APOCENTER GLOW IN ECCENTRIC DEBRIS DISKS: IMPLICATIONS FOR FOMALHAUT AND ERIDANI

We first estimate the linear number density, ℓ, of an eccentric ring of particles as a function of longitude, f, in the ring. ${\ell }(f)$ is the most relevant quantity for the many images of disks that are unresolved in the radial direction. We assume that the particles form an annulus about the star, and that their eccentricity, inclination, and semimajor axis distributions are centered, respectively, on values e, 0, a and have widths of ${\rm{\Delta }}e\lt e\ll 1$, ${\rm{\Delta }}i\simeq {\rm{\Delta }}e$, ${\rm{\Delta }}a\simeq a{\rm{\Delta }}e$. Because we are considering a ring that appears eccentric overall and has a fractional radial width no larger than ${\rm{\Delta }}e$, we must constrain the longitude of pericenter ϖ to a distribution centered on $\varpi =0$ with width

$$\begin{eqnarray}&&{\rm{\Delta }}\varpi \simeq \sqrt{1-{e}^{2}}\,{\rm{\Delta }}e/e.\end{eqnarray} \tag{ 1 }$$

Also, we assume the particles' orbital phases and longitudes of the ascending node are distributed uniformly.

For a single particle in a stable elliptical orbit given by

$$\begin{eqnarray}&&r=\displaystyle \frac{a(1-{e}^{2})}{1+e\cos f},\end{eqnarray} \tag{ 2 }$$

the time-averaged linear number density along the orbit is inversely proportional to the local Keplerian velocity: ${{\ell }}_{\mathrm{single}}(f)\propto 1/v(f)$, where v(f) is given by

$$\begin{eqnarray}v(f) & = & {\left(\displaystyle \frac{{{GM}}_{* }}{a}\displaystyle \frac{1+2e\cos f+{e}^{2}}{1-{e}^{2}}\right)}^{1/2}\\ & \simeq & {v}_{0}\left(1+e\cos f+{e}^{2}\left(1-\displaystyle \frac{{\cos }^{2}f}{8}\right)\right)\\ {v}_{0} & = & \sqrt{{{GM}}_{* }/a}.\end{eqnarray} \tag{ 3 }$$

Here v₀ is simply the orbital velocity of a circular orbit with the same a. We consider only mildly eccentric disks ($e\ll 1$), so to lowest order in e the linear number density in the disk scales as

$$\begin{eqnarray}&&{{\ell }}_{\mathrm{single}}(f)\propto \displaystyle \frac{1}{v(f)}\propto 1-e\cos f.\end{eqnarray} \tag{ 4 }$$

In short, because particles orbit faster at pericenter and slower at apocenter, their number density decreases at pericenter and increases at apocenter by the same fractional amount e. Marsh et al. (2005) also calculated ${\ell }(f)$ by a somewhat different method and found a similar enhancement at apocenter.

In a disk of such particles, the overall linear density ${\ell }(f)$ is the sum of all the individual densities ${{\ell }}_{\mathrm{single}}(f-\varpi )$. For a disk with particle orbit elements within the ranges given above, the linear density dependence on f should remain close to that of ${{\ell }}_{\mathrm{single}}(f)$, that is,

$$\begin{eqnarray}&&{\ell }(f)\mathop{\propto }\limits_{\sim }1-e\cos f,\end{eqnarray} \tag{ 5 }$$

with slight variations due to the finite widths of the distributions of orbit elements.

To check our scaling argument above, we performed each Monte Carlo disk simulation by randomly drawing $5\times {10}^{4}$ disk particle orbits from orbit element distributions as follows. The semimajor axes, eccentricities, longitudes of pericenter, and inclinations were drawn from Gaussian distributions with center values and widths (σ) given by

• 1.  
  typical eccentricity e and eccentricity width ${\rm{\Delta }}e$ chosen for each disk;
• 2.  
  typical semimajor axis a = 1 and semimajor axis width ${\rm{\Delta }}a={\rm{\Delta }}e;$
• 3.  
  typical longitude of pericenter $\varpi =0$ and corresponding width ${\rm{\Delta }}\varpi$ given by Equation (1);
• 4.  
  typical inclination i = 0 and inclination width ${\rm{\Delta }}i={\rm{\Delta }}e$.

The particles' longitudes of the ascending node and mean anomalies were chosen from uniform distributions over $[0,2\pi )$.

We then measured the linear density as a function of longitude for each simulation, an example of which is shown in Figure 1. As expected, ℓ varies nearly sinusoidally with f with amplitude about equal to the eccentricity.

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In the above calculations we neglected short-term density variations. In particular, we ignored azimuthal variations in dust production caused by differences in the frequency and severity of collisions between larger planetesimals induced, for example, by perturbations from planets orbiting in the system (see, for example, Nesvold & Kuchner 2015). These should be unimportant in our work here on gravitationally bound particles in a steady-state disk, since any effects of azimuthal variations in their production rate should smear out within an orbit period.⁶ In the next section, we describe a numerical test of this effect.

We assume a passively heated disk containing dust of sizes $\{s\}$ with size distribution ${dN}/{ds}\propto {s}^{-q}$. We assume the disk is optically thin so that the thermal equilibrium condition for each grain is

$$\begin{eqnarray}&&\int d\lambda A(s,\lambda ){L}_{* }\displaystyle \frac{\pi {s}^{2}}{4\pi {r}^{2}}=\int d\lambda A(s,\lambda )4\pi {s}^{2}{\sigma }_{\mathrm{SB}}{T}^{4}(s,f)\end{eqnarray} \tag{ 6 }$$

where $r=a(1-{e}^{2})/(1+e\cos f)$ is the orbital radius, T is the dust's effective temperature, and the absorptivity $A(s,\lambda )$ is 1 when $s\gt \lambda$ and scales as ${(s/\lambda )}^{\beta }$ otherwise. We solve this iteratively for sizes s over a grid of wavelengths λ including the star's blackbody peak and sum over the sizes s, with weights assigned according to dN/ds, to find the flux at each of the desired wavelengths.

Because it is so well-observed at wavelengths from UV to millimeter-wave radio, the Fomalhaut disk (eccentricity ∼0.1, Kalas et al. 2005) provides an excellent test for our models. As discussed in Section 1, previous works have consistently found that Fomalhaut exhibits pericenter glow—i.e., is brightest at its southeast limb—at wavelengths shorter than about 250 μm (see, for example, Kalas et al. 2005; Acke et al. 2012, and references therein). However, observations at longer submillimeter wavelengths consistently suggest that the northwest limb appears brighter than would be expected in a uniform ring (Holland et al. 2003; Marsh et al. 2005; Boley et al. 2012).

We applied our dust reradiation model to the Fomalhaut system using stellar temperature ${T}_{* }=8590$ K, stellar radius ${R}_{* }=1.28\times {10}^{11}$ cm, disk semimajor axis a = 133 au, radial width ∼20 au, and eccentricity 0.1. To approximate the results of our semi-analytic model, we vary the linear mass density of the disk sinusoidally with longitude with amplitude 0.1 (simulating a disk with eccentricity ∼0.1). We explored a grid of absorptivity laws $1.0\leqslant \beta \leqslant 2.0$ and dust size distributions $3\leqslant q\leqslant 4$ to predict the apocenter/pericenter glow from visual to millimeter wavelengths. Figure 2 shows some examples of our results. Each panel shows the flux as a function of longitude, integrated over the radial width of the disk and normalized to pericenter, for 12 different wavelengths shown as different-colored curves. At the longer wavelengths, the simulated ring exhibits apocenter glow rather than pericenter glow. The location of the peak in flux shifts from apocenter (longitude = π) to pericenter (longitude = 0) with decreasing wavelength. The shortest wavelength to show apocenter glow in Figure 2 is typically 160 or 70 μm.

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We show a summary view of the apocenter/pericenter flux contrast in Figure 3, which displays the range of apocenter/pericenter flux ratios occurring across our grid of models as a function of wavelength. Again, the modeled fluxes were integrated over the radial width of the disk, since azimuthal width variations are not well resolved in the images. Here, models exhibiting pericenter glow lie below the black dotted line while models with apocenter glow lie above it. At short wavelengths, all of our models exhibit pericenter glow, while at long wavelengths, they all exhibit apocenter glow. This illustrates the competition between higher temperatures at pericenter discussed in detail by Wyatt et al. (1999), which for Fomalhaut affect smaller particles more strongly and dominate in the shorter-wavelength emission, and the higher densities at apocenter, whose signature emerges strongly in the long-wavelength emission. Far-infrared and submillimeter observations of Fomalhaut appear broadly consistent with the range of apocenter/pericenter flux ratios our models predict, though flux ratio values based on ALMA and ATCA observations longward of 350 μm are currently not precise enough to include in Figure 3.

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A sufficiently large beam can dilute the effect of apocenter glow or blur it with background galaxies or other disk features like clumps, or even disk ansae, which themselves can be limb-brightened if the disk is not face-on. For example, in their discussion of pericenter glow in the Keck image of HR 4796 from Telesco et al. (2000), Wyatt et al. (1999) calculate the effects of beam size and study the disappearance of disk asymmetry as pericenter/apocenter moves away from the disk ansae. Because the beam size in the Fomalhaut images is much smaller relative to the disk size than in the Telesco et al. (2000) HR 4796 observations, we believe these dilution/confusion effects are less important for the Fomalhaut data of Acke et al. (2012) and Marsh et al. (2005).⁷ Indeed, both sets of disk images appear ring-shaped rather than dumbbell-shaped/double-lobed as in the earlier HR 4796 images. Also, for Fomalhaut Acke et al. (2012) measure the disk major axis to coincide within 1° with the line of nodes in the sky plane, so it is highly unlikely that pericenter orientation masks the pericenter/apocenter flux ratio.

In these relatively well-resolved images, we believe we can reasonably estimate the uncertainty in the flux ratio due to beam size by convolving a one-dimensional Gaussian of width equal to the beam radius with a cosine representing the sinusoidal flux variation with longitude and quoting the difference in the pre- and post-convolution amplitudes. In order to compensate here for the apparent increase in flux at the disk ansae due to Fomalhaut's 66° inclination, we convolve the cosine with a one-dimensional Gaussian of width equal to the beam radius divided by $\cos \ 66^\circ$. For the Herschel data this gives an extra flux uncertainty of ∼5% of the difference between the pericenter and apocenter fluxes at 70 μm and ∼13% of this difference at 160 μm, smaller than the size of the plot symbols. Marsh et al. (2005) incorporated beam/orientation effects in their data analysis, so we infer they are included in the quoted uncertainties.⁸

Precisely where the observed flux ratios fall among our models can be a revealing diagnostic of disk properties. Figure 4 shows the pericenter to apocenter flux ratio at 70 and at 160 μm for our Fomalhaut models as a function of β and q. A Herschel image of the Fomalhaut disk at 70 μm shows pericenter glow with an apocenter/pericenter flux ratio of about 0.7 (Acke et al. 2012). This value corresponds approximately to the line $\beta \simeq 1.9q-3.2$. At 160 μm, the Fomalhaut ring also shows pericenter glow (Acke et al. 2012), but the derived apocenter/pericenter flux ratio is about 0.89, corresponding approximately to the line $\beta \gtrsim 1.8q-4.7$.

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Observations of Fomalhaut at 7 mm by Ricci et al. (2012) provide an additional constraint: $q=3.48\pm 0.14$. Combining these constraints with our findings above from the Herschel 70 and 160 μm data and our models yields the range $\beta \simeq 1.4$ to 1.7, somewhat higher than the $\beta =1.1$ obtained by Dent et al. (2000) in their Fomalhaut model. Large values of β often point to a lack of larger, millimeter-sized grains in the grain distribution. For example, in the outer regions of T Tauri disks, where grain growth has not yet occurred, β is often in the range of 1.7–2 (Pérez et al. 2012). The millimeter dust opacity slope for the ISM yields $\beta \approx 1.7$ (Li & Draine 2001). However, the Fomalhaut system is too old for lack of grain growth to be likely. Explaining the Fomalhaut disk's strong pericenter glow may require additional physics not included in our models, for example radiation from unbound grains or planet–disk interactions more complex than those we considered (e.g., resonant dynamics). Indeed, Acke et al. (2012) find that unbound grains contribute about a quarter of the non-stellar flux in their 70 μm Fomalhaut system simulations.

Finally, 350 μm observations by Marsh et al. (2005) using SHARC II at the CSO indicate somewhat stronger apocenter glow than our models predict. Either a very steep size distribution ($q\gt 4$), a very shallow absorptivity law ($\beta \lt 1$) or, perhaps least unlikely, a larger eccentricity $e\gt 0.1$ is required in our models to reproduce the Marsh et al. (2005) result. In fact, Acke et al. (2012) measure e = 0.125 and e = 0.17, respectively, from their 70 and 160 μm images.

Resolved images of the Eridani cold dust disk ($a\simeq 61\,\mathrm{au}$ Greaves et al. 2014) were recently obtained at several submillimeter/millimeter wavelengths with Herschel, SCUBA, and MAMBO (Greaves et al. 2014; Lestrade & Thilliez 2015). These images also display azimuthal asymmetry, providing an independent constraint on our models. Because  Eridani (K2V, ${T}_{* }\simeq 5084$ K, ${R}_{* }\simeq 5.12\times {10}^{10}$ cm, Kovtyukh et al. 2003) is much cooler than Fomalhaut, none of its associated dust is subject to radiation pressure or stellar wind blowout (Reidemeister et al. 2011). Small grains may potentially provide much of the disk's surface area. Following Reidemeister et al. (2011), we included grains down to 0.1 μm in our models of  Eridani . As with Fomalhaut, we produced a grid of models with $3\leqslant q\leqslant 4$ and $1\leqslant \beta \leqslant 3$. For  Eridani, we also examined a range of disk eccentricities $0.02\leqslant e\leqslant 0.25$, allowing the amplitude of the surface density variation with longitude to scale linearly with eccentricity, as demonstrated in our semi-analytic model.

Figure 5 shows our modeling results of  Eridani with e = 0.1, the disk eccentricity favored by Greaves et al. (2014), plotted together with the observed south to north flux ratios. As these fluxes were reported in a variety of formats in the discovery papers, we converted them as follows. Greaves et al. (2014) report that their 160, 250, and 350 μm flux ratios differ from unity by 3.0, 3.7, and 2.5 times the spread measured in a 9 pixel grid around each point. Assuming a roughly Gaussian distribution of pixel fluxes, the spread among 9 pixels would be ≃2σ, where σ is the Gaussian width parameter. We produced our plotted uncertainties by assigning an uncertainty of $\pm \sigma$ to the north and the south fluxes and propagating errors. Although  Eridani is nearly face-on, with an inclination of ∼26°, we estimate using the convolution method described in Section 4.2 that the large Herschel beam size relative to the disk size adds uncertainties of 27%, 42%, and 71% of the pericenter-apocenter flux difference for 250, 350, and 500 μm, respectively. Lestrade & Thilliez (2015) report in their Figure 5 the 850 and 1200 μm fluxes as a function of azimuth. For each wavelength, we took the five points closest to due north and due south, averaged them to get the north and south fluxes, and took the larger of the spread in the points or the uncertainties plotted by Lestrade & Thilliez (2015) as the overall uncertainty in the north/south fluxes. We then propagated errors to get the uncertainties plotted in Figure 5. These overwhelm the flux uncertainties due to beam size.

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Because the  Eridani disk has no independent pericenter direction determination, the flux asymmetry may be interpreted as either pericenter or apocenter glow. A pericenter glow interpretation of the Herschel 160, 250, and 350 μm observations is consistent with the range of flux ratios given by our models as long as $e\geqslant 0.02$. However, plots analogous to those in Figure 4 indicate that with a pericenter glow interpretation, the 160 μm data consistently favors large q (a steep size distribution) and small β (a shallower absorptivity law), while the 350 μm favors the opposite end of our parameter range, small q and larger β. By contrast, an apocenter glow interpretation of the Herschel measurement is compatible with our models only if $e\gtrsim 0.2$. In this case, all three Herschel flux ratios are consistent with a steep size distribution (large q) and a shallower absorptivity law (small β). While our models consistently predict apocenter glow only longward of 850 μm, the precision of the flux ratios gleaned from the Lestrade & Thilliez (2015) data are such that our models appear broadly consistent with either a pericenter or an apocenter glow interpretation.

Figure 4 suggests that with its maximum operating wavelength of 28.3 μm, the James Webb Space Telescope (JWST) may not be able to add much to our understanding of pericenter/apocenter asymmetries in Fomalhaut: our Fomalhaut models are degenerate in this band. However, JWST could play an important role for warmer disks. As a final application for our dust reradiation model, we studied disks with semimajor axes of 10, 20, and 30 au around a Fomalhaut-like A star. Figure 6 shows a summary of the results. Due to the higher effective temperature of the dust, we expect the transition wavelength between pericenter and apocenter glow to occur in the far-infrared rather than the submillimeter bands that are important for Fomalhaut and Eri. The longest JWST MIRI bands are well-placed to constrain β and q in such disks around A stars by measuring pericenter/apocenter asymmetries. Even barely resolved observations at 24–30 μm could constrain the size distribution and eccentricities of those disks.

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