DARK MATTER SUB-HALO COUNTS VIA STAR STREAM CROSSINGS

An extensive set of restricted n-body simulations guides our approximate evaluation of the linear gap rate, Equation (1), in a general context. The background potential is an NFW density profile as fitted to the Springel et al. (2008) simulation of a Milky Way- or M31-like galaxy. The stream is idealized as a circular ring at either 30 or 60 kpc of 10⁴ mass-less particles. The ring of particles is initiated on tangential orbits at the local circular velocity, $v_c(r)=\sqrt{-\nabla \Phi \cdot r}$. That is, there are no random velocities, although the particles could be considered the guiding centers of a set of particles. Only one sub-halo is sent toward the ring in each simulation. An individual sub-halo is adequately modeled as a Hernquist (1990) sphere with the mass-scale radius relation found in the Aquarius simulations, since for our purposes the exact details of the internal mass distribution are not important. The sub-halo motion is simply at a fixed velocity in a straight line, so it crosses the ring only once, which avoids the complications of a sub-halo orbiting and having multiple ring crossings. A single sub-halo is started at a distance of the ring radius from the ring. The time step is 0.001 unit of r/v_c(r) at 30 kpc or about 1.4 × 10⁵ yr. The ring of particles responds to the sub-halo and background halo potential but has no self-gravity. A series of simulations is done varying the angle of the tangent with respect to the ring in the direction of motion over θ = (41, 60, 90, 104, 120, 139) deg, and the angle with respect to the plane of the ring of ϕ = (0, 30, 60, 90, 270, 300, 330) deg. Together these two distributions non-redundantly coarsely sample an isotropic velocity dispersion field. The encounter velocities are varied over (0.3, 0.5, 0.8, 1.2, 1.6) times the circular velocity. For an isotropic velocity dispersion of sub-halos in our potential the typical ring encounter velocity is close to v_c(r). The simulations are done for 10⁶, 10⁷, 10⁸, and 10⁹ M_☉ sub-halos. In total 9310 encounters are simulated for a ring.

The outcome of a typical set of encounters after 6.92 Gyr is shown in Figure 1. For purposes of easily visible illustration we chose a relatively massive sub-halo, 10⁸ M_☉, which has a scale radius of 0.83 kpc. The sub-halo is moving at 1.2v_c. The top two rows show impact parameters of 0.06 kpc, where the sub-halo significantly penetrates the ring. The bottom row shows an encounter with an impact parameter of 1.5 kpc, about twice the sub-halo scale radius. The outcome at late times in all cases is that the ring is primarily perturbed perpendicular to the plane with loops pulled out. Note that in Figure 1 the vertical extent of the loops is stretched by a factor of 220. The ratio of the upward and downward loop heights depends on the angle ϕ, as illustrated in the two upper rows. The upward or downward extension also varies with time as the perturbed particles pass through the orbital plane. For the large b relative to the R_s passage of in the bottom row of Figure 1, the vertical deflection of the ring particles is somewhat larger than in the upper two rows, but there is essentially no visible gap in the ring density. The top two rows suggest that the density gap depth and length is insensitive to the angle ϕ, the angle relative to the orbital plane.

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The gaps in these completely cold rings have a very simple morphology, whereas the gaps created in more realistic stellar streams with internal velocity dispersion and a finite width are much more complex (Carlberg et al. 2011; Yoon et al. 2011). In particular, real gaps are often slanted to the centerline. Old streams can be subject to so many encounters that the remaining segments of the stream are relatively dense clumps smaller than the gaps between. If the progenitor is on a fairly epicyclic orbit the tidal stripping episodes occur at perigalacticon and lead to additional structure in the stream. Therefore, the simplified model presented here is intended to be a statistical guide to the analysis of streams, not a detailed description of any particular stream.

We measure the xy projected linear density relative to the unperturbed density using the distance Δd_xy between every third particle. The relative density is then nΔd_xy/(6πr), where n is the number of particles in the ring and r is its radius. Figure 2 shows that the density profile of the gap is effectively independent of ϕ, the encounter angle with respect to the plane. The particles that are stretched out in the loop pile up at the edges of the gap. The dependence of the depth of the density dip in the gap with b is shown in Figure 3, showing that although a loop is pulled out there is little density dip for encounters significantly beyond the sub-halo scale radius.

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2.1.1. Critical Impact Parameter

The relative density at the bottom of the dip is displayed as a function of the ratio of the impact parameter to the scale radius, b/R_s(M), in Figure 4. The density measurements are averaged over all angles, θ and ϕ. The results are displayed for velocities of 0.8, 1.2, and 1.6 v_c. Since the ring particles are moving at the circular velocity, there is a range of encounter velocities with the particles in the ring. If the sub-halo velocity is significantly below the circular velocity, there will be no very low velocity encounters in the frame of an individual star and the sub-halo, so on the average the gaps will be less deep. However, since encounters below 0.8v_c comprise only about 10% of the encounters, we leave them out of the average.

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We need to determine the maximum impact parameter that creates a visible gap. We will model the behavior of the density dip as a function of b/R_s(M) as a step function at a value α(M, r, t), with visible gaps created for b ⩽ αR_s(M) and no gaps for larger b values. We evaluate the value of α = b/R_s(M) at one-third of the unperturbed density. The resulting critical values of b/R_s(M) are function of both mass and time, as shown in Figure 5. Using a density dip of half yields larger values of α, but its use gives values within our overall modeling uncertainties. The function α(M, r, t) has a significant time behavior. At early times, roughly before 2 Gyr, the gaps are not well developed and we exclude them from the fit. Over a Hubble time the value of α nearly doubles. The time behavior is mainly relevant for dynamically young streams or the young part of streams that are still being created where we will expect the stream to be far less gapped than our average value.

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Fitting to the data of Figures 4 and 5, we find

$$\begin{eqnarray} \alpha _{33\%}(M,r,t) &=&(1.71 + [0.28 + 0.10 \log _{10}{M_8}] [t -7 ])\nonumber\\ && \times \left( {r\over {30\,{\rm kpc}}} \right)^{0.23} M_8^{0.12}, \end{eqnarray} \tag{ 2 }$$

where t is in Gyr and normally we will simply use the value at 7 Gyr as a reasonable average value and ignore the time dependence for this initial study.

2.1.2. The Gap Creation Rate

We now have the elements to evaluate the gap creation rate, Equation (1). The maximum impact parameter which creates a gap is b_max = α(M, r, t)R_s(M), where we will use the 7 Gyr value of α, roughly the average over a Hubble time. The mean approach speed to the ring is the integral over v_⊥, which is $\sqrt{2/\pi }\sigma \simeq 0.80\sigma$ for a Gaussian velocity distribution. The rate integral then becomes an integral entirely in the mass distribution,

$$\begin{equation} {\mathcal R}_\cup (\hat{M},r) = 0.80 \pi \sigma n(r) \int _{\hat{M}}^\infty N(M) \alpha _{33\%}(M) R_s(M) dM. \end{equation} \tag{ 3 }$$

The Aquarius simulations (Springel et al. 2008) find that R_s ≃ 0.83M^0.43₈ kpc and that the mass distribution function is N(M)dM = 234M^−1.9₈dM₈. The mass function gives the total number of sub-halos contained within the volume, V₅₀, that has a mean interior density 50 times the critical density which has a radius of 433 kpc (Springel et al. 2008). Putting these pieces together we find

$$\begin{equation} {\mathcal R}_\cup (\hat{M},r) = 490 \,\sigma V_{50}^{-1} {n(r)\over n_0} \int _{\hat{M}_8}^\infty {M_{8}}^{-1.9} \alpha _{33}(M_8){M_{8}}^{0.43} \, dM_{8}, \end{equation} \tag{ 4 }$$

where $\hat{M}_8\equiv \hat{M}/10^8 {\,M_\odot }$ is the smallest mass halo that creates a visible gap. The normalizing density in the V₅₀ of the radial density variation, n(r), of the sub-halo population volume is n₀. The mass integral is dominated by the smaller masses so we have set the upper mass limit to infinity. Evaluating the integral, choosing a local σ = 120 km s⁻¹, and converting to astronomical units, we find

$$\begin{eqnarray} {\mathcal R}_\cup (\hat{M},r) &=& 0.0066\, \left({r\over {100\,{\rm kpc}}}\right)^{0.23}{n(r)/n_0\over {6}}\nonumber\\ && \times {\sigma \over {120 {\,\hbox{km\,s}^{-1}}}} \hat{M}_{8}^{-0.35} \, {\rm kpc}^{-1}\,{\rm Gyr}^{-1}, \end{eqnarray} \tag{ 5 }$$

where we have used the Springel et al. (2008), finding (their Figure 11) that the sub-halo density at 100 kpc is six times the mean within the V₅₀ volume. The rate at which gaps are created increases as the minimum mass sub-halo that can create a noticeable gap goes down, i.e., $\mathcal {R}_\cup \propto {\hat{M}}^{-0.35}$. Halo-to-halo sub-halo number differences (Chen et al. 2011) will lead to ±50% or so variation in the rate in a specific halo.

An important observable quantity is the length of the gaps. We measure the length as the distance in the orbital plane between the two outside points at one-third of the vertical maximum of a loop. The length of a gap should be related to the distance that the loop of ring particles is pulled out of the orbital plane, that is, as the vertical extension of the ring increases, so does the length of the visible gap. This means we first need to understand the dynamics of the vertical perturbations. The impact approximation and the tidal approximation are the two general approaches used to describe the interaction of a satellite, such as a sub-halo, with a set of particles, the stars in the stream. The impact approximation is best suited to calculating deflections of individual stars from the rapid passage of a satellite and the tidal approximation is best suited to problems involving the differential gravitational field across a finite sized object. Both approaches are useful for calculating the gravitational heating of the perturbed objects and at their root both have the same underlying gravitational dynamics.

The impact approximation assumes that both a sub-halo and a star stream orbit can be approximated as straight lines and sets aside the possibility that orbital resonances play a significant role. A sub-halo of mass M moving past a star at a relative velocity v with a distance of closest approach b will induce a velocity perpendicular to its direction of motion in the direction of closest approach of δv_⊥ = 2GM/bv. Since many of the encounters partially intersect the stream, the relevant mass will be M(b), the mass enclosed within the impact parameter, not the total mass of a sub-halo. For an orbit that is approximately circular at radius r and taking δv_⊥ to be in the vertical direction, conservation of angular momentum requires that the maximum orbital deviation out of the plane is z_m = rδv_⊥/v_c.

We need to average over sub-halos crossing at all angles to the ring tangent and to the orbital plane. To a first approximation, forces in the horizontal plane do not add to the angular momentum around the normal to the plane of the orbit, but do act to coherently increase the epicyclic motion around the guiding center, but that makes little difference to the stream density. A sub-halo moving an angle to the orbital plane initially pulls the stream down and then up, which evolves into a paired set of peaks perpendicular to the orbital plane, as shown in Figure 1. The vertical extent of the perturbation is measured as the vertical distance from the positive peak to the negative peak, rather than the measurement of the absolute value of the offset from the initial orbital plane for a horizontal sub-halo passage. The resulting Z⁺ − Z⁻ distribution is shown as a binned average in Figure 6. The mean is about 30% higher than the prediction with about a factor-of-two scatter above and below the line. For our use the scale offset is not an issue.

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The tidal approximation is an alternative description of the encounter of a sub-halo with a stream. The Hill radius, $R_{{\rm Hill}}=r\sqrt [3] {M/(3M_t)}$, inside of which the gravitational field of the sub-halo, mass M, begins to dominate over the gravitational field of the total mass interior to the orbit of the stream, M_t. The plot of the vertical deflections against Hill radius is shown in Figure 7 using the same data and same vertical scale as Figure 6. Although the vertical perturbation and the Hill radius are strongly correlated a larger scatter remains than with the impact approximation. Including a power of b shows that R_Hillb^−1/3 removes some of the scatter. This is hardly surprising since with the addition of v this becomes the impact approximation formula, multiplied by a constant. An alternative tidal quantity is the differential tidal field, which is proportional to M/b³. A plot of the vertical deflections against M/b³ is mainly scatter. We conclude that the impact approximation is better at predicting the vertical deflections than the tidal approximation.

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2.2.1. The Characteristic Velocities

Star streams have a mean particle motion close to the circular velocity. Therefore, the speed of an encounter between a sub-halo and a star in the stream, which is required for the impact approximation, will on the average be higher than the mean circular velocity. A consequence is that low velocity encounters of sub-halos with the stream are extremely improbable. Taking the mean encounter along the stream to be v_c and the velocity dispersion in the two axes perpendicular to the stream as σ, the typical relative velocity is $\sqrt{v_c^2+ 2\sigma ^2}$. Using σ ≃ v_c/1.8 in the outer part of a galactic halo then the typical sub-halo star relative encounter velocity is 1.27v_c. The full triple velocity integral has the complication that there is a singularity at a relative velocity of zero. Physically, this would correspond to the velocity of the stream's progenitor sub-halo. Inserting a very small "core" velocity of about one meter per second, we find that the mean encounter velocity is 1.077v_c. Combining these factors we find that on the average:

$$\begin{equation} z_m = {2GM r \over {1.08 v_c ^2b }}. \end{equation} \tag{ 6 }$$

2.2.2. Gap Lengths

The length of a sub-halo-induced gap as a function of the sub-halo parameters is needed to determine what gaps are large enough to be visible. We measure the gap length at one-third of the maximum vertical height as a conservative approach. The basis of a relation is visible in Figure 8 in which gap lengths relative to z_m are plotted for a ring at 60 kpc. We see that for substantially penetrating encounters, b/R_s ⩽ 2, the ratio of the gap length to the vertical extent, z_m = Z⁺ − Z⁻, is roughly a constant at a given mass. A ring at 30 kpc has ratios of length to vertical extent almost exactly twice as large. That is, the lengths of the gaps are nearly independent of radius, but since z_m includes an r factor the ratio is dependent on the radius. Defining β(M) as the length of the gap relative to the value of z_m we can fit for the weak mass dependence:

$$\begin{equation} \beta _{33\%}(M,r) = 60 \left({r\over {30\,\rm kpc}}\right)^{-0.63} M_8^{-0.11}. \end{equation} \tag{ 7 }$$

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At radii below 30 kpc the β value becomes somewhat larger than Equation (7) gives. Combining the average β with the impact Equation (6) for the vertical extent the gap length at 33% of full height is

$$\begin{equation} l(M) = \beta _{33\%}(M,r) {2 G M(R_s)r\over { 1.08 R_s(M) v_c^2}}. \end{equation} \tag{ 8 }$$

Approximating a sub-halo density profile as a Hernquist profile, we find that the mass inside the scale radius is M(R_s) = 1/4M (Hernquist 1990). Evaluating the numerical factors we find

$$\begin{equation} l(M_8) = {8.3}\left({r\over 30\,{\rm kpc}}\right)^{0.37} M_8^{0.41}\,{\rm kpc}. \end{equation} \tag{ 9 }$$

We now have an equation that gives the typical gap length as a function of sub-halo mass averaged over the velocity distribution. The gap length–mass relation can either be used as a consistency test or taken a step further and be used to eliminate the unobservable minimum mass parameter from the rate equation in favor of a minimum gap length.

The width of the stream and the minimum visible gap size are linked through orbital dynamics. The stream width is largely the result of stars emerging through the tidal Lagrangian points with low initial velocity dispersion. The epicyclic approximation describes the subsequent motion of the stars with small velocity dispersions around a guiding center. An epicycle has comparable amplitudes in the direction of the motion and perpendicular to the motion; hence the stream width and the random motion along the stream are of nearly equal size. Provided that the phase angles of the motion are not significantly correlated, any induced gap narrower than the stream width will be blurred out. Therefore, we make the assumption that the width of the stream is equal the minimum gap size and use it to estimate the minimum mass sub-halo that creates a gap:

$$\begin{equation} w=l(\hat{M}) . \end{equation} \tag{ 10 }$$

Solving Equation (10) for $\hat{M}$ we substitute it into Equation (5) to give the gap creation rate as a function of the width of stream,

$$\begin{eqnarray} {\mathcal R}_\cup (w,r) &=& 0.059 \left({n(r)/n_0\over {6}}\right)\nonumber\\ && \times \left({ r\over {\rm 100\, kpc}}\right) ^{0.55} w^{-0.85}\,{\rm kpc}^{-1}\,{\rm Gyr}^{-1}, \quad\ \ \end{eqnarray} \tag{ 11 }$$

for w in kiloparsecs, where we have used Equations (2) and (7). Equation (11) recasts the gap rate in terms of an observable, assuming that the CDM halo substructure model gives the correct density of sub-halos. Over the range of 10–100 kpc where streams are generally found the factors of n(r)r counteract each other to reduce the radial variation. Equation (11) is an observational test of the CDM sub-halo prediction.

The overall M31 PAndAS imaging project and the streams are described in McConnachie et al. (2009) and Richardson et al. (2011), and the details of the NW stream are described in Carlberg et al. (2011). Here we use the linear density profile to estimate the number of gaps. In Figure 9, we show the density field as a function of the angle around the star stream, with all quantities measured as reported in Carlberg et al. (2011). The plots are made with a moving average over 5 and 11 bins of 025 or roughly 2.5 and 5.5 kpc, respectively. To determine what scales in the stream appear to be physically correlated, we plot the cross-correlation of the stream density minus the mean value with itself in Figure 10. We see that the stream densities are positively correlated over the first three bins of 025, after which the fluctuations are essentially random. This correlation length is comparable to the width of the stream and helps validate the assumption that leads to Equation (10). Therefore, the smallest size scale we search for gaps in these data is about 1° or 2 kpc.

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To count the gaps we need to have a good measure of the level of statistical fluctuations. The cross-correlation of the data with itself, at shifts beyond 5 bins where there is no intrinsic signal, provides an accurate internal estimate of the variance of the density field. The product of two normally distributed random variables is distributed like a modified Bessel function of the second kind and has a large kurtosis of 9, compared to a normal distribution with a kurtosis of 3. However, the sum of many products (115 in our 025 sample) converges to a Gaussian distribution in accord with the central limit theorem. The kurtosis of n sums of products is 3 + 6/n and therefore declines fairly quickly to the Gaussian value of 3 with increasing n. The variance around the mean density of the data within the stream region, prior to background subtraction, is 1.22 (stars/kpc)², and a comparable 1.33 in the background region. The background subtracted mean stream density has a variance of 6.16 (stars/kpc)². The F ratio of the measured variance over the randomly expected variance is 6.16/(1.22+1.33) = 2.42, which has a random chance of occurrence of less than 2 × 10⁻⁵. Larger bins give even higher significance that the measured stream density has variations well in excess of what is expected from random fluctuations. The conclusion and the levels of significance are in accord with the results presented in Carlberg et al. (2011) but use an independent line of argument which does not rely on an explicit calculation of shot noise statistics.

An estimate of the number of gaps is simply to count the number of times the density crosses down and up through some density threshold, requiring that the gap be at least the size of the smooth length. For a five-point moving average at (−1σ, 0, +1σ) density levels we find (11,11,15) gaps of mean length (7.1,6.9,15.1) bins or (1.8, 1.7, 3.8) kpc. Taking the stream length to be 200 kpc and assuming an age of 10 Gyr, we find from the 12 ± 2 gaps that the M31 NW stream has been subject to a mean gap creation rate of 0.006% ± 50% gaps kpc⁻¹ Gyr⁻¹. In principle the predicted rate also includes any sufficiently massive object, but the detectable mass is well above the range for globular clusters and we do not expect molecular clouds along the large radius orbit of the NW stream.

The rate equation, Equation (5), applied to the M31 NW stream requires a minimum sub-halo mass of $\hat{M}=1.3\times 10^8{\,M_\odot }$ and implies a total of 205 sub-halos, which is a factor of 7.3 higher than 28 known dwarf galaxies of M31 (Richardson et al. 2011) but includes objects out to 433 kpc, of which the only the inner 150 kpc radius volume has been carefully searched. Therefore, the M31 stream does not demand a huge dark matter sub-halo population by itself, but this analysis does recover the well-known discrepancy which begins to appear for even fairly massive sub-halos.

A narrow stream is predicted to have more visible gaps relative to a wider stream at a larger radius. The globular cluster Pal 5 has both leading and trailing streams with the northern, trailing arm having the most data in the Sloan Digital Sky Survey (SDSS). With data made available before Data Release 1 (DR1), Odenkirchen et al. (2003) found the stream to be about 6° long and measured the width to be 0.11 kpc. With the better coverage of SDSS DR4, Grillmair & Dionatos (2006) found that the northern arm was about 185 and provided a surface density contour plot showing the considerable variation in the surface density of the stream. The density variations closest to the cluster are convincingly explained as epicyclic pileups of stars tidally stripped from the cluster (Küpper et al. 2010, 2011); however, beyond about 1 kpc (25) the variation in phase angles smooths out this behavior. From the published Figure 3 of Grillmair & Dionatos (2006) we count regions in the stream below the lowest contour to find that there are about five gaps over the outer 16° of the northern tail of Pal 5. Odenkirchen et al. (2003) estimated that the 5°–6° of stream took about 2 Gyr to fill out, which scaled to the 185 length gives an age of about 7 Gyr. With these values we estimate the linear gap creation rate in the northern stream of Pal 5 to be about 0.11 kpc⁻¹ Gyr⁻¹ to which we assign an error of 50% based on uncertainties in the gap counts and stream age.

Grillmair (2011) finds that the EBS is 0.17 kpc wide and located at a galactocentric distance of about 15 kpc. Figure 4 of his paper plots the density along the 18° of the stream with an estimate error of 30% in each bin of 05. We estimate that there are between about 4 and 12 gaps along the stream. We will take the stream age to be approximately 7 Gyr to derive a gap creation rate of 0.24 gaps kpc⁻¹ Gyr⁻¹, with about a factor-of-two uncertainty in this value.

The Orphan Stream (Grillmair 2006; Belokurov et al. 2006, 2007) has received considerable attention, culminating with a detailed orbital model (Newberg et al. 2010). Increasing the mean distance from 20 to 30 kpc revises Grillmair's (2006) estimated width of 700 pc–1 kpc. Gaps are marginally present in Figure 5 of Newberg et al. (2010) at −16° and −31°, particularly when compared to the bottom panel of their Figure 17. We estimate that there are two gaps present, although we will allow a 100% error on this estimate. Newberg et al. (2010) present a model that estimates the age of the stream to be about 3.9 Gyr. If we used the dynamical age of 3.9 Gyr rather than a uniform 7 Gyr the correction would raise the Orphan data into better agreement with the prediction.

The estimated rate of gap creation in the EBS stream with Equation (5) implies a minimum halo mass of $\hat{M}\simeq 1\times 10^5 {\,M_\odot }$. The derived minimum mass implies a total of 1 × 10⁵ sub-halos, although we note that both the mass and in turn the implied total number of sub-halos are sensitive to the gap rate estimate and the details of the model in Equation (5). Possible complications for the EBS stream is that the sub-halos are subject to erosion by the disk and bulge (D'Onghia et al. 2010) and the stream may encounter smaller scale structures such as spiral arms and molecular clouds which could add to gaps in the stream.