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One of the main objectives of stream surveys in nearby spiral galaxies is to compare the observations with cosmological simulations to ascertain whether the frequency and surface brightness of the detected stellar streams are consistent with those predicted by models. Observational modeling and theoretical understanding of such diffuse and intricate features requires specifically tailored cosmological numerical simulations. There are two main difficulties for these models: i) sufficiently fine mass and spatial resolution is needed to recover complex and delicate tidal features around Milky Way mass halos; and ii) a sufficient volume is required to build a statistically meaningful sample of host galaxies. To simultaneously meet these two requirements, state-of-the-art cosmological simulations are needed. For this reason, there are still a limited number of models of the stellar halos of Milky Way-like galaxies. Examples include the models by Bullock & Johnston (2005) (also described in Chapter 5 of this volume) and high resolution models of individual stellar halos for Milky Way-like galaxies based on the (dark matter only) Aquarius suite of simulations.

Numerical simulations (e.g., Johnston et al. 2008, Cooper et al. 2010) can be used as a guide to what we may expect; models suggest that remnants of recent (0-8 Gyr ago) accretion events, which correspond to the last few tens of percent of mass accretion for a Milky Way-like spiral, should remain visible as substructures in presently observable stellar halos. These models also make predictions about what can be physically inferred from an external, “global” view of stellar halos. For example, from the shapes of debris remnants, we can infer the basic dynamical properties of the progenitor – Johnston et al. (2008) demonstrates that tidal debris of different morphologies each occupy different regions in the time of accretion vs. orbital eccentricity/energy plane (see their Fig. 3), and that surface brightness also gives evidence of both the time of accretion and the luminosity of a remnant's progenitor (see Fig. 4 of Johnston et al.). While this gives some general insight into properties of the accretion events, there is considerable degeneracy in the inference of such properties, which may best be thought of as providing reasonable estimates upon which to base specific modeling of the accretion events.

Figure 6 compares the predicted morphologies of model debris structures from Johnston et al. (2008) to observed structures in external galaxies, demonstrating that we see examples of the remnants predicted by the models in the local Universe. The classifications of debris structures suggested by Johnston et al. include: “great circles” – streams that arise from satellites on nearly circular orbits that were accreted ∼ 6−10 Gyr ago; “cloudy” morphologies (also known as “shells” or “plumes”) resulting from recently-accreted (less than ∼ 8 Gyr ago) satellites that were on rather destructive, radial orbits; and “mixed”-type tidal remnants from ancient accretion events (more than ∼ 10 Gyr ago) that have had time to phase mix and become nondescript. While these simulations make detailed predictions about the total number, frequencies, and specific properties of halo substructures, there is no analogous observational data set to which these simulations can be compared en masse. As demonstrated in Figure 6, some observational examples of different stream morphologies have been identified in the local Universe, including “great circles” in NGC 5907 (Shang et al. 1998, Martínez-Delgado et al. 2008) and M63 (Chonis et al. 2011, see also Figure 1.1), the “plume”-like feature (or “umbrella”) in NGC 4651 (Martínez-Delgado et al. 2010), and a feature of “mixed” morphology around NGC 1055 (Martínez-Delgado et al. 2010).

Figure 6

Figure 6. Bottom row: externally-viewed snapshots showing the surface brightness of individual accretion relics in models of Milky Way-like stellar halos by Johnston et al. (2008). The three main morphological types identified in this study are illustrated: “great circle” streams (or “arcs”; left panel) arise from satellites accreted ∼ 6-10 Gyr ago on nearly circular orbits; “cloudy” morphologies (also dubbed “shells” or “plumes”; middle panel) arise from accretion events within the past ∼ 8 Gyr that fell in on eccentric orbits; and “mixed”-type tidal remnants (right) arise from ancient (more than 10 Gyr ago) accretion events that have had ample time to fully mix along their orbits. Top row: observational archetypes of each type of tidal debris from the survey by Martínez-Delgado et al. (2010): great circle stream in NGC 5907 (left), shell-like features around NGC 4651 (middle); and “mixed” debris near NGC 5866 (right panel).

Generally, it is easiest to infer the physical properties of great circle streams, as the great circle is a reasonable tracer of the progenitor's orbit. Analytical relationships derived by Johnston et al. (2001) may be used to estimate the accretion time and total (dark matter+stellar+gas) initial mass of the progenitor of a stream on a great circle. Surface photometry across streams in multiple filters can be used to infer the stellar populations and total stellar mass for the stream (similarly to integrated light mapping of stellar mass surface density in external galaxies; e.g., Zibetti et al. 2009). Variations in the optical colors along the stream can be used to infer changes in the mean properties of the stellar populations, but are on the whole less diagnostic than similar studies using resolved stars in the Local Group. However, extraction of meaningful properties from the integrated light of streams requires S/N > 5-10 above the local background fluctuations over a sufficiently wide area to cover the full stream. Thus, not only must the image be deep, but the backgrounds need to be well characterized. This can be seen in Figure 1, which shows that while hints of debris may be identifiable in SDSS imaging (e.g., Beaton et al. 2014), extraction of physical properties requires deep, well-characterized imaging.

Cooper et al. (2010) coupled the Aquarius simulations to a state-of-the-art semi-analytic model known as GALFORM, which computes the properties (mass, size, star formation history and chemical abundance) of the galaxy forming in each dark matter halo. The GALFORM model is constrained through statistical comparisons to collective properties of the cosmological galaxy population (for example, optical and infrared luminosity functions), and by requiring that the surviving counterparts reproduce the observed size-luminosity relationship for Milky Way dwarfs. To meet these constraints, this technique demands fine-grained simulations such as Aquarius in order to adequately resolve the star-forming cores of satellite halos. This approach results in a set of dynamically self-consistent N-body realizations of stellar halos and their associated tidal streams at a resolution beyond the reach of current hydro-dynamical simulations (e.g., Abadi et al. 2006), and without the need to invoke many of the approximations required by previous models (e.g., Bullock & Johnston 2005). The individual star formation history of each satellite (and hence properties such as stellar mass, luminosity and net metallicity) can be studied alongside the full phase-space evolution of its stars.

Current models have sufficient numbers of particles to resolve the main contributors of bright, coherent substructures that are similar to tidal features detected in the current imaging surveys (e.g., Martínez-Delgado et al. 2010). Thus it is possible to make sky-projected snapshots of these stellar halos from different viewing angles and a selected photometric band. Each model halo has experienced a unique merging history and provides predicted surface brightness, morphologies and overall distribution of the observable streams, survival of the progenitors and stellar populations (or mean colors), that can be compared with observational data. These snapshots can be used as the input source for creating a mock catalogue of synthetic images, which can be generated by adding simulated streams to real images including all the observational effects of the telescopes (e.g., typical sky noise, flat-field corrections, surface brightness limits, etc.) and contamination from other galactic substructures (e.g., the stellar disk that is not directly simulated by models of this type). An example of this technique using the Bullock & Johnston (2005) models as input is shown in Figure 7. These mock observations give preliminary predictions for the level of substructure detectable in the stellar halo of a nearby spiral for the typical surface brightness limits of current imaging surveys. Moreover, additional observational properties, like chemical compositions (both [Fe/H], [α/Fe]), can be explored as they relate to the other properties of substructure, the most significant being the time since accretion and luminosity (or stellar mass), as was done by, e.g., Font et al. (2006).

Figure 7

Figure 7. Expected halo streams around a Milky Way-like galaxy from a simulation (Bullock & Johnston 2005). The figure shows an external perspective of one realization of a simulation within the hierarchical framework, with streams resulting from tidally disrupted satellites. The snapshot on the left is 300 kpc on a side (the virial radius for a Milky Way sized galaxy), and illustrates the result of a typical accretion history for a Milky Way-like galaxy. Right panels: theoretical predictions for the detectable tidal features in the same halo as the left panel, but assuming three different surface brightness (SB) detection limits: A: µlim = 28, Blim = 29 and C: µlim = 30 mag arcsec−2. Each snapshot is 100 kpc on a side. No discernible substructure is predicted for surveys with SB limits brighter than ∼ 27–28 mag arcsec−2 (e.g., POSS-II and SDSS). This result also shows as the number of tidal features visible on the outskirts of spirals depends dramatically on the SB limit of the observations. Moreover, the brightest substructures tend to be from the most massive satellites, which sample relatively rare accretion events (Johnston et al. 2008, Gilbert et al. 2009).

The Bullock & Johnston (2005) halos all have some structure visible at surface brightnesses of ∼ 27-28 mag arcsec−2. To surface brightnesses of ∼ 29 mag arcsec−2 there is ∼ 1 visible stream in each simulated halo, and typically about 2 visible streams above ∼ 30 mag arcsec−2. The majority of the substructure, and thus the majority of the accretion history, is at surface brightnesses fainter than ∼ 30 mag arcsec−2. The degeneracy between the luminosity of the satellite, its accretion time, and its surface brightness is studied in depth by Johnston et al. (2008). Our inability to see the fainter streams, which were either accreted earlier or come from lower luminosity progenitors, implies that our view of halo substructure beyond the Local Group will be dominated by the most recent and/or most massive accretion event, and, in either case, the most metal rich populations (Johnston et al. 2008, Gilbert et al. 2009). Moreover, more massive accretion events tend to preferentially populate the innermost regions of the halo (Rproj < 30 kpc), as dynamical friction, which is more effective for more massive satellites, will cause the orbit of the progenitor to degrade (Johnston et al. 2008). Thus, our view of extragalactic tidal streams at relatively shallow surface brightness limits (i.e., those of POSS and SDSS) is highly biased to a specific subset of accretion events that are relatively rare for Milky Way sized galaxies.

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