Accuracy of Analytic Potentials for Orbits of Satellites Around a Milky Way-like Galaxy: Comparison with N-Body Simulations

Abstract

To study the orbits of satellites, a galaxy can be modeled either by means of a static gravitational potential or by live N-body particles. Analytic potentials allow for fast calculations but are idealized and non-responsive. On the other hand, N-body simulations are more realistic but demand higher computational cost. Our goal is to characterize the regimes in which analytic potentials provide a sufficient approximation and those where N-bodies are necessary. We perform two sets of simulations, using both Gala and Gadget, in order to closely compare the orbital evolution of satellites around a Milky Way-like galaxy. Focusing on the periods when the satellite has not yet been severely disrupted by tidal forces, we find that the orbits of satellites up to ${10}^8{\mathrm{M}}_{\odot }$ can be reliably computed with analytic potentials to within 5% error if they are circular or moderately eccentric. If the satellite is as massive as ${10}^9{\mathrm{M}}_{\odot }$ then errors of 9% are to be expected. However, if the orbital radius is smaller than 30 kpc then the results may not be relied upon with the same accuracy beyond 1–2 Gyr.

1. Introduction

Stellar streams are tidal debris left by dwarf galaxies or globular clusters. The study of stellar streams around the Milky Way and, consequently, of their orbits has gained considerable interest due to the outstanding results from the Gaia mission in the last decade [1,2,3,4,5]. The first streams, discovered around the turn of the century, were Sagittarius [6,7] and Palomar 5 [8,9]. Currently, over 120 streams are known [10,11,12,13].

The morphological and kinematic data from stellar streams can be used to constrain the orbits of their progenitors, including in cases where the progenitor is a globular cluster [14]. Additionally, they can be used to estimate the mass of the Milky Way [15].

In order to perform an orbital calculation, a given model of the galaxy needs to be assumed. Analytic models of the Milky Way have been tailored [16,17,18], including multiple components. Such time-independent models are commonly used to study the dynamics of satellite galaxies and globular clusters. However, disregarding the accretion history of the Milky Way is known to introduce significant uncertainties [19,20]. Recently, more realistic Milky Way models have been proposed, taking into account the presence not only of the Sagittarius dwarf spheroidal (Sgr) but also of the Large Magellanic Cloud (LMC) [21,22]. Simulations indicate that gravitational acceleration due to the presence of the LMC induces a response from the Milky Way that, in turn, may significantly affect the orbits of satellites such as the Sgr dwarf; see Gómez et al. [23]. The arrival of the Gaia Sausage–Enceladus (GSE) galaxy has also been modeled with simulations; see Naidu et al. [24].

Analytic potentials and self-consistent N-body simulations are two commonly adopted approaches to studying the dynamics of satellite galaxies. The fundamental distinction between them is that N-body particles are responsive while an analytic potential is static. This means that the particles that represent the mass of the main galaxy move self-consistently according to a time-dependent evolving gravitational potential. Furthermore, the total gravitational potential in an N-body simulation takes into account not only the mass of the main galaxy but also the mass of the satellite itself, which may not be negligible. The forces are computed at each time step using the current mass distribution.

In contrast, if the main galaxy is represented by a fixed analytic potential then there is no time evolution of its internal structure. If one is focused mainly on the satellite orbit then the analytic potential may be a sufficient approximation. It is expected that the internal distribution of mass in the stellar disk would have a small impact on the global orbital properties. For example, whether the disk mass is axisymmetric or instead redistributed in spiral arms or a bar should, in principle, have a small impact on the satellite orbit as a whole. Yet, there are known secondary effects, such as gaps in the streams induced by the rotation of the bar [25]. In any case, a static disk ceases to be a good approximation if the satellite comes too close to it, and even more so if the satellite punctures or crosses the region of the stellar disk itself. Furthermore, the presence of substructures in the halo may also affect the orbits for stars and, thus, the clustering of streams in action space [26]. Perhaps more importantly, reflex motion of the host galaxy is understood to require time-dependent treatment of the perturbation of the gravitational potential [22,23]. In a time-independent potential, the center of mass of the Milky Way is artificially fixed and the reflex motion due to the pull of the LMC cannot be captured.

In terms of the physical results of the calculations, N-body simulations are generally regarded as being more realistic due to their time-dependence and the self-consistent response of the particles. However, from a practical point of view, N-body simulations are far more costly. In an analytic galactic model, once the potentials are chosen the gravitational forces are immediately known at any given coordinate. If the satellite is represented as a point mass then the orbital calculations are nearly instantaneous. This is valuable if one wishes to carry out large sets of orbits covering the parameter space of interest. The same cannot be said for N-body simulations. Even if modern collisionless N-body codes are extremely efficient, preparing, performing and processing thousands of runs would be challenging in practice.

Additionally, the disruption of satellites may be explored in simulations where the satellite is represented by N-body particles but the galaxy remains a fixed potential. This intermediate approach may be regarded as a compromise in terms of realism and practical expedience, and it is commonly adopted if the focus of the analysis is on the development of the stellar streams. The advantage is that the number of particles in the satellite can be relatively small and, thus, the number of mutual gravitational forces to be computed is drastically reduced if the stellar disk and dark matter halo do not contribute. Nevertheless, even if this approach is focused on the tidal forces that give rise to the streams it still assumes that the global orbit is accurately computed. This may not necessarily be the case under certain conditions. Additionally, other methods exist to mimic the formation of tidal streams, such as so-called particle-sprays [27], which rely on tracer particles.

Thus, it would be useful to provide quantitative constraints regarding the conditions under which an analytic potential is sufficient and the regimes wherein full N-body simulations are needed. It is anticipated that three main parameters should be critical in this context: satellite mass, orbital radius and time scale. One expects that a low-mass satellite in a distant orbit will behave approximately as a test particle, while a massive satellite colliding with the disk would render the analytic approximation invalid. It is also expected that the longer the simulations progress, the larger the deviations will grow. The goal of this work is to map the regimes of interest and quantify the errors associated with the orbital deviations. Here, we aim to take a step in this direction by exploring a limited region of the possible parameter space.

This paper is organized as follows. In Section 2, we give the initial conditions of the simulations. Section 3 present the analysis and results. In Section 4, we offer discussion and conclusions.

2. Simulations

The physical system we wish to model consists of a small satellite galaxy orbiting a Milky Way-like galaxy. We calculated the orbits of the satellites with two approaches. First, the main galaxy is represented as a fixed analytic potential. In this case, the satellite is a point mass. In the second approach, both the main galaxy and the satellite are represented as N-body particles. In this section, we will describe the methods employed, the parameters of the initial conditions and the sets of models.

2.1. Analytic Potentials

In the first approach, the main galaxy consists of the sum of three components: halo, disk and bulge. The masses and sizes of these components are given as follows. The dark matter halo is represented by a Hernquist potential Hernquist [28] with mass $M_{\mathrm{h}}=1\times {10}^{12}{\mathrm{M}}_{\odot }$ and scale length $a_{\mathrm{h}}=47$ kpc. The bulge is also modeled as a Hernquist potential with $M_{\mathrm{b}}=1\times {10}^{10}{\mathrm{M}}_{\odot }$ and $a_{\mathrm{b}}=1.5$ kpc. The disk has an exponential profile with mass $M_{\mathrm{d}}=5\times {10}^{10}{\mathrm{M}}_{\odot }$. The radial and vertical scale lengths of the disk are, respectively, $R_{\mathrm{d}}=3.5$ kpc and $z_{\mathrm{d}}=0.7$ kpc. In practice, this disk potential is obtained indirectly as a sum of three Miyamoto–Nagai [29] disk potentials that are meant to approximate the potential generated by a double exponential disk; see [30]. With these choices of parameters, the main galaxy is approximately comparable to the Milky Way.

The satellite galaxy is set on a polar orbit around the main galaxy, i.e., while the galactic disk lies on the $xy$ plane, the orbit of the satellite is perpendicular to it, confined to the $yz$ plane by construction. To achieve this, the satellite is placed on the z axis at a given distance above the center of the main galaxy. Its initial velocity points entirely in the y direction. In one set of models (‘circular’ orbits), the value of $v_y$ is chosen to be the circular velocity at that radius. In the other set of models (‘eccentric’ orbits), the initial radii are kept as before, but now the initial velocity is a fixed factor of 0.7 of what the corresponding circular velocity would have been. This results in non-circular orbits. In other words, these are orbits with a given fraction of the angular momentum of the circular cases. We can use the first pericenter and apocenter to estimate the eccentricity, even though these orbits are not truly elliptical. The initial eccentricities are approximately 0.3. For each set, six initial radii are adopted in the range 20–70 kpc. Their values and their corresponding initial velocities $v_y$ are given in

Table 1. Initial radii and initial velocities of the orbits.

	Circular	Eccentric
${\mathbf{R}}_{\mathbf{0}}$	${\mathbf{v}}_{\mathbf{y}}$	${\mathbf{v}}_{\mathbf{y}}$
(kpc)	$\mathbf{\left(}\mathbf{km}{\mathbf{s}}^{-\mathbf{1}}\mathbf{\right)}$	$\mathbf{\left(}\mathbf{km}{\mathbf{s}}^{-\mathbf{1}}\mathbf{\right)}$
20	168	117
30	166	116
40	164	115
50	161	112
60	157	110
70	154	108

. There is naturally small variation in the range of velocities, since the rotation curve is approximately flat in that radial range.

These potentials and initial conditions were constructed using the Gala1 package [18] and were then integrated forward in time for 3 Gyr. The Cartesian coordinates of the orbits as a function of time will be used for the subsequent comparisons in the remainder of the paper.

2.2. N-Body Simulations

Here, we present the properties of the N-body simulations. The initial conditions of the main galaxy were created with the same profiles and the same parameters, i.e., masses and scale lengths identical to those described in Section 2.1. These N-body initial conditions were realized with the galstep2 code [31]. The numbers of particles were $N_{\mathrm{h}}=2\times {10}^5$, $N_{\mathrm{b}}=4\times {10}^4$ and $N_{\mathrm{d}}=1\times {10}^5$, for the halo, bulge and disk, respectively.

In the N-body simulations, the satellite galaxy was not a simple point mass; rather, it was represented by a distribution of particles. To model the mass of a spheroidal galaxy, we adopted a simple Plummer profile [32]. In this approximation, the Plummer sphere is meant to represent the entire mass of the galaxy, without discriminating between baryons and dark matter. We created three versions of the satellite galaxy, varying only the masses: ${10}^7$, ${10}^8$ and ${10}^9{\mathrm{M}}_{\odot }$. In all cases, the Plummer scale length was $a=0.5$ kpc, meaning that their central concentrations were not the same. Likewise, in all three cases, the number of particles was $2\times {10}^4$. This uniform choice ensured an adequate numerical resolution for all the simulations, and it meant that the satellite mass was more finely resolved than the main galaxy. Alternatively, an exceedingly high number of particles would have been demanded if we had imposed both a fixed mass resolution across different simulations and a minimum number for the lowest-mass satellite. The initial conditions for the N-body satellite were created with the Plummer IC3 code.

Prior to the actual runs, both the main galaxy and the satellite were separately relaxed in isolation for a period of 1 Gyr. This ensured that any numerical transients would have dissipated before the beginning of the actual simulation. Importantly, we verified that all the Plummer sphere models were stable in isolation. Next, the satellites were placed in the same initial configurations as described in Table 1. The N-body simulations were carried out with the Gadget-4 code [33,34] for 3 Gyr.

Table 2. Labels and parameters of the N-body simulations.

Label	${\mathit{M}}_{\mathbf{sat}}$	${\mathit{R}}_0$	Orbit Type
	(${\mathrm{M}}_{\odot }$)	(kpc)
M7R20c	${10}^7$	20	circular
M7R30c	${10}^7$	30	circular
M7R40c	${10}^7$	40	circular
M7R50c	${10}^7$	50	circular
M7R60c	${10}^7$	60	circular
M7R70c	${10}^7$	70	circular
M8R20c	${10}^8$	20	circular
M8R30c	${10}^8$	30	circular
M8R40c	${10}^8$	40	circular
M8R50c	${10}^8$	50	circular
M8R60c	${10}^8$	60	circular
M8R70c	${10}^8$	70	circular
M9R20c	${10}^9$	20	circular
M9R30c	${10}^9$	30	circular
M9R40c	${10}^9$	40	circular
M9R50c	${10}^9$	50	circular
M9R60c	${10}^9$	60	circular
M9R70c	${10}^9$	70	circular
M7R20e	${10}^7$	20	eccentric
M7R30e	${10}^7$	30	eccentric
M7R40e	${10}^7$	40	eccentric
M7R50e	${10}^7$	50	eccentric
M7R60e	${10}^7$	60	eccentric
M7R70e	${10}^7$	70	eccentric
M8R20e	${10}^8$	20	eccentric
M8R30e	${10}^8$	30	eccentric
M8R40e	${10}^8$	40	eccentric
M8R50e	${10}^8$	50	eccentric
M8R60e	${10}^8$	60	eccentric
M8R70e	${10}^8$	70	eccentric
M9R20e	${10}^9$	20	eccentric
M9R30e	${10}^9$	30	eccentric
M9R40e	${10}^9$	40	eccentric
M9R50e	${10}^9$	50	eccentric
M9R60e	${10}^9$	60	eccentric
M9R70e	${10}^9$	70	eccentric

presents the complete list of N-body simulations: each of the three satellites was set on six initial radii, resulting in 18 runs for the circular case and another 18 runs for the eccentric case.

3. Analysis and Results

3.1. Circular Orbits

Our analysis is focused on the comparison between the analytic orbits and the N-body orbits. In the case of the analytic approach, the trajectory is obtained simply by the successive coordinates of a point mass, calculated with Gala. In the case of N-body satellite, the trajectory could be obtained in principle from the coordinates of its center-of-mass. However, tidal forces give rise to stellar streams, causing the satellite to be partially disrupted. In some cases, a dense core remains; in other cases the disruption is severe.

The formation of stellar streams is illustrated in Figure 1, which shows the time evolution of three models (M7R30c, M8R30c and M9R30c). These are all circular models with radius 30 kpc. Each column of Figure 1 gives the time evolution of a given satellite mass, with the blue/green/purple points corresponding to ${10}^7$, ${10}^8$ and ${10}^9{\mathrm{M}}_{\odot }$, respectively. Different levels of disruption are perceptible already in the first Gyr of evolution, in the sense that the purple points remain mostly coherent, while the green and blue ones develop significant streams. By the end of the simulation, the M9 satellite remains bound. The M8 satellite still retains a discernible dense core, in spite of its very long streams. In the M7 case, however, the disruption was such that a clearly detectable remnant is no longer present; the density is roughly the same along the blue stream.

These considerations point to the difficulty of locating the core of the satellite galaxy in such intense disruption cases. At some point in the evolution, the center of the satellite is intrinsically ill-defined. For the models and times where the core is indeed detectable, the peak-finding method consists in the following. We compute the local density at the position of each particle, sort them by density and draw a sphere around the position of the particle of highest density. The center of mass is computed using all the particles within that sphere. The radius of the sphere is shrunk and the process repeated iteratively until the center of the sphere converges within a given tolerance. This method successfully finds the density peak by progressively eliminating the contribution of the streams, whose arcs would otherwise cause the center-of-mass to be found outside the true core. In the severely disrupted galaxies, this method is not always reliable. Even though a density peak is always found, the density structure of some streams is such that there is not one single peak but rather a chain of comparably dense knots. In these cases, finding a given peak is not meaningful. As an alternative to finding the maximum density, other possibilities include finding the minimum potential at each time or keeping track of the IDs of the most bound particles (at $t=0$). Because of the severe disruption of the M7 models, all methods gave roughly similar results in the end.

Being able to properly track the position of the satellite, we can compare the orbits. Figure 2 compares all the models in the circular cases. The three masses are shown as columns; the six radii as rows. The gray dotted lines are the analytic orbits, and the colored lines are the N-body simulations. Towards the lower left of Figure 2, there are a few ill-determined points that were kept and will be dealt with later. A noticeable feature of Figure 2 is that most of the orbits remain rather stable. The departures are seen mostly towards the right (higher satellite masses) and towards the bottom (smaller orbital radii). These will be quantified next.

Figure 3 presents the galactocentric distance of the satellite as a function of time. From top to bottom, each panel corresponds to an initial radius, from 70 kpc to 20 kpc. Note that the vertical scales of the panels are of the same range in kpc. The differently colored lines are the satellite masses. The comparison with the analytic orbits in this case is straightforward because the radii are constant in time in the circular models. The purple lines in Figure 3 confirm quantitatively that the M9 models are the ones that diverge the most and the soonest.

In order to capture a global indication of the degree of departure between the analytic and N-body orbits, we computed a simple relative difference $|R_N-R_{\mathrm{ana}}|/R_0$, where $R_N$ is the galactocentric radius of the satellite as calculated from the particles in the N-body simulation at each time, $R_{\mathrm{ana}}$ is the radius from the analytic orbit, and $R_0$ is the initial radius. For the circular cases, $R_{\mathrm{ana}}=R_0$ at all times, but not for the eccentric cases.

Figure 4 displays the mean relative errors with their deviations for all circular models. They are shown for each initial radius. For the low satellite masses M7 and M8, most of the models have mean errors below $\sim 5\%$, except M8R20c, which is slightly larger. The most pronounced difference is in the M9 models, where M9R20c reaches more than 15% error. It is also interesting that in the M9 cases, at least, the error decreases towards larger orbital radii. Nevertheless, M9R70c still gives larger errors than its counterparts of lower mass at the same radius.

3.2. Eccentric Orbits

We now turn to the analysis of the eccentric orbits. Likewise, we present first an illustrative example of the time evolution of 3 out of the 18 eccentric models. These are shown in Figure 5, which displays models of initial radius 30 kpc, with the three masses. Regarding the disruption of the satellites, one notices a behavior similar to what had been seen in Figure 1. Namely, the M9 satellite remains bound, the M8 develops long streams while retaining a core, and the streams of M7 cannot be said to have one distinctive core.

It is understandable that the degrees of disruption are approximately comparable, since the internal structures of the satellites themselves are the same as in the circular cases, as are the initial radii of the equivalent subset. One meaningful difference is that here in the eccentric orbits the satellites visit regions of the potential more internal than 30 kpc. There, they undergo stronger tidal forces that tend to disrupt them more than in the circular case.

A striking feature in Figure 5 is that eccentric orbits are, in general, not closed. It is worth noting that the streams are not a trail of the orbit. In other words, the shape of the streams at a given time does not trace the past loops of the satellite trajectory. Only under certain circumstances do they temporarily match, often close to the pericenter, but not in general.

Figure 6 present the full comparison of all the eccentric orbits, contrasting the analytic orbits in dashed gray lines to the N-body orbits in color. As was the case with the previous analysis, the ill-determined density peaks are also kept in the plot. They tend to occur at the smaller radii of M7 and M8. Here again, in Figure 6, we find that the main result is that the departures between the orbits are more pronounced towards the M9 side and towards small radii.

The galactocentric distances of the satellites are shown as a function of time in Figure 7. Now, for the eccentric orbits the analytic black line is itself not constant but oscillatory. It remains true that the purple lines are the ones that generally diverge the most. Regarding the initial radii of the models, there is a clear tendency in the sense that the R70 cases follow the analytic orbit quite well until the end of the simulation, but the departures increase systematically towards smaller radii.

The gradual offset between the curves of Figure 7 presents an additional difficulty. The direct subtraction of the curves would lead to large errors at times when an expressive phase difference has developed. One possible alternative way to capture the similarity between the curves would be to compute the normalized cross-correlation (NCC), which would reveal whether the signals have similar periods. The cross-correlation is a sliding inner product, which serves to quantify the similarity between two time series even if they are displaced relative to one another, e.g., [35,36]. Given two time series $(t_1,y_1)$ and $(t_2,y_2)$, the normalized cross-correlation coefficient is defined as

$$NCC={\displaystyle \frac{{\sum }_i\left(y_1\left(t_i\right)-\overline{y_1}\right)\left(y_2\left(t_i\right)-\overline{y_2}\right)}{\sqrt{{\sum }_i{\left(y_1\left(t_i\right)-\overline{y_1}\right)}^2{\sum }_i{\left(y_2\left(t_i\right)-\overline{y_2}\right)}^2}}}$$

(1)

For example, the NCC coefficient is quite close to unity in all the R70 and R60 cases. In the poorly matched cases of R30 and R20, NCC reaches as little as 0.3–0.4. Unlike the pointwise subtraction, the NCC coefficient takes the entire signal into account. However, we can apply specific cuts to recompute the NCC coefficient of the curves of Figure 7 in the following way. For example, we find that M9R20e gives NCC $=0.3$. This poor match is probably due in large part to the later times. We can then cut the last times and recompute NCC, taking into account only the new interval $t<t_{\mathrm{cut}}$. If this process is repeated for several cuts, we can plot the recomputed NCC as a function of $t_{\mathrm{cut}}$ and then ask at which cut the NCC would give an arbitrary threshold of NCC $=0.95$. The results of this procedure are shown in Figure 8. In the example of M9R20e, we find that if times later than 1.0 Gyr are excluded then the NCC would be 0.95. In Figure 8, the final values of each curve are the full NCC if no cut has been applied (i.e., if the cut is $t<3$ Gyr). By applying this procedure, we found in each model the timescale during which there is a good match between the orbits.

With this information, we can compute the mean relative errors in two ways. First, in the same way that was applied in the circular cases, taking all points into account. These are the full symbols shown in Figure 9. A similar behavior is found, with slightly larger errors. As expected, the largest errors within the eccentric models are in M9 and in the smaller radii. Secondly, we have also recalculated the mean relative errors of the eccentric models, but now taking into account only $t<t_{\mathrm{cut}}$, with the cuts provided by the NCC $=0.95$ criterium from Figure 8. The recalculated errors are the open symbols with lighter colors in Figure 9. The outstanding errors decrease and become more comparable to the other points.

Finally, we can compare the results of both sets of models together. In Figure 10, the mean relative errors are shown for all 36 models using the uncut data, i.e., all the times. Here, we see that the largest errors are found in the bottom-right corner (large mass, small radii). The errors would reach as much as 17% and 23% in the worst cases of circular and eccentric models, respectively. In contrast, Figure 11 presents the same structure, but recalculated using the $t<t_{\mathrm{cut}}$ restriction. Notice that the color bar is the same as in Figure 10. To allow for a fair comparison between the sets of models, the same cuts were also applied to the circular models. Otherwise, the means would have been calculated with considerably different numbers of points. If the comparison is limited to these times, then Figure 11 shows that the maximum errors are of 9% in the extreme case M9R20 for both orbital types. More specifically, we see in Figure 11 that the greatest deviations are present in M9, which is the most massive satellite, with errors from 5 to 9%. In the other models, the errors are found within 1–4% and 1–5% in circular and eccentric orbits, respectively.

4. Discussion and Conclusions

In this paper, we have analyzed a variety of polar orbits of a satellite galaxy around a Milky Way-like central galaxy. The sets of simulations cover different orbital radii and different satellite masses. Both circular and eccentric orbits were explored. We have compared the orbits computed with N-body simulations and with fixed analytic potentials.

Regarding the formation of stellar streams, we found systematic differences between the satellites with masses ${10}^7$, ${10}^8$ and ${10}^9{\mathrm{M}}_{\odot }$. The most massive satellite always remains bound as a coherent structure. The intermediate-mass satellite develops extended stellar streams, while retaining a central core. The low-mass satellite becomes fully disrupted by tidal forces. This systematic trend is already perceptible as early as $t=0.5$ Gyr and grows more pronounced in time. Illustrative examples were presented in the paper for the case of orbits with initial radius 30 kpc, but this general behavior also holds for all of the 36 simulated models. The degree of disruption in our simulations is more tightly connected to the intrinsic properties of the satellites themselves than to the orbital properties. Since the three Plummer spheres have the same scale length, this means that the low-mass satellite is less centrally concentrated, while the massive satellite is more centrally concentrated. For this reason, the satellite of ${10}^7{\mathrm{M}}_{\odot }$ was naturally more prone to be disrupted.

An important caveat of this work is the limitation imposed by adopting fixed scale lengths in Plummer spheres that represent the satellite galaxies. This uniform choice leads to central concentrations that are somewhat artificial. More realistic initial conditions could have been created by imposing a mass-dependent scale radius inspired by empirical scaling relations from dwarf galaxies [37,38,39]. On the one hand, the global orbital properties depend more strongly on the masses and not on the inner density of the satellite; and orbits are the main focus of this paper. On the other hand, the disruption of the satellite and the generation of streams indeed depend importantly on the central concentration. In this sense, the conclusions regarding stream formation should be regarded with caution and not generalized, since they rely on rather artificial setups. Although the gravitational dynamics of the streams are computed reliably, they stem from initial conditions that may not be realistic across all explored masses. Systematically exploring the effects of concentrations was beyond the scope of the present paper, since introducing the Plummer scale length as an additional dimension would significantly increase the number of combinations in the parameter space of the simulations. Exploring the resilience of a dwarf galaxy against tidal disruption as a function of central density is a compelling topic for future systematic investigation.

It is interesting to notice that the stream is composed of a leading tail and a trailing tail. Bearing in mind that the motion of the satellite is clockwise in Figure 1, it is clear that the leading tail is inside the circular orbit, and the trailing tail outside. This is because on the leading side the stars that are moving ahead have slightly larger orbital energies.

The disruption of the satellite introduces a practical difficulty in determining its center. We have applied an iterative algorithm that successfully locates the density peak even when the streams are well developed. However, in the low-mass cases there is no longer one single density peak to be found, but the stream is constituted rather by a string of clumps with similar density. In this sense, there is no longer a well defined center.

To confront the evolution of the analytic versus N-body orbits, we compared the galactocentric distance of the satellite as a function of time. For the circular orbits, straightforward mean differences are immediately applicable as a metric. For the eccentric cases, it happens that the small phase differences hinder the direct calculation of radial differences. The calculation of normalized cross-correlation coefficients proved useful to capture the similarity of orbits with the same period even when they were slightly lagged. With this method, we were also able to quantify the time intervals during which the orbits match within a given accuracy.

The main result of this work is a quantitative description of the errors when comparing analytic versus N-body orbits. The satellite whose orbits diverge the most are always M9, the most massive in our analysis. With the M9 masses, the orbital errors are in the range 5–9%. For the remainder of the models, the errors are generally limited to 1–4% in circular orbits and 1–5% in eccentric ones. It should be emphasized that these summary statistics hold only if the time cuts are applied. In other words, R30 and R20 orbits are only this reliable within the first ∼1–2 Gyr. Furthermore, in the massive M9 case, the R50 and R40 cases also need to be considered only in the first ∼2 Gyr. For the radii in question here, these times correspond to roughly three orbital periods.

In the case of the non-circular orbits, it is true that the satellite visits radii somewhat smaller than the initial radius. In fact, such proximity to the galactic center, even if momentary, is what causes larger deviations in the eccentric models. Nevertheless, we only explored one moderate eccentricity, meaning that the results cannot be generalized to arbitrarily elongated orbits. For example, a nearly radial orbit that leads the satellite to fall into the center of the galaxy would clearly violate the assumptions of the parameter space explored here.

In summary, this means that orbits of satellites up to ${10}^8{\mathrm{M}}_{\odot }$ can be reliably computed with analytic potentials to within 5% error if they are circular or moderately eccentric. However, if the orbital radius is smaller than 30 kpc then the results may not be relied upon with the same accuracy beyond 1–2 Gyr. If the satellite is as massive as ${10}^9{\mathrm{M}}_{\odot }$ then errors of 9% are to be expected, and eccentric orbits even as large as 50 kpc will cease to be reliable beyond 1–2 Gyr. For such cases, and others beyond the parameters given here, full N-body simulations would be preferable.

Orbits in the range 10–20 kpc were not presented in this work, because placing satellites of masses ${10}^{7-9}{\mathrm{M}}_{\odot }$ instantaneously at such small radii would lead to unreasonable initial conditions. However, in the regime of globular cluster masses similar analyses could be carried out to provide quantitative constraints on the orbital accuracy.