A Complete ALMA Map of the Fomalhaut Debris Disk

Figure 1 (left panel) shows the primary-beam-corrected ALMA 1.3 mm continuum image of Fomalhaut. With natural weighting, the rms noise level is 14 μJy/beam and the synthesized beam size is 156 × 115 (12 × 9 au at 7.7 pc) with a position angle of −87°. The right panel of Figure 1 shows the ALMA 1.3 mm image overlaid as contours on a Hubble Space Telescope (HST) STIS coronagraphic image of optical scattered light (Kalas et al. 2013). The millimeter continuum emission structure appears to match well with the narrow belt structure observed in the previous HST image. Overall, the new ALMA image shows emission from three components: (1) a narrow, eccentric ring (30σ), (2) an unresolved central point source at the stellar position (54σ), and (3) an unresolved point source on the eastern side of the disk (10σ). Most strikingly, we note a significant flux difference between the apocenter (NW) and pericenter (SE) sides of the disk of ∼65 μJy (>5σ), which we attribute to "apocenter glow," a result of the disk's eccentricity (Pan et al. 2016; see Section 4.1 for further discussion).

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We attribute the unresolved point source in the southeast quadrant to a background galaxy. The total flux density for this source is 0.150 ± 0.014 mJy, determined by fitting a point-source model to the visibilities using the uvmodelfit task in CASA. Recent deep ALMA surveys have built up statistics on the number of faint background sources expected in a given field of view (Hatsukade et al. 2013; Carniani et al. 2015). Given these (sub)millimeter source counts, the number of sources with flux density of >0.15 mJy expected within our field of view is ${2.6}_{-1.9}^{+5.7}$. The measured position of this point source is $\alpha ={22}^{{\rm{h}}}{57}^{{\rm{m}}}40.766$, δ = −29°37'32309 (J2000). This region has been imaged with HST/STIS in the optical (GO-13726; PI Kalas), where the nearest background source is 068 west and 003 north of the ALMA position. Given that the ALMA beam radius is ∼078 along R.A., it is likely that the ALMA source is the same background object as observed in optical data.

Given the clear observed eccentricity in the Fomalhaut debris disk, we construct models that account for the orbital parameters of particles in the disk. A particle orbiting within a circumstellar disk has both a proper and forced eccentricity, e_p and e_f, respectively, as well as a proper and forced argument of periastron, ω_p and ω_f. We begin by populating the complex eccentricity plane defined by these four parameters following Wyatt et al. (1999). The forced eccentricity and argument of periastron, e_f and ω_f, are imposed on the particles by the massive perturber forcing the eccentricity in the disk and are free parameters in our model. The proper eccentricity is also left as a free parameter, e_p, and describes the additional scatter in the eccentricity of each particle's orbit; the ω_p associated with a given e_p is assumed to be randomly distributed from 0 to 2π. By assuming a semimajor axis, a, for each particle and random mean anomalies, we iterate to find the true anomaly, f, using the newtonm code from ast2body (Vallado 2007). Then, the radial orbital locations of each particle can be found simply using

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

To create our 2D model, we complete this calculation for 10⁴ individual disk particles. By creating a 2D model, we assume that the disk structure has a negligible vertical component. This assumption is motivated by the result from Boley et al. (2012) that the vertical scale height of the disk is described by an opening angle of ∼1° from the midplane. Adding a vertical component to the model would likely loosen the constraints that we are able to place on the width of the belt (see Section 4.2.2 for further discussion).

To create an image, we bin the determined orbital locations into a 2D histogram with the bin size equal to the desired pixel scale and impose a radial temperature profile, $T\propto {r}^{-0.5}$. The belt semimajor axis (${R}_{\mathrm{belt}}$) and range of semimajor axes (Δa) are both free parameters. In this eccentric disk model, the belt semimajor axis is the mean inner edge location, ${R}_{\mathrm{belt}}=({R}_{\mathrm{per}}+{R}_{\mathrm{apo}})/2$, where ${R}_{\mathrm{per}}$ and ${R}_{\mathrm{apo}}$ are the radial location of the disk inner edge at pericenter and apocenter, respectively. The total flux density of the disk is normalized to ${F}_{\mathrm{belt}}=\int {I}_{\nu }d{\rm{\Omega }}$. A point source with flux density, ${F}_{\mathrm{star}}$, is added to account for the central stellar emission. In addition to fitting for both fluxes, we fit for the geometry of the disk (inclination, i, and position angle, PA), as well as offsets of the stellar position from the pointing center of the observations (Δα and Δδ).

For a given model image, we compute synthetic model visibilities using vis_sample,¹⁵ a python implementation of the Miriad uvmodel task. Following our previous approach (e.g., MacGregor et al. 2013, 2016a), we evaluate these model visibilities using a χ² likelihood function that incorporates the statistical weights on each visibility measurement. This iterative process makes use of the emcee Markov chain Monte Charlo (MCMC) package (Foreman-Mackey et al. 2013). Given the affine-invariant nature of this ensemble sampler, we are able to explore the uncertainties and determine the 1D marginalized probability distribution for each independent model parameter.

Table 2 presents the best-fit model (reduced χ² = 1.1) parameters and their 1σ (68%) uncertainties. Figure 2 shows the ALMA 1.3 mm data (left panel) along with the best-fit model displayed at full resolution and imaged like the ALMA data (center panels). The rightmost panel shows the imaged residuals resulting from subtracting this best-fit model from the data, which are mostly noise. The only significant peak corresponds to the background galaxy discussed in Section 3.1. The full MCMC output is shown in Figure 8 in the Appendix.

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Table 2.  Best-fit Model Parameters

Parameter	Description	Best-fit Value
${F}_{\mathrm{belt}}$	Total flux density (mJy)	24.7 ± 0.1
${F}_{\mathrm{star}}$	Total stellar flux (mJy)	0.75 ± 0.02
${R}_{\mathrm{belt}}$	Belt inner edge (au)	136.3 ± 0.9
${\rm{\Delta }}a$	Range of semimajor axes (au)	12.2 ± 1.6
ΔR	Belt FWHM (au)	13.5 ± 1.8
i	Disk inclination (deg)	65.6 ± 0.3
PA	Disk position angle (deg)	337.9 ± 0.3
ef	Forced eccentricity	0.12 ± 0.01
ep	Proper eccentricity	0.06 ± 0.04
${\omega }_{f}$	Forced argument of periastron (deg)	22.5 ± 4.3
Δα	R.A. offset (arcsec)	0.08 ± 0.01
Δδ	Decl. offset (arcsec)	0.06 ± 0.01

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The total belt flux density determined for the best-fit model is 24.7 ± 0.1 mJy (with an additional 10% uncertainty from flux calibration), consistent with previous flux measurements at slightly shorter wavelengths. Boley et al. (2012) determine a total flux density at 860 μm of ∼85 mJy, estimated from ALMA observations of the NW half of the ring. Holland et al. (1998) and Holland et al. (2003) determine flux densities of 81 ± 7.2 mJy and 97 ± 5 mJy from SCUBA imaging at 450 and 850 μm, respectively. Assuming a millimeter spectral index of ∼2.7 (Ricci et al. 2012), the measurement from Boley et al. (2012) extrapolates to ∼27 mJy at 1.3 mm, consistent with our results within the uncertainties. Using ALMA observations at 233 GHz (∼1.3 mm), White et al. (2017) obtain a flux density of ${30.8}_{-3.0}^{+3.4}$ mJy (all uncertainties from White et al. 2017 are in the 95% credible range) by fitting directly to the visibilities and ${26.3}_{-4.7}^{+4.5}$ mJy by fitting in the image plane, again consistent with our results within the mutual uncertainties. For optically thin dust emission, the total dust mass is given by ${M}_{\mathrm{dust}}={F}_{\nu }{D}^{2}/({\kappa }_{\nu }{B}_{\nu }({T}_{\mathrm{dust}}))$, where D = 7.66 pc is the distance, ${B}_{\nu }({T}_{\mathrm{dust}})$ is the Planck function at the dust temperature, ${T}_{\mathrm{dust}}$, and ${\kappa }_{\nu }$ is the dust opacity. We assume a dust opacity at 1.3 mm of ${\kappa }_{\nu }=2.3$ cm² g⁻¹ (Beckwith et al. 1990), which may be a source of systematic uncertainty. Given the best-fit radial location of the disk, 136.3 ± 0.9 au, the radiative equilibrium temperature is ∼48 K. Thus, the total mass of the dust belt is 0.015 ± 0.010 M_⊕.

We find good agreement with all previous determinations between the belt semimajor axis, eccentricity, inclination, and position angle for our best-fit model with previous results. The best-fit belt semimajor axis from our modeling is 136.3 ± 0.9 au. At pericenter, the inner edge of the belt is located at a radial distance of ${R}_{\mathrm{per}}=(1-e){R}_{\mathrm{belt}}=119.9\pm 0.8$ au. At apocenter, the inner edge of the belt is at ${R}_{\mathrm{apo}}=(1+e){R}_{\mathrm{belt}}\,=152.6\pm 1.0$ au. HST imaging yields a value of 136.28 ± 0.28 au (Kalas et al. 2005, 2013), while Acke et al. (2012) obtain 137.5 ± 0.9 au from Herschel observations. Boley et al. (2012) determine a semimajor axis of ${135}_{-1.5}^{+1.0}$ from ALMA imaging of the NW half of the disk, and White et al. (2017) determine a belt center location of ${139}_{-3}^{+2}$ au from their model fits. These same observational studies yield inclination and position angles in the range of 65°–67° and 336°–350°, respectively. We obtain robust constraints on both angles of i = 656 ± 03 and PA = 3379 ± 03. The best-fit eccentricity is 0.12 ± 0.01, consistent with both the Herschel result of 0.125 ± 0.006 and the HST and previous ALMA results of 0.12 ± 0.03.

Our new ALMA image is the first conclusive observational evidence for apocenter glow. The Keplerian orbital velocity in an eccentric disk is slower at apocenter than at pericenter, producing an overdensity of material at apocenter. At mid-infrared wavelengths, the observed flux is strongly dependent on the grain temperature; grains at pericenter glow more brightly since they receive more flux from the star, masking the apocenter overdensity. This effect is evident as "pericenter glow" (Wyatt et al. 1999) in Herschel images of the Fomalhaut disk at 70 μm, where the SE (pericenter) side of the disk appears brighter (Acke et al. 2012). In contrast, previous imaging of the Fomalhaut debris disk at longer far-infrared to millimeter wavelengths suggests a slight excess (<3σ) of emission at the NW (apocenter) side of the disk, farthest from the star (Holland et al. 2003; Marsh et al. 2005; Ricci et al. 2012). To explain this phenomenon, Pan et al. (2016) construct a model of "apocenter glow" where the enhancement of the surface density of the disk at apocenter results in wavelength-dependent surface brightness variations. At millimeter wavelengths, larger grains dominate the emission. Since these grains radiate efficiently at the blackbody peak, the pericenter–apocenter temperature difference has less impact on the total flux. As a result, the larger surface density at apocenter dominates and the apocenter appears brighter.

Figure 3 shows the apocenter-to-pericenter flux ratio for Fomalhaut as a function of wavelength, including our new ALMA measurement at 1.3 mm of 1.10 ± 0.02. Plotted together with the observational results are curves showing the smallest (purple dotted line) and largest (red solid line) apocenter-to-pericenter flux ratios obtained with a grid of simulated Fomalhaut disks. A detailed description of the disk simulations is given by Pan et al. (2016); here, we include a brief overview. We created disks with the forced eccentricity e_f, radial location R_belt, and semimajor axis range Δa given in Table 2 orbiting stars with effective temperature ${T}_{* }=8590$ K and radius ${R}_{* }=1.28\times {10}^{11}$ cm (Mamajek 2012). We populated the disks with particles of sizes a following power-law size distributions ${dn}/{da}\propto {r}^{-q}$ and grain absorptivities $Q\propto {a}^{-\beta }$. We drew each model disk's q and β values from a grid covering the ranges 3 ≤ q ≤ 4, 1 ≤ β ≤ 3. We then calculated the radially integrated disk brightness as a function of longitude assuming passively heated, optically thin disks in thermal equilibrium. The ALMA flux ratio measurement falls well within the range obtained in our model grid.

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As Figure 3 suggests, the observed apocenter-to-pericenter flux ratios can be diagnostic of disk grain properties, including β, the grain absorptivity, and q, the size distribution power-law index. The long-wavelength spectral index, ${\alpha }_{\mathrm{mm}}$, of dust emission constrains the size distribution of dust grains in the disk. Again assuming that the differential number of grains of size a is a power law, ${dn}/{da}\propto {a}^{-q}$, then $q=({\alpha }_{\mathrm{mm}}-{\alpha }_{\mathrm{Pl}})/{\beta }_{s}+3$ (Ricci et al. 2012; MacGregor et al. 2016b). Here, ${\alpha }_{\mathrm{Pl}}=1.88\pm 0.02$ (see discussion in MacGregor et al. 2016b) and ${\beta }_{s}=1.8\pm 0.2$, the dust opacity spectral index in the small particle limit for interstellar grain materials (Draine 2006). We note that different assumptions for the dust opacity can produce steeper grain size distributions (Gáspár et al. 2012). Ricci et al. (2012) measured the flux density of Fomalhaut at 6.66 mm with ATCA. By pairing our new ALMA flux density with this previous measurement, we determine ${\alpha }_{\mathrm{mm}}=2.71\pm 0.11$ and thus q = 3.46 ± 0.09. This result is consistent with the determination of White et al. (2017) of ${\alpha }_{\mathrm{mm}}=2.73\pm 0.13$ and q = 3.50 ± 0.14. Using the flux ratios measured at 70 μm, 160 μm, and 1.3 mm, respectively, and with a slight extension in the parameter range for our models, our 1σ uncertainty range in q implies 0.9 < β < 1.6, 0.7 < β < 1.5, and 0.7 < β.¹⁶ The overlap between these indicates an allowed range of 0.9 < β < 1.5, consistent within 1σ with the $\beta \simeq (q-3){\beta }_{s}$ quoted by Draine (2006).

4.2.1. Constraints on Azimuthal Belt Structure

The ALMA 1.3 mm mosaic map of the outer Fomalhaut debris disk was designed to cover the complete ring with equal sensitivity, allowing us to examine azimuthal structure along the belt. After subtracting our best-fit model, any azimuthal structure should be clearly visible in the imaged residuals. Figure 2 shows the resulting residuals, and no significant peaks are visible along the disk. Figure 4 shows the azimuthal profile of the disk in the sky plane. The mean brightness is calculated in small annular sections of 10° around the ring starting in the north and moving counterclockwise to the east. Uncertainties are obtained by dividing the rms noise of the image by the square root of the number of beams in each annular sector. The two disk ansae are visible as two peaks in the SE and NW, and apocenter glow is indicated by the significant brightness difference between these two peaks. No other significant peaks or fluctuations are present. We note a slight brightness difference (<3σ) between the NE and SW sides of the disk (along the direction of the disk minor axis). The median belt flux density measured between 170° and 270° (SW side) is 0.11 ± 0.01 mJy arcsec⁻² and 0.13 ± 0.01 mJy arcsec⁻² between 0° and 100° (NE side). A similar dimming of the SW side of the disk is seen by Boley et al. (2012), which they interpret as resulting from a loss of sensitivity at the edges of the ALMA primary beam. However, it is likely that this slight asymmetry between the NE and SW sides of the disk presents further evidence for apocenter glow. The expected overdensity of particles at apocenter forms an arc, which would cover much of the eastern side of the disk given the observed disk geometry (Pan et al. 2016; Pearce & Wyatt 2014).

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Previous imaging surveys at optical to infrared wavelengths have indicated several azimuthal features, which our millimeter observations do not confirm. Kalas et al. (2013) demonstrate that the dust belt has a ∼50% deficit of optical scattered light in an azimuthal wedge at position angle ∼331°, just north of the current location of Fomalhaut b. The sky-plane width of the gap is 2'' (∼15 au), corresponding to a deprojected width of ∼50 au. One possibility is that the gap in scattered light represents a deficit of material, where grains collect on horseshoe orbits on either side of a planet embedded in the gap. The brightness deficit could also result from self-shadowing in an optically thick, vertically thin belt. Millimeter observations are minimally affected by optical depth effects and should reveal the true surface density of grains. Since our ALMA observations do not detect the same 331° gap, it is likely that this feature results from a shadowing effect. However, we cannot rule out smaller structures ≲10 au that would remain unresolved in our current map.

SCUBA imaging at 450 μm shows evidence for an arc of emission at position angle ∼141° interior to the outer belt at ∼100 au separation from the star (Holland et al. 2003). Boley et al. (2012) note a broadening of the disk width on the northwestern side of the belt, to the right of the disk ansae. We do not confirm either of these features in our ALMA map. White et al. (2017) also note that the disk appears azimuthally smooth.

4.2.2. Determining the Belt Width

The FWHM width of our best-fit model is 13.5 ± 1.8 au. Boley et al. (2012) estimate a half-maximum width for the disk of ∼11.4 au given a power-law belt model and ∼16 au given a Gaussian model, consistent with our results. White et al. (2017) determine a belt width of 13 ± 3 au from their recent ALMA data. Figure 5 shows the surface brightness of our ALMA image in four cuts from the star along both the disk major (SE and NW sides) and minor (SW and NE) axes. We do not see any fluctuation in width along the belt. The flux difference between apocenter and pericenter is evident. Also of note is the offset of the star from the disk centroid to the SW by ∼030 (∼2.3 au) in R.A. and ∼14 (∼10.7 au) in decl.

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Given the best-fit parameters of our 2D model, we can constrain the fractional width of the belt to be ΔR/R = 0.10 ± 0.01. Adding a vertical component to the model likely adds to the uncertainty of this constraint. The Fomalhaut debris disk is similarly narrow to the main classical Kuiper Belt in our own solar system, which is radially confined between the 3:2 and 2:1 orbital resonances with Neptune, implying a fractional width of ∼0.18 (Hahn & Malhotra 2005). In contrast, both the HD 107146 (Ricci et al. 2015) and η Corvi (Marino et al. 2017) debris disks appear much broader with fractional widths of >0.3. Boley et al. (2012) propose that the narrow ring observed in Fomalhaut may also result from interactions with planets, namely, two shepherding planets on the inner and outer edges of the belt. If the structure of the belt is indeed due to truncation by interior and exterior planets, we would expect to see sharp edges. However, given the resolution of our observations (∼10 au) compared with the width of the belt (∼14 au), we are unable to place any strong constraints on the sharpness of the disk edges.

In our models, there are two parameters that contribute to the width of the belt: the range of semimajor axes assigned to the particles (Δa) and the proper or intrinsic eccentricity (e_p) of a particle's orbit. As expected, these parameters are highly degenerate, and we are unable to place strong constraints on either of these parameters independently given the moderate resolution of our observations. The best-fit values for both parameters are Δa = 12.2 ± 1.6 au and e_p = 0.06 ± 0.04. Figure 6 shows the MCMC output for Δa and e_p; the degeneracy between the two parameters is clearly seen by the slope in the contours. Altering the proper eccentricity of the particles predicts azimuthal variations in the width of the belt. For a low proper eccentricity (${e}_{p}\sim 0.01$), the particle orbits are apsidally aligned and the belt appears narrower at pericenter than at apocenter. For a high proper eccentricity (${e}_{p}\sim 0.1$), the width of the belt is closer to uniform around the entire circumference of the ring. Future ALMA observations of the disk apocenter and pericenter locations, but with higher resolution, could distinguish between these two cases and place the first robust constraints on the proper eccentricity of the Fomalhaut debris disk. Whereas White et al. (2017) have higher angular resolution in their recent ALMA observations (synthesized beam of 0329 × 0234), the two disk ansae are at the edge of the primary beam of their single ALMA pointing and do not allow for this analysis.

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4.2.3. Geometry of the Disk: The Argument of Periastron

Fomalhaut b was first discovered through HST direct imaging (Kalas et al. 2008) at a location consistent with theoretical predictions for a massive planet orbiting interior to the eccentric debris disk (Quillen 2006; Chiang et al. 2009). However, follow-up observations at later epochs revealed that Fomalhaut b is instead on a highly eccentric, possibly ring-crossing orbit (Kalas et al. 2013; Beust et al. 2014). Furthermore, this object appears brighter at optical wavelengths than in the infrared, contrary to predictions from models of planetary atmospheres. Kennedy & Wyatt (2011) discuss the possibility of a collisional swarm of irregular satellites surrounding a $\sim 10\,{M}_{\oplus }$ planet. Alternatively, Fomalhaut b may instead be a dust cloud generated through collisions between larger planetesimals (Currie et al. 2012; Galicher et al. 2013; Kalas et al. 2013; Kenyon et al. 2014; Tamayo 2014; Lawler et al. 2015). To date, the true nature of Fomalhaut b remains uncertain.

If Fomalhaut b is indeed a dust cloud, our ALMA observations provide useful constraints on its possible dust mass. We can place a robust 3σ upper limit on the flux density of 0.042 mJy, assuming a point source. Following the approach for optically thin emission described in Section 3.3, we can determine an upper limit on the potential dust mass. The current separation of Fomalhaut b is ∼125 au. In radiative equilibrium, this location implies a dust temperature of ∼51 K. The resulting upper limit on the dust mass is $\lt 0.0019\,{M}_{\mathrm{Moon}}$ ($\lt 1.40\times {10}^{23}$ g), which is consistent with estimates of the ${10}^{18}\mbox{--}{10}^{21}$ g in total submicron dust mass needed to account for the scattered light (Kalas et al. 2008). We can also consider optically thick dust emission and instead derive an upper limit on the size of the dust clump: ${R}_{\mathrm{dust}}=\sqrt{{F}_{\nu }D/(\pi {B}_{\nu }({T}_{\mathrm{dust}}))}$. Given the upper limit of ${F}_{\nu }\lt 0.042$ mJy, ${R}_{\mathrm{dust}}$ must be <0.021 au for an optically thick clump.

The best-fit flux density for the central star is 0.75 ± 0.02 mJy (with an additional 10% uncertainty for flux calibration). CHARA measurements of the stellar bolometric flux robustly determine the effective temperature to be 8459 ± 44 K (Boyajian et al. 2013). Given this effective temperature, a PHOENIX stellar atmosphere model (Husser et al. 2013) predicts a flux density of ∼1.3 mJy at 1.3 mm (with 5% uncertainty), in excess of our flux measurement. At long wavelengths, however, this stellar model is essentially a Rayleigh–Jeans extrapolation. Boley et al. (2012) measure a stellar flux of ∼4.4 mJy at 850 μm with ALMA in Cycle 0, which extrapolates to ∼1.8 mJy at 1.3 mm, consistent with atmospheric model predictions, but not consistent with our ALMA flux. It is important to note, however, that this measurement is strongly influenced by the primary beam correction applied to the data, since the star is located at the edge of the single pointing. ALMA Cycle 1 observations at 870 μm by Su et al. (2016) detect a central point source as well, with a lower flux density of 1.789 ± 0.037 mJy. Extrapolating to 1.3 mm, this measurement yields an expected flux density of 0.80 ± 0.02, more comparable to our result. White et al. (2017) also determine a low stellar flux density of 0.90 ± 0.15 mJy from recent ALMA observations at 1.3 mm (233 GHz, a slightly higher frequency than our observations). Figure 7 shows the flux density spectrum (top) and brightness temperature (bottom) of Fomalhaut from Herschel (Acke et al. 2012), ALMA (this work; Su et al. 2016; White et al. 2017), and ATCA (Ricci et al. 2012). To calculate the brightness temperature, we follow Liseau et al. (2016) and adopt a photospheric radius of 1.842 ± 0.019 R_⊙ (Mamajek 2012). The stellar flux density at infrared wavelengths from Herschel is inferred, since the measured flux includes contributions from both the star and the inner belt, which are unresolved in these observations. Given the possible contribution from an inner warm belt at all wavelengths, we quote only upper limits on the brightness temperature. At the Herschel wavelengths, the brightness temperature is mostly consistent with the effective temperature. However, at millimeter wavelengths, the brightness temperature dips to <6600 K and <6200 K at 870 μm and 1.3 mm, respectively. The ATCA flux measurement at 6.66 mm suggests a brightness temperature of <17,900 K.

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It is clear that the brightness temperature of Fomalhaut is significantly lower than the measured photospheric effective temperature at millimeter wavelengths before increasing again at longer, centimeter wavelengths. Similar behavior is seen for a number of K and M giants by Harper et al. (2013). With the advent of ALMA, there are a growing number of stars with robust millimeter flux measurements. Excess emission at long wavelengths has been observed for several other Sun-like stars, including α Cen A/B, Eridani, and τ Ceti (Liseau et al. 2015, 2016; MacGregor et al. 2015, 2016a), which is consistent with emission from a hot stellar chromosphere. Liseau et al. (2016) observe a temperature minimum, like we observe for Fomalhaut, for both α Cen A and B at shorter, submillimeter wavelengths with ALMA, which they attribute to a change in the sign of the temperature gradient above the stellar photosphere, as is seen in our own Sun. At 1.3 mm wavelength, the flux densities of these stars have recovered and their brightness temperatures are similar to their effective temperature. For Fomalhaut, it seems likely that the flux density measured by ATCA at 6.66 mm is dominated by chromospheric emission. However, the long-wavelength spectrum of A-type stars, like Fomalhaut, is further complicated by ionized stellar winds, which flatten the spectral slope at radio wavelengths (Aufdenberg et al. 2002). The ability to measure these behaviors with ALMA will enable advances in our understanding of stellar radiative transfer and chromospheres and of stellar winds.

For the Fomalhaut system, understanding the stellar flux contribution at long wavelengths is especially critical. Spitzer and Herschel observations reveal excess emission at infrared wavelengths (Stapelfeldt et al. 2004; Acke et al. 2012), which is attributed to a warm inner dust belt similar to the asteroid belt in our solar system (Su et al. 2013). However, no inner belt has been detected or resolved with ALMA (Su et al. 2016). Robustly determining the spectral energy distribution of the star at long wavelengths will help to determine the nature of such an inner asteroid belt.