FOMALHAUT b: INDEPENDENT ANALYSIS OF THE HUBBLE SPACE TELESCOPE PUBLIC ARCHIVE DATA

For each filter and epoch, we derive the photometry by integrating the flux density of the PSF template that best fits the data (Section 4.2). As the PSF template is generated in a 2.5 × 2.5 arcsec² image, we use a 125 radius aperture for the ACS data and a 1'' radius for the STIS data. The fractions of the PSF-integrated energy inside these apertures are 0.960, 0.961, 0.918, and 0.996 for the F435W, F606W, F814W Sirianni et al. (2005), and CLEAR/STIS (STIS handbook, chap. 14/CCDClearImaging) filters, respectively. The F110W flux upper limit is derived by estimating the 5σ noise in the area where Fomalhaut b is expected to be located, after convolving the image by a 0.4 arcsec diameter aperture (aperture matching the WFC3 photplam parameter). To convert the estimated flux densities F_λ in erg s⁻¹ cm⁻² Å⁻¹ to flux densities F_ν in erg s⁻¹ cm⁻² Hz⁻¹ (i.e., 10²³ Jy), we use the photplam keyword recorded by the ACS, WFC3, and STIS pipelines in the fits headers:

$$\begin{equation} F_\nu = F_\lambda \ \mathrm{photplam}^2\ 10^{-18.4768}. \end{equation} \tag{ 2 }$$

The resulting flux densities (μJy) are given in Table 3. The error bars σ_ν in percentage are the inverse of the S/Ns. In these ratios, the signal is the integrated flux density inside a 025 radius aperture centered on Fomalhaut b and the noise is the square root of the total variance of the residual noise after subtraction of the best PSF in the same area. K08 express their Fomalhaut b photometry and upper limits in Vega magnitudes. We convert their measurements to μJy (last column in Table 3) using the ACS handbook (Section 5.1.1).

Most of the flux densities (Table 3) are consistent with K08 values except our F606W/2006 point, which is ∼2σ brighter (σ is the quadratic sum of K08 error bars and ours). Moreover, our error bars are larger than K08's. Thus, even if we still detect a variability in the F606W filter between 2004 and 2006, it is not as significant (1.7σ_G) as it is in K08 (5σ_K–6σ_K)—where σ_G and σ_K are our error bars and K08's, respectively. We also find that the flux density measured in the CLEAR filter (its bandpass roughly corresponds to F435W+F606W+F814W) is consistent with the three ACS flux densities given the large error bar. Finally, we (marginally) detect Fomalhaut b at F435W, unlike K08, who have an upper limit.

We plot the photometry of our detections (crosses) in Figure 7 along with 5σ upper limits from the literature (K08; Marengo et al. 2009; J12) at various wavelengths. J12 find that to comply with their 4.5 μm upper limit, the planetary mass upper limit is 1 M_J at 400 Myr. Thus, we compare the measurements with a model of a cloud-free atmosphere for a 1 M_J planet at 400 Myr with the solar metallicity (Siegel & Burrows 2012) (full line in Figure 7). Our new F110W upper limit is consistent with the expected planet flux at that wavelength given the J12 Spitzer 4.5 μm upper limit. It is clear that a planet-only model for Fomalhaut b is not consistent with the visible observations. K08 proposed two other models: a cloud of dust (Section 4.3.2) or a disk of dust around a Jovian planet (Section 4.3.3). We revisit these two models in light of our updated photometry.

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We consider the model introduced in K08 with a 0.53 AU diameter cloud composed of dust grains with a differential size distribution dn/da∝(a/a₀)^−3.5 where the radius a goes from a_min to 1000 μm. Using Mie theory, K08 calculate the apparent magnitudes of such a cloud composed of water ice (density = 1, m_ice) or refractory carbonaceous material (density = 2.2, m_LG) with a_min = 0.01 μm (hereafter m^0.01) or 8 μm (hereafter m⁸). The total mass in grains is adjusted such that the integrated light in F814W from the model matches K08's observations (K08's and our photometry in F814W are in agreement). We convert the Vega magnitudes provided in K08's Table S3 to flux densities in μJy (Table 4). The last line gives the error between the expected flux densities F_{e, ν} and the observed densities F_ν:

$$\begin{equation} \epsilon =\sqrt{\sum _\nu \frac{(F_\nu -F_{e,\nu })^2}{\sigma _\nu ^2}}. \end{equation} \tag{ 3 }$$

Table 4. Expected Flux Densities F_{e, ν} (μJy) Derived from the Cloud Models ($m^{0.01}_\mathrm{ice}$, $m^{0.01}_\mathrm{LG}$, $m^{8}_\mathrm{ice}$, and $m^{8}_\mathrm{LG}$) Proposed in K08

Filter	$m^{0.01}_\mathrm{ice}$	$m^{0.01}_\mathrm{LG}$	$m^{8}_\mathrm{ice}$	$m^{8}_\mathrm{LG}$
F435W	0.71	0.58	0.63	0.46
F606W	0.55	0.50	0.46	0.45
F814W	0.37	0.37	0.37	0.37
	2.0	1.3	1.7	1.0

Notes. The last line gives the difference between F_{e, ν} and our measured photometry F_ν (see the text for details).

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K08 reject the possibility that Fomalhaut b can be explained by one of these cloud models because (1) they do not detect the object at F435W (they do not reject $m^{8}_\mathrm{LG}$ for this reason), (2) the red color they observe does not match the model, and (3) they cannot explain the F606W variability. All these reasons do not apply to our new photometry because (1) we detect Fomalhaut b at F435W, (2) the expected flux densities match the observed flux densities within 1.7σ for three of the four models ( < 1.7), and (3) the F606W variability is not significant in our images. K08 also explain that such a cloud could result from a collision of two planetesimals and that the probability of such an event is lower at the Fomalhaut b position than closer to the star or closer to the belt. However, as suggested by J12, the probability of a collision is not the probability of its detection because the speckle noise and the high brightness of the ring may prevent detections of such clouds close to the star and the belt, respectively. Moreover, the collision could have occurred inside the ring of dust and the resulting materials could have moved from the ring to the current position of Fomalhaut b. Finally, K08 argue that such dust clouds would be sheared due to differential gravitational forces and rapidly spatially resolved by HST. However, assuming a cloud with diameter 0.5 AU (maximum size for an unresolved source) only subject to gravitational forces from the star and no initial velocity, we find that its image would be larger than 2 pixels (∼FWHM) and 4 pixels after ∼100 yr and ∼200 yr, respectively. It would take ∼500 yr to shear the cloud of dust so that it could be spatially resolved in the HST images with no doubt. Thus, we find no strong arguments to reject K08 models of a dust cloud with radius ∼0.5 AU, composed of water ice or refractory carbonaceous small grains, and younger than ∼500 yr. We overplot a line that gives the expected fluxes for the $m^{8}_\mathrm{LG}$ model in Figure 7.

A second scenario proposed by K08 is an unseen Jovian planet surrounded by a disk of dust with a radius of 16–35 planet radius. As the K08 photometry is close to ours and K08 only work out rough numbers (they could not constrain all the parameters with only two photometric points), the 16–35 planet radius disk surrounding an undetected Jupiter-like planet is consistent with our photometry. J12 reject this model because (1) it does not explain the F606W variability and (2) the belt geometry would be strongly affected considering a ring-crossing orbit for Fomalhaut b (Kalas et al. 2010). As we do not find a significant F606W variability and our new astrometry cannot reject an orbit that does not cross the ring, we cannot rule out this model using the K08 arguments. J12 also consider that if the spin of the star is aligned with the plane of the disk, the northwest side of the disk is closer to Earth than the southeast side. In that case, Fomalhaut b is between its star and Earth in the radial direction and J12 claim that it would be difficult to explain how an optically thick disk can reflect so much light toward Earth. It is true if we observe the non-illuminated side of the disk, but we can imagine an inclined disk such that we observe the illuminated part of the disk even if Fomalhaut b stands between Fomalhaut and Earth.

Given that a possible model for Fomalhaut b involves a cloud of dust, it would be possible that the object has slowly expanded in time. We test here the possibility that the Fomalhaut b images are slightly spatially resolved.

First, we combine all the Fomalhaut b ACS images weighting the images by the S/Ns of the detections (linear and quadratic weighting give very similar results), and we fit a two-dimensional (2D) Gaussian function to the combined image. The best Gaussian function FWHM is 6 ± 1 pixels.

Then, we test how our processing can widen the image of a point-like source. For each filter/epoch of ACS observations, we extract a small sub-image close to Fomalhaut b (at 50 pixels maximum from Fomalhaut b). We add this noise to the PSF templates generated in Section 4.2 adjusting the noise level to reach the same S/Ns as we have for the Fomalhaut b detections. We combine the four epoch/filter images weighting by the S/Ns, and we fit a 2D Gaussian function to the combined image. Applying this analysis for noises picked at eight different locations in each filter/epoch image, we find that the PSF FWHM estimation is 2.8 ± 0.5 pixels. We repeat the same full analysis replacing the PSF templates by the detected southwest background source images, and we find that the background source image FWHM is 3.8 ± 0.5 pixels. Assuming that this source is not spatially resolved, we conclude that our data processing can widen the image of a point-like source by ∼1 ± 0.7 pixels.

We now model an extended object assuming a uniform intensity distribution over a disk with radius R. We convolve the object model by the PSF templates (Section 4.2) and obtain the object image templates for all epochs/filters. We adjust the S/Ns of the detections adding noise sub-images picked around the Fomalhaut b images. We combine the images accounting for the S/Ns and fit a 2D Gaussian function. FWHMs found for sources with R between 0.39 and 0.78 AU are at less than 1σ (estimated from noises picked at eight different locations) from the 6 pixel FWHM measured for the Fomalhaut b image.

Finally, we find that the Fomalhaut b image FWHM is ∼2σ from the widening induced by our data processing, suggesting that Fomalhaut could be resolved, but it is not yet conclusive. We also find that a basic model of an extended source could explain the measured Fomalhaut b extension. It is clear this low S/N analysis is not sufficient to fully conclude whether Fomalhaut b is or is not spatially extended; new observations are required. However, in the rest of Section 4.4, we consider an extended source for which the intensity distribution is a uniform disk with radius 0.58 AU (3 ACS pixels).

Assuming the image is resolved, we investigate what instrumental effect or data processing could explain such an extended source.

The ACS/HRC PSF is contaminated by a halo for red sources, especially at F814W (Section 5.1.4 in the ACS handbook). The halo that adds to the "normal" PSF has a diameter (42–2.36 λ) pixels and contains a total fractional intensity 2 (λ − 0.45)³ for the wavelength λ in μm. A 10 pixel diameter halo requires a dominant flux at λ ∼ 11 μm from this expression, which does not make sense because it is well outside the sensitive bandpass of the detector. Moreover, even if the S/N is low, we do not observe in the F814W image a PSF plus a halo but only an extended image.

A second explanation for such an image could be a misregistering of the raw images. In that case, after the rotations that put north up in the ADI process, all the Fomalhaut b images would not fall at the exact same position, resulting in a blurred image. If this happens, any source in the field of view would be affected the same way. This effect is included in the estimated widening induced by our processing (Section 4.4.1).

The last instrumental effect that we foresee is a differential geometric distortion of ≳ 1 pixel at the Fomalhaut b position between the images of a same sequence. The ACS pipeline corrects for the distortions with an accuracy of 0.01 pixels (Section 10.3 in the ACS handbook). Thus, it would require differential distortions 100 times larger than the pipeline accuracy at the Fomalhaut b position but almost no distortions at the background source position, which is roughly at the same angular separation from the star. This scenario seems very unlikely.

Finally, we find no instrumental effects that could explain the possible spatial resolution of Fomalhaut b in our images. Since the current paper was submitted, Kalas et al. (2013) mentioned that Fomalhaut b appears slightly extended in the 2012 images, which is qualitatively consistent with our analysis of the three earlier epochs. However, we insist that more observations with higher S/N are needed to establish whether or not Fomalhaut b is extended in the HST images.

For each filter/epoch, we consider the template T_o for a 1.16 AU diameter object. We adjust its flux to minimize the residual noise in a 025 radius aperture when we subtract it from the observations. We follow the steps described in Section 4.3.1 to convert the flux densities to Jy and estimate the error bars. The results are given in Table 5.

As expected, the fluxes are larger than in the case of a point source. Moreover, the flux variation at F606W is larger than in the point source case, but it is still less than 2.5σ_G, thus not yet significant. An unfortunately situated speckle at less than 3 pixels from Fomalhaut b could explain this variation. Finally, the flux density measured in the large band of STIS is consistent with the average flux density measured in the ACS filters within 1.8σ_G. In the case in which we resolve Fomalhaut b, we plot the photometry of our detections (crosses) in Figure 8.

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In Figure 8, we add the fluxes derived from the $m^{8}_\mathrm{LG}$ K08 model of a cloud of refractory carbonaceous material (Section 4.3.2). We multiply the three fluxes by 0.47/0.38, i.e., we adjust the F814W flux and assume the ratios between filters are the same. The model seems to be in good agreement with the data.

In the case of a spatially resolved Fomalhaut b, we propose one basic model that assumes that Fomalhaut b is the result of the collision of two Kuiper Belt objects (KBOs; Section 4.4.4) and we adapt a model of a circumplanetary swarm of satellites (Section 4.4.5) proposed by Kennedy & Wyatt (2011).

In this section, we propose a basic model to roughly estimate the size and the amount of light that is scattered by a cloud of dust produced by the collision of two KBOs. The objective is not to derive the exact radius, mass, and velocity of the KBOs that could create Fomalhaut b but to show that the collision of two KBOs is not completely inconsistent with the observations. First, we estimate the total grain mass that can explain the fluxes received from Fomalhaut b. Then, we show that the amount of dust can be the result of a collision of two 50 km radius colliders. We evaluate the rate of collisions of two such KBOs around Fomalhaut. Finally, we estimate when the collision may have occurred to reproduce the size of the Fomalhaut b images.

If the particles of dust are spheres with radius a and if the cross section of the particles equals their geometric albedo, the mass M_d of a cloud of dust that lies at a distance D from its star is (Jura et al. 1995)

$$\begin{equation} M_d \gtrsim \frac{16\,\pi }{3}\,\rho \,D^2\,a\,\frac{L_{{\rm sc}}}{L_*}, \end{equation} \tag{ 4 }$$

where ρ is the mass density of the dust grains, and L_sc and L_* are the luminosity of the light scattered by the cloud and the luminosity of the star, respectively. Instead of estimating the ratio of the luminosities, we work with the fluxes F_sc and F_* received at the telescope. We estimate F_sc from the measured fluxes (F_i) in the ACS filters (i = F435W, F606W, and F814W; Table 5)

$$\begin{equation} F_{sc} = \sum _i\,F_i \Delta \nu _i \end{equation} \tag{ 5 }$$

with Δν_i the bandwidths of the filters. Our estimation of F_sc only includes the scattered energy in the F435W, F606W, and F814W bandpasses. F_* has to be calculated for the same bandpass. Assuming a Planck law, F_* in the bandpass [λ_min = 435–50 nm, λ_max = 825 + 115 nm] is

$$\begin{equation} F_* = F_{*,{\rm tot}}\,\frac{\int _{u_{{\rm min}}}^{u_{{\rm max}}} u^3/(\exp {u}-1)\,{d}u}{\int _0^{\infty } u^3/(\exp {u}-1)\,{d}u}, \end{equation} \tag{ 6 }$$

where u_{min, max} = h c/(k T λ_{max, min}) with the Planck constant h, the speed of light in vacuum c, the Boltzmann constant k, the stellar effective temperature T (8751 K; Di Folco et al. 2004), and the stellar flux F_{*, tot} received at the telescope (8.914e−6 erg cm⁻²; Kalas et al. 2008). For D ∼ 120 AU and dust grains with radius a = 10 μm and ρ = 2 g cm⁻³, we find from Equations (4), (5), and (6) that the total grain mass needed to reproduce the photometry of Fomalhaut b is M_d ∼ 4.10¹⁹ g.

Jewitt (2012) estimates the mass m_e of particles that are ejected after a collision of two KBOs with a mass M_kbo, a radius r, a density ρ, and a relative velocity U

$$\begin{equation} \frac{m_\mathrm{e}}{M_\mathrm{kbo}}=A\,\left[r\,\sqrt{\frac{8\,\pi \,G\,\rho }{3}}\right]^{-1.5}\,U^{1.5}, \end{equation} \tag{ 7 }$$

where A equals 0.01 and G = 6.67 10⁻¹¹ m³ kg⁻¹ s⁻² is the gravitational constant. Jewitt (2012) assumes the particles have radii in the range 0.1 μm ≲ a ≲ 0.1 m with a power-law distribution in radii with index ∼3.5 (see also Kadono et al. 2010). For typical KBOs in the ring, U is the orbital velocity times h_r/(2D) with h_r the full vertical height of the ring at radius D. With h_r ∼ 3.5 AU (Kalas et al. 2005) at D ∼ 120 AU, U is close to 60 m s⁻¹. Assuming KBOs with radius r = 50 km, the total debris mass m_e after the collision is roughly 1% of the mass M_kbo of one of the two colliders with a density ρ = 2 g cm⁻³. Given the approximations in the models, the expected mass of dust (1% M_kbo) that is ejected after a collision of two 50 km radius KBOs is consistent with the mass estimated from the photometry of Fomalhaut b (4% M_kbo for a 50 km radius KBO).

We assume a maximum post-collision outflow velocity at infinity equal to the escape velocity, $r\,\sqrt{8\,\pi \,G\,\rho /3}$. Considering this upper limit, the diameter s of the cloud is $2\,r\,t\,\sqrt{8\,\pi \,G\,\rho /3}$ at the date t after the collision and reaches the observed size s = 1.16 AU (Section 4.4.1) after ∼50 yr, which is then a lower limit to the time since the collision of the putative KBOs. We can also estimate from the expansion expression that, after ∼150 yr, the source would have a diameter ∼3.5 AU and would be ∼10 times fainter than the current detections assuming the same amount of reflecting dust. It would not be detected in our images. Thus, if Fomalhaut b is the product of a collision, the event should have occurred between ∼50 and 150 yr ago to be consistent with our detections. This range is consistent with the possible trajectories that put Fomalhaut b inside the ring of dust 50–150 yr ago (Section 4.2). As the size of the possible extended source is very approximative, we keep in mind that these numbers are coarse estimations.

Finally, we evaluate the rate of a collision of two r = 50 km KBOs inside the ring of dust. We first find the collision time, which reads

$$\begin{equation} t_\mathrm{col} = \frac{1}{4\,\pi \,n\,r^2}\,\frac{1}{U} \end{equation} \tag{ 8 }$$

with n being the number of KBOs with radius r per unit volume. To estimate n we need the mass M_disk of the debris disk around Fomalhaut. We assume M_disk = 40 M_Earth because (1) estimates of the mass of the Sun's early Kuiper Belt are 40 M_Earth (Schlichting & Sari 2006), and (2) estimates of the mass of the Vega debris belt are 10 M_Earth in objects with radii <100 km (Müller et al. 2010) whereas the Vega IR luminosity is four times fainter than that of Fomalhaut. Simplifying by setting the radii of all the KBOs to 50 km does not qualitatively alter the collision frequency estimated below. Under these conditions and considering the belt surrounding Fomalhaut A has a volume 2 π D ΔD h_r with ΔD ∼ 0.13 D (Kalas et al. 2005), the number of KBOs is

$$\begin{equation} n\,2\,\pi \,D\,\Delta D\,h_r=\frac{M_\mathrm{disk}}{M_\mathrm{kbo}}\sim 2\times 10^8. \end{equation} \tag{ 9 }$$

Using $U=\pi \,h\sqrt{M_*/M_\odot \,(1\ \mathrm{AU}/D)^3}$ and Equations (8) and (9), we can write

$$\begin{equation} t_\mathrm{col}=\frac{0.13}{2\,\pi }\left(\frac{D}{r}\right)^2\frac{M_\mathrm{kbo}}{M_\mathrm{disk}}\sqrt{\left(\frac{D}{1\ \mathrm{AU}}\right)^3\frac{M_\odot }{M_*}}. \end{equation} \tag{ 10 }$$

Finally, the rate κ of collisions of two KBOs with radii r is the ratio of the number of KBOs (Equation (9)) to t_col

$$\begin{equation} \kappa =\frac{2\,\pi }{0.13}\left(\frac{r}{D}\right)^2\left(\frac{M_\mathrm{disk}}{M_\mathrm{kbo}}\right)^2\sqrt{\left(\frac{1\ \mathrm{AU}}{D}\right)^3\frac{M_*}{M_\odot }}. \end{equation} \tag{ 11 }$$

Equation (11) with M_disk = 40 M_Earth and M_* = 2 M_☉ indicates that ∼1 collision of two r = 50 km KBOs occurs every century in the ring around Fomalhaut A. The rate is low enough to explain that we detect only one event around Fomalhaut as each event would be detectable during ∼200 yr in our images. At the same time, it is high enough to make such a ∼50–150 yr old event plausible.

In summary, we conclude that it is plausible that Fomalhaut b is a cloud of dust that was produced ∼50–150 yr ago inside the dust belt by the collision of two KBOs with radii ∼50 km.

Kennedy & Wyatt (2011, KW11) propose a model of circumplanetary satellite swarms that they apply to Fomalhaut b. They find that the planet mass can be ∼2–100 M_Earth surrounded by a swarm that lies at 0.1–0.4 Hill radii. The swarm mass would be of the order of a few lunar masses. But these numbers are derived from K08 photometry of an unresolved source.

Here, we use the same model under the same assumptions (body size distribution and maximum/minimum body sizes, dust density, etc.) but we consider a swarm of satellites with diameter 1.16 AU, our photometry (Table 5), and a star with age 440 Myr instead of 200 Myr (Mamajek 2012). We do not describe the model as it is done in KW11. We only use the meaningful equations to constrain the planet mass and the swarm mass and size following the steps in Sections 3.3.1 and 3.3.2 of KW11. First, we derive the total cross-sectional area of dust σ_tot from our photometry: σ_tot = 6.12 × 10⁻⁴ AU², assuming a geometric albedo 0.08, a phase function 0.32 (Lambert sphere at maximum extension from its host star), the star effective temperature 8751 K (Di Folco et al. 2004), and a stellar luminosity 6.34 × 10²⁷ W (K08).

As we assume that we resolve Fomalhaut b, we can write 2 η R_Hill = s, where η is the semimajor axis of the satellites of the swarm relative to the Hill radius R_Hill at the Fomalhaut b separation (118 AU) and s is the swarm diameter. As explained in Section 4.4.1, the size of the extended source s = 1.16 AU is approximate and at F814W, the image of Fomalhaut b could be reproduced by a source with radius up to s = 2.32 AU. Thus, we consider 1.16 AU <s < 2.32 AU. Using the Hill radius expression (Equation (1) in KW11) and 2 solar masses for Fomalhaut (KW11), we derive two constraints: $0.61/M_\mathrm{pl}^{1/3}<\eta <1.22/M_\mathrm{pl}^{1/3}$, where M_pl is the planet mass expressed in Earth masses.

Considering a collision-limited satellite swarm around Fomalhaut b (i.e., swarm has just started to suffer collisions) that reproduces the observed σ_tot, it imposes a minimum limit for the satellite semimajor axis $\eta >0.29/M_\mathrm{pl}^{0.12}$ for a 440 Myr system (i.e., a 440 Myr collision time; see KW11 for details).

KW11 also study the collision velocities that are required to destroy a large object at the Fomalhaut b position. Assuming a steady-state collisional cascade and a two-phase size distribution for the particles, KW11 link the collision velocity to the swarm size η and the planet mass M_pl. Using their equations in the case of a resolved object we set a constraint that reads $\eta >0.69/M_\mathrm{pl}^{0.46}$ (KW11).

Moreover, KW11 assume that satellite orbits with η > 0.5 are not stable and do not consider them. Finally, we account for the 1 M_J upper limit that Janson et al. (2012) put from the non-detection at 4.5 μm for a 400 Myr system (close enough to 440 ± 40 Myr proposed by Mamajek 2012). We plot all the constraints in Figure 9, which gives the semimajor axis η of the satellites against the planetary mass M_pl. The parameters for which KW11's model can reproduce the photometry of a 1.16 AU source are within the dashed area. The minimum and maximum planetary masses are ∼2 M_Earth and 1 M_J, and the swarm has a total mass 2–11 M_Moon and lies at 0.15–0.5 Hill radii around the planet.

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In the case of an unresolved object (Figure 7 in KW11), KW11 find that the mass of the planet (<100 M_Earth) is not sufficient for it to have a significant gaseous envelope and enable mechanisms that could explain the migration of Fomalhaut b that presumably originates somewhere closer to the star. KW11 also argue that a single planet with mass <100 M_Earth—which is similar to or less than the mass of the main debris ring (1–300 M_Earth; Wyatt & Dent 2002; Chiang et al. 2009)—is unlikely responsible for shaping the dust belt. In our case of a source with diameter 1.16 AU, the range of the planetary mass goes up to 1 M_J and a Jupiter-like planet can have a significant gaseous envelope and shape the dust belt.