective parameters with all other parameters free. We searched for the best fit within the island of stability in Run 3, a subset of which contains 82 models with τ > 10^{8} yr and χ^{2} between 95.56 and 95.60 for 109 degrees of freedom. The parameters of the bestfitting model are listed separately in Table 2. In the bottom part of the table, we list the astrocentric semimajor axes and planetary masses derived from the observed periods and LTTamplitudes, assuming coplanar edgeon orbits. With masses sin i_{o} M_{o} = 7.0 M_{Jup} and sin i_{i} M_{i} = 1.7 M_{Jup}, the two companions to NN Ser qualify as giant planets for a wide range of inclinations.
Fig. 6 Residuals for the outer planet after subtracting the contribution of the inner planet (top) and vice versa (bottom). The abscissa divisions are the same for the two panels. The residuals from the fit are identical to those of Fig. 5. 
Fig. 7 Temporal evolution of selected osculating orbital elements of a model, starting from the bestfit Keplerian parameters of Table 2. The first 3050 yr of the greater than 10^{8} yr lifetime are displayed. The lefthand panels show the eccentricities and arguments of periapse for the motion of the center of mass of the binary, the top righthand panel shows the resonant angle Θ_{1}, and the bottom righthand panel the minimum separation between the two planets for successive orbits of the inner planet. 
Figures 5 and 6 show the data along with the bestfit model of Table 2. The ordinate O − C_{lin} in the top panel of Fig. 5 is the deviation of the observed mideclipse times from the underlying linear ephemeris of the binary (1)which combines the fit parameters T_{bin} and P_{bin}. The 1σ errors quoted in parentheses reflect the width of the stability island. The residuals O − C_{ell} of the 76 individual MONET mideclipse times from the fit with two elliptical orbits are listed in the last column of Table 1. They have an rms value of 0.34 s. For clarity in the presentation, we show the MONET data in Figs. 5 and 6 only in the form of the weighted mean values for the 19 groups of mideclipse times introduced in Sect. 2 (cyanblue filled circles). Their rms value is 0.14 s. If a third periodicity hides in the residuals, its amplitude does not exceed 0.25 s. Figure 6 shows the contributions O − C_{i} of the outer and O − C_{o} of the inner planet to the LTT signal. The data points in these graphs are obtained by subtracting the LTT contribution of the mutually other planet in addition to the linear term from the observed mideclipse times. It is remarkable how well the twoplanet model fits the data, since the observations now cover nearly a complete orbit of the outer planet and two orbits of the inner one.
4.2. Temporal evolution of the planetary system of NN Ser
In this section, we explore the temporal evolution of the secularly stable twoplanet models that start with orbital elements defined by the Keplerian fits to the data. We find that all longlived models of Fig. 3 (central and righthand panels) adhere to the 2:1 meanmotion resonance. This holds for the models in the stability island, but also for the solutions that correspond to the skerries surrounding it. Their isolated character probably results from the increasing difficulty of finding a set of parameters that secures resonance, as the start parameters deviate from their optimal values. In all these solutions, the meanmotion resonant variable Θ_{1} = λ_{i} − 2 λ_{o} + ω_{i}, a function of the planet longitudes λ_{i} and λ_{o}, librates about zero and the secular variable Δω = ω_{i} − ω_{o} circulates. This agrees with the expected behavior of a twoplanet system with a mass ratio M_{o}/M_{i} > 1 (Michtchenko et al. 2008).
Figure 7 shows the evolution of selected parameters for the bestfitting model of Table 2 over the first 3050 yr of the lifetime, which exceeds 10^{8} yr. The two principal periods of the system (Rein & Papaloizou 2009) are 105 yr and 736 yr. Both are prominent in the librations of Θ_{1}. In the shorter period, lowamplitude antiphased oscillations of the semimajor axes occur (not shown). In the longer period, Δω circulates, antiphased oscillations of the eccentricities take place, and the minimum separation between the two planets reoccurs. The protection mechanism due to the 2:1 resonance always keeps the separation above 2.15 AU, effectively limiting the mutual gravitational interaction (bottom right panel). As an illustration, we show the orbits at the time of peak eccentricity e_{i} in Fig. 8. The system is locked deep in the 2:1 meanmotion resonance, as demonstrated by the nearzero mean values of Θ_{1}, averaged over successive 736yr intervals. For the first 20 000 yr of the evolution, the 26 values of ⟨ Θ_{1} ⟩ can be represented by an underlying linear and a superposed sinusoidal variation with a period of 3450 yr and an amplitude of 0.86°. The linear component has a fitted slope of deg yr^{1}, entirely consistent with zero. All longlived solutions that start from fits in the stability island behave similarly to the bestfit model, in the sense that all of them have Θ_{1} librating and Δω circulating. This also holds for the longlived outliers in the central panel of Fig. 3 that have τ > 10^{8} yr. They differ in the secular period, which ranges from 330 to 1700 yr.
Fig. 8 Astrocentric orbits for t = 471 − 487 yr after the begin of the calculation. The locations of the periapses are marked “P”, the solid dots indicate conjugation, and the open circles opposition. Orbital motion is counterclockwise. 
All models considered thus far involved prograde coplanar edgeon orbits. We calculated a few models with different inclinations i_{i} and i_{o} of the inner and outer planet as a first step toward a more comprehensive study of NN Ser’s planetary system. A common tilt in the planetary orbits, which enhances the planetary masses is limited to 25°, beyond which instability increases rapidly. Similarly, the calculations limited the mutual tilt of the two orbits with respect to each other to 5°.
5. Summary and discussion
We have presented 69 new mideclipse times of NN Ser, which cover the minimum and subsequent recovery of the O − C_{lin} residuals expected from our previous model in Paper I. The data allowed us to derive a significantly improved twoplanet model for NN Ser based on the interpretation of the observed eclipsetime variations in terms of the LTT effect. Combined with extensive stability calculations, we find that the only model that simultaneously fits the data and is secularly stable involves two planets locked in the 2:1 meanmotion resonance. We did not find any good fits that are nonresonant and stable. Apart from differences in the periods and mass ratio, the NN Ser system bears resemblance to the HD 128311 planetary system (Sándor & Kley 2006; Rein & Papaloizou 2009). The best fit implies eccentricities e_{o} = 0.14 and e_{i} = 0.22 for the outer and inner planets, respectively, and a mass ratio M_{o}/M_{i} = 4 for coplanar orbits. As expected theoretically for a system with a more massive outer planet (Michtchenko et al. 2008), the temporal evolution of the model that starts from our best fit involves a libration of the resonant variable Θ_{1} and a circulation of Δω. Preliminary stability calculations for tilted orbits suggest that deviations from coplanarity with the binary orbit cannot be large.
Eclipsetime variations in PCEB seem to be ubiquitous (e.g. Beuermann et al. 2012; Parsons et al. 2010b; Qian et al. 2012, and references therein) and the plausibility of explaining them by the LTT effect has increased by the definitive discovery of planets orbiting nonevolved close binaries with KEPLER (e.g. Doyle et al. 2011; Orosz et al. 2012; Welsh et al. 2012). That the orbital periods in the two types of systems differ by one to two orders of magnitude is a selection effect, because the LTT effect increases with the orbital period, while the radial velocities and transit probabilities decrease. Contrary to radial velocities measured by line shifts, however, the interpretation of the eclipsetime variations in terms of a displacement of the binary along the line of sight is not unique, since eclipsetime variations can also be produced by mechanisms internal to the binary (e.g. Applegate 1992). Most authors, however, consider this action as inadequate for explaining the magnitude of the observed variations (e.g. Brinkworth et al. 2006; Chen 2009; Watson & Marsh 2010). Doubts about the LTT interpretation nevertheless remain, raised by such problematic cases as HU Aqr (Goździewski et al. 2012; Horner et al. 2011; Hinse et al. 2012) and QS Vir (Parsons et al. 2010b). Resolving these cases remains an important task. Presently, NN Ser represents the best documented case in favor of the LTT hypothesis with an agreement between data and model at the 100 ms level, and it is also the prime contender for an observational proof of the required strict periodicity of the signal, given enough time.
The evolutionary history of planetary systems orbiting PCEB may be complex. Planets either existed before the CE and had to pass through the envelope or they were formed in it. In the former case, the orbit of any preexisting planet is severely affected by the loss of typically more than one half of the mass of the
central object. In the latter case, formation depends critically on the conditions in the expanding envelope. In both scenarios, migration may drive a planet pair into resonance, but predicting the properties of the emerging postCE system is a difficult task. Establishing the structure of systems like NN Ser may provide the observational basis for such a program.
There is no official nomenclature for the naming of exoplanets. At the time of Paper I, the A&A editor considered assigning the first planet discovered around a binary the small letter “b” as illogical and suggested that the two planets orbiting the binary NN Ser should be named “c” and “d”, “ab” being the binary components. In the nomenclature proposed by Hessman et al. (2010), the planets in NN Ser would be labeled (AB)b and (AB)c. Since neither convention has been officially adopted, we prefer the neutral designations “outer” and “inner planet”, which avoids any misunderstanding, as long as there are only two planets.
Acknowledgments
We thank the referee for the careful reading of the paper and the constructive and helpful comments, which served to significantly improve the presentation. This work is based in part on data obtained with the MOnitoring NEtwork of Telescopes (MONET), funded by the Alfried Krupp von Bohlen und Halbach Foundation, Essen, and operated by the GeorgAugustUniversität Göttingen, the McDonald Observatory of the University of Texas at Austin, and the South African Astronomical Observatory. We thank George Miller (UT Austin), Bernhard Starck (Gymnasium Frankenberg), Ronald Schünecke (Evangelisches Gymnasium Lippstadt), Paul Breitenstein (Gymnasium Münster) and Alexander Schmelev (Institut für Astrophysik Göttingen) for taking a total of ten of the eclipses listed in Table 1, some of them as part of the MONET PlanetFinders program that involves high school teachers and students.
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Online material
All Tables
All Figures
Fig. 1 Residuals of the mideclipse times of NN Ser since 2007 relative to the linear ephemeris of Eq. (1), with the bestfitting model of Fig. 5 and Table 2 shown as solid curve. See text for further explanation. 

In the text 
Fig. 2 Temporal variation of the mideclipse times of NN Ser as predicted by our bestfitting dynamical twoplanet model of Table 2 relative to the underlying linear ephemeris of the binary and folded over the orbital period of the outer planet. The first two orbits, covering a time interval of 30 yr (black curves), and the next 29 orbits (30–500 yr, red curves) are displayed. 

In the text 
Fig. 3 Results of fitting the Keplerian twoplanet model to the eclipsetime data of NN Ser. Left: χ^{2} distribution in the e_{o}, e_{i} plane. The + sign indicates the minimum χ^{2} and the contour lines refer to the increments Δ χ^{2} indicated in the legend. Center: maximum lifetime τ of models with χ^{2} at the 99.9% confidence level (Δ χ^{2} = + 13.8 for two degrees of freedom). For ease of comparison, the χ^{2} contours of the left panel are included. Right: same for models with χ^{2} at the 68.3% confidence level (Δ χ^{2} = + 2.3). 

In the text 
Fig. 4 Histograms of the orbital periods P_{o} and P_{i} of the outer and inner planets of NN Ser and the period ratio P_{o}/P_{i} for solutions with a lifetime exceeding 10^{6} yr and a χ^{2} corresponding to the 99.9% confidence level (yellow) or the 68.3% confidence level (green) for two degrees of freedom. 

In the text 
Fig. 5 Best fit of the Keplerian twoplanet model to the eclipsetime variations of NN Ser. The lower panels display the residuals from the fit for two different time intervals. Data points with errors larger than 1 s are omitted. 

In the text 
Fig. 6 Residuals for the outer planet after subtracting the contribution of the inner planet (top) and vice versa (bottom). The abscissa divisions are the same for the two panels. The residuals from the fit are identical to those of Fig. 5. 

In the text 
Fig. 7 Temporal evolution of selected osculating orbital elements of a model, starting from the bestfit Keplerian parameters of Table 2. The first 3050 yr of the greater than 10^{8} yr lifetime are displayed. The lefthand panels show the eccentricities and arguments of periapse for the motion of the center of mass of the binary, the top righthand panel shows the resonant angle Θ_{1}, and the bottom righthand panel the minimum separation between the two planets for successive orbits of the inner planet. 

In the text 
Fig. 8 Astrocentric orbits for t = 471 − 487 yr after the begin of the calculation. The locations of the periapses are marked “P”, the solid dots indicate conjugation, and the open circles opposition. Orbital motion is counterclockwise. 

In the text 
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