 By this criteria any satellites with long periods are excluded, which are falling in for the first time. Whenever we write period instead of half period, we mean the half period multiplied by two. Figure 1. Evolution of the mass and scale radius of the VL1 main halo (solid line). The scale radius is more difficult to constrain for high redshifts. The dashed line is the best fitting power law going back 9 Gyrs in time.

The orbit recovery models
We consider the following models for our orbit integration: • The fiducial model (F ) uses a static, spherical, NFW potential (Navarro et al. 1996). In F, we take parameters from the best fit to the VL1 main halo at redshift z = 0: M200 = 1.77 × 10 12 M⊙, R200 = 389 kpc and rc = 24.6 kpc (Diemand et al. 2007a).
• The dynamical friction model (DF ) uses the same potential as in model F with dynamical friction forces added (Chandrasekhar 1943). We use a radially varying Coulomb logarithm as in Hashimoto et al. (2003). We also explore the impact of mass loss of the infalling satellites, and the mass growth of the VL1 halo with time (see Figure 1). We fit simple functions to the subhalo mass loss history, and VL1 main halo mass growth with time as in Zhao (2004).
• The triaxial model (T ): This uses a triaxial NFW potential as in Kuhlen et al. (2007) and Guedes et al. (2009). We use axis ratios: q = 0.83 and s = 0.8, as measured for the VL1 main halo at outer radii at z = 0.
• The double halo mass model (2M ): This uses the potential as in model F, but doubling the mass of the main halo (doubling the scale radius has a negligible effect).
These different models allow us to assess the importance of systematic effects on our orbit recovery for real Milky Way dwarf data. In practice, only model F can be realistically applied to real data since we do not know the mass loss history of the Milky Way dwarfs, nor the mass growth history or shape of the Milky Way halo. We use models DF, T and 2M to explore the systematic impact of this poor knowledge on our orbit recovery.

Results
In this section, we measure how well we can recover the orbits of subhalos in our VL1 data set. We use each of the the recovery models presented in §2.1, with and without measurement errors and assess our results using criteria (i), (ii) and (iii) in §1.

Perfect data
To assess the model systematics, we first consider the case without position and velocity measurement uncertainties.   2, 3 and 4 show our recovery for each of the models F, DF, T and 2M for the '1D' pericentres rp and apocentres ra, the 3D pericentres rp and apocentres ra and the half period t 1/2 over N orbits. The coloured bars show the percentage of subhalos with fractional error Q = 0 − 0.1 (blue), 0.1 − 0.3 (cyan), 0.3 − 0.5 (green), 0.5 − 0.8 (red) and > 0.8 (magenta). We define the relative error Q as: for the '1D' pericentre (left) and 3D pericentre (right), where rp,t is the true pericentre and rp is the recovered pericentre. Q is defined similarly for the other orbit quantities ra, ra and t 1/2. Note that in comparing rp and rp,t, the limited time resolution of the VL1 simulation can lead to an additional error (that causes us to always over estimate rp,t).
We tested the effect of this, finding that only orbits with rp 20 kpc are affected. The additional error due to finite time resolution is small compared to our model uncertainties.

Recovering the 1D pericentre and apocentre
From Figure 2, we see that the last apocentre is well recovered in our fiducial model F. No satellites have their last apocentre recovered with greater than 30% error. This is perhaps to be expected. Subhalos are most likely to be at apocentre now and so their present position already gives a reasonable constraint on ra. The DF and T models both improve the recovery for the most recent apocentre. This suggests that the primary reason for our error in the most recent apocentre recovery is a combination of wrong halo shape and dynamical friction, where the halo shape is more significant. These effects will be swamped by other effects for the apocentre recovery further back in time (c.f. 2.2.3). By contrast, model 2M gives a poor performance, with only ∼ 20% of the subhalos having better than 10% error on the last apocentre. This demonstrates that it is critical to have a good estimate of the mass of the Milky Way interior to the orbits of its dwarfs.
The last pericentres are significantly less well recovered than the apocentres, with only 20% having an error better than 10% in our model F. Here the halo shape plays a major role, with model T giving a significantly better performance (∼ 40% recovered to better than 10%); dynamical friction (model DF ) has little impact on the recovery. To better understand why the pericentre recovery fails, some typical example orbits from each Qp bin are given in Figure 5. As can be seen, orbits that have a well recovered last rp are short period orbits. These are well recovered in model F and well recovered in the triaxial model T that has the correct current halo shape. However, long period orbits sample the potential over several giga years backwards in time. Here even our triaxial model (fit to the VL1 halo at redshift z = 0) fails and the rp recovery is poor (see the magenta orbit).

recovering orbits over several periods: The problem of group infall
Recovering orbits over several periods backwards in time is more challenging. The fraction of orbits with better than 10% error in ra in our triaxial model T drops by ∼ 40% by Na = 2; while there are no well recovered rp in our model F by Np = 3. The triaxial model T gives an improvement in the recovered rp, but no improvement in ra. Note that there are some fluctuations in this trend due to small number statistics in the Na = 3 and Na = 4 bins. So individual orbits, that are improved in one of the models have a higher impact than they would have in a larger sample.
To understand why the errors on ra grow so rapidly backwards in time, we give some example orbits recovered at Na = 2 in Figure 6. As with the rp (Figure 5), the longer period orbits are more poorly recovered.
There is a significant class of orbits -50% -with 0.1 < Qa < 0.3 (cyan) that are less well recovered at Na = 2. We explore a typical orbit in this class in more detail in Figure 7. The middle panel shows our orbit recovery for the most massive subhalo in this group in model F (black dashed line), T (magenta dashed line), DF (blue dash-dotted line) and an orbit integrated with dynamical friction, but without mass loss of the satellite nor mass evolution of the main halo (green dotted line). Notice that none of our models give a good fit to the orbit. Accounting for dynamical friction and mass loss (see right panel) as well as using the triaxial potential significantly overestimates the apocentre. We investigated this orbit further and find this satellite to be part of a loose group. The left panel shows the orbit of the satellite (black) and all subhalos that were initially close in phase space to this satellite (red). This grouping was found by determining all satellites closer than 4rt, where rt is the tidal radius defined in Diemand et al. (2007a), for at least six time outputs. These 'loose groups' usually break up at first pericentre, which causes an energy change that results in lower 2nd apocentre than we would expect from any of our integration models.
75% of the the satellites in the cyan bin were found to be falling into the galaxy as part of a group. This suggests that the group infall statistics recently determined by Li & Helmi (2008) may be a lower bound. We defer a detailed analysis of the statistics of loose group infall to future work, but note here that it appears to be responsible for many of our poor orbit determinations at Na 2.

recovering the orbital phase
In Figure 4, we show how well we recover the half orbital period over N 1/2 orbits assuming perfect data. Notice that the period is well-recovered up to even two orbits backwards in time. However, in determining the phase of the orbit, such period errors accumulate. This makes it challenging to try and match pericentre passages with observed star formation histories for the dwarfs. By N 1/2 = 2, our typical phase error is 0.6 Gyrs, at N 1/2 = 3 it is already 0.8 Gyr. Recall that this is for perfect data that have no measurement errors. With current proper motion errors, recovering the orbital phase is simply not possible (see §2.2.6). Figure 3 shows how well we recover the 3D pericentre and apocentre for perfect data. Here having an accurate halo shape is vital. Our results in model F, even for a single orbit, are poor. Even in our triaxial model T the results