RHADaMAnTe: An Astro Code to Estimate the Spectral Energy Distribution of a Curved Wall of a Gap Opened by a Forming Planet in a Protoplanetary Disk

Abstract

When a star is born, a protoplanetary disk made of gas and dust surrounds the star. The disk can show gaps opened by different astrophysical mechanisms. The gap has a wall emitting radiation, which contributes to the spectral energy distribution (SED) of the whole system (star, disk and planet) in the IR band. As these newborn stars are far away from us, it is difficult to know whether the gap is opened by a forming planet. I have developed RHADaMAnTe, a computational astro code based on the geometry of the wall of a gap coming from hydrodynamics 3D simulations of protoplanetary disks. With this code, it is possible to make models of disks to estimate the synthetic SEDs of the wall and prove whether the gap was opened by a forming planet. An implementation of this code was used to study the stellar system LkCa 15. It was found that a planet of 10 Jupiter masses is capable of opening a gap with a curved wall with a height of 12.9 AU. However, the synthetic SED does not fit to Spitzer IRS SED (${\chi }^2$∼4.5) from $5\mathsf{\mu }$m to $35\mathsf{\mu }$m. This implies that there is an optically thin region inside the gap.

1. Introduction

In a small fraction of young stellar objects (YSOs) surrounded by disks, observations have discovered a low excess radiation in the near-infrared but a high excess in longer wavelengths. This has been interpreted as an evidence that these disks called transitional disks (TDs) have central holes which have practically no dust [1]. More recently, disks showing a significant excess in the near-infrared have been discovered. Such an excess indicates the presence of an optically thick inner disk. This inner disk is separated from an outer disk which also has a high optical depth. In this way, the spectral energy distribution (SED) suggests the incipient development of a gap between both disks; these disks are called pre-transitional disks (Pre-TDs) [2]. Several physical mechanisms have been suggested to explain the gaps or holes in protoplanetary disks. The one implemented in this work is driven by forming giant planets.

A key element that produces characteristic features in the SEDs of protoplanetary disks is the outer wall of the gap or hole facing the star. To simplify SED wall models, it is often assumed that the wall is uniform in the vertical direction and frontally irradiated by the central star [2,3]. However, this assumption is physically wrong; in [4], it was found that the wall of the gap in LkCa 15 is curved. Furthermore, for dust sublimation walls, located near from the star, it has been proposed that the wall is curved, where the dust-grain growth and its fall into the mid-plane of the disk, and the high dependence of gas density on sublimation temperature, are the physical mechanisms responsible for such curvature [5].

To model the structure and SED of gaps or holes in protoplanetary disks, many free available radiative codes, such as ProDiMo [6], DALI [7], or RADMC-3D [8], are implemented. However, these codes do not consider the contribution of the wall to the total SED, and they are focused on the thermo-chemical gas/dust modeling of the disk.

In order to create the synthetic SED of a protoplanetary disk considering the contribution of the outer curved wall of the gap or hole, to explain the strong infrared excess emission detected at wavelengths around ∼$10\mathsf{\mu }$m, I have developed the computational code called rhadamante. This code is based on an older code which suggests that the inner vertical wall of the outer disk can explain the mid-infrared spectrum of the low-mass pre-main-sequence star CoKu Tau/4 [9].

To test the code, a model of a truncated dusty disk is presented—a disk with an inner hole—that accounts for the Spitzer Infrared Spectrograph observations of the low-mass pre-main-sequence star LkCa 15. In this model, the mid-infrared spectral energy distribution (between 10 and 25 µm) arises from the inner curved wall of the gap in the disk.

1.1. Dust in Protoplanetary Disks

Dust is a pretty important component of protoplanetary disks surrounding young stars. The growth of dust grains from sizes of microns to centimeters or larger grains is the first step in planet formation.

The dust grains in protoplanetary disks follow a size distribution based on a single power law $f\left(a\right)\propto a^p$ ([10], MNR), where the maximum dust size, $a_{max}$, the minimum dust size, $a_{min}$, and the power-law index, p, are different for each grain species.

The dust composition in protoplanetary disks has been extensively studied through mid-IR observations. This dusty mixture includes silicates (mainly), carbonaceous grains, poly-cyclic aromatic hydrocarbons, and sulfide-bearing grains [11].

Silicate grains are the best-understood dust component. The most abundant crystalline silicates are olivine, which is magnesium-rich (Fo₉₀), and pyroxene. The olivine series ranges from forsterite, Mg₂SiO₄ (denoted as Fo₁₀₀), to the fayalite, Fe₂SiO₄ (denoted as Fo₀). While the pyroxene series ranges from the enstatite, MgSiO₃ (denoted En₁₀₀), to the ferrosilite, FeSiO₃ (denoted En₀).

The modeling of the observed spectra expects amorphous silicate grains to exist in protoplanetary disks [12]. The composition of these grains are glass with embedded metals and sulfides, and series ranging from ferromagnesian silica to Fe-Mg-bearing aluminosilica. These grains are difficult to observe directly from infrared spectroscopy. Their observed spectral signature is a combination of grain composition, shape, size, and structure, making it difficult to isolate the pure amorphous silicate signal.

Carbonaceous grains, including amorphous and graphite elemental carbon, are difficult to detect in the infrared. However, grain modeling suggests these grains are needed in order to explain the observed infrared spectra of protoplanetary disks [12].

Nano–diamonds, from sizes of <2 nm to ∼10 nm, are found in protoplanetary disks. Diamond emission coming from the inner region of the disk (∼10–50 AU) at 3.43 and 3.53 $\mathsf{\mu }$m has been detected in disks (see, e.g., [13]).

The presence of poly-cyclic aromatic hydrocarbons (PAHs) has been detected in the surface layers of some protoplanetary disks (see, e.g., [14]). Disks surrounding higher-mass stars, such as Herbig stars, show more PAH emission [15] than disks surrounding lower-mass stars, such as T Tauri stars [16]. Protoplanetary disks with a flaring outer surface show significantly more PAH emission [15]. It follows that PAHs exist in all disks, but they can only be detected, as infrared emission, when ultraviolet radiation from the central star is able to excite them. The discovery of weak PAH features in T Tauri stars supports this idea (see, e.g., [16]).

Other dusty components in protoplanetary disks are iron–nickel sulfide grains (FeS, NiS) and water ice (H₂O). Sulfide emission around 23.5 $\mathsf{\mu }$m has been detected in the emission spectra of protoplanetary disks [17], while water-ice emission has been identified at 3 $\mathsf{\mu }$m [18], 44 $\mathsf{\mu }$m [19], 60 $\mathsf{\mu }$m [20], and 62 $\mathsf{\mu }$m [21].

1.2. LkCa 15

LkCa 15 is a K5-type [22] T Tauri star located in the nearby ($145\pm 15\mathrm{pc}$) the Taurus–Auriga star-forming region [23]. The mass of the central star is $0.97\pm 0.03M_{\odot }$[22]; it has an effective temperature of 4370 K [24] and a radius of $1.6R_{\odot }$[25]. Three planet candidates have been detected: LkCa 15b (semi major axis $a=14.7\pm 2.1\mathrm{AU}$) [26], LkCa 15c ($a=18.6\pm 2.5\mathrm{AU}$), and LkCa 15d ($a=18.0{\pm }_{5.4}^{6.7}\mathrm{AU}$), with masses lower than 5–10 $M_{♃}$, for the two first planets, and ≤$0.5M_{♃}$ for the third one [27].

Observations of the far-ultraviolet (1100–2200 Å) radiation field and the near-to mid-IR (3–13.5 $\mathsf{\mu }$m) spectral energy distribution of LkCa 15, from the Space Telescope Imaging Spectrograph (STIS), indicate the existence of an inner disk gap of a few astronomical units [28].

LkCa 15 has an inner disk, a gap, and an outer disk [29]. Using the Spitzer data, LkCa 15 has been classified as a pre-transitional disk [30], and it has been showed that the inner hole is not devoid of dust between 0.1 and 5 AU. The outer disk extends from 46 to 800 AU [31]. Models of the LkCa 15 disk SED show that the inner edge of the disk has a radius of ∼58 AU [25]; this outer disk has a mass of ∼$50$${\mathrm{M}}_{♃}$ [32].

Recent observations from Gemini NIRI suggest a multiple-planets scenario to open a gap as large as the one observed in the LkCa 15 disk [24]. However, such observations cannot confirm or falsify the existence of the ∼$6M_{♃}$ planet candidate reported in [26]. This assumption leads me to assume a mass of ∼$10M_{♃}$ for the planet candidate in the current work.

2. Geometry of the Wall Projected on the Sky

To find the two-dimensional geometry of the wall of a gap, I implement the radiative and geometrical code artemise [4]. This code analyses a tri-dimensional simulation of the disk–planet interaction, considering the radiation from the star follows a radial direction through the disk. The wall is found at the points $(x,y,z)$ where the disk optical depth is ${\tau }_{wall}=\frac{2}{3}$. Simulations are carried out with fargo-3D code [33] under some specific parameters of the young stellar object to be studied.

2.1. Inclined Walls

artemise output is a collection of points in the $xz$-plane. To obtain the curved wall, such points must be continuously connected. The easiest way to achieve this is by using straight line segments, as seen in Figure 6. Next, each line segment is rotated around the z axis, and a conic wall is obtained (see Figure 1). The projections on the plane of the sky of all these conic walls (see Figure 2) are needed to calculate the projected area of the curved wall, which is required to estimate the emission of the wall.

Definition 1.

An inclined two-dimensional wall is a line segment with boundaries $(R_{\mathrm{wall}}^{\mathrm{up}},H_{\mathrm{wall}}^{\mathrm{up}})$ and $(R_{\mathrm{wall}}^{\mathrm{down}},H_{\mathrm{wall}}^{\mathrm{down}})$, with $R_{\mathrm{wall}}^{\mathrm{up}}>R_{\mathrm{wall}}^{\mathrm{down}}$ and $H_{\mathrm{wall}}^{\mathrm{down}}=-H_{\mathrm{wall}}^{\mathrm{up}}$, as seen in Figure 1a. By rotating this line segment around z-axis, a conic ring lying in the Euclidean space $(x,y,z)$ is obtained, as seen in Figure 1b. This ring is the tri-dimensional conic wall.

In the coordinate system $(x,y,z)$, the star is centered at the origin; here, the z-axis is the disk rotation axis, and the plane $(x,y)$ is the disk mid-plane. For simplicity, I also consider the cylindrical coordinate system $(R,\theta ,z)$, such that all points on the wall’s upper boundary have the coordinates z, $x=R_{\mathrm{wall}}^{\mathrm{up}}cos\left(\theta \right)$, and $y=R_{\mathrm{wall}}^{\mathrm{up}}sin\left(\theta \right)$, whereas all points on the wall’s lower boundary have the coordinates z, $x=R_{\mathrm{wall}}^{\mathrm{down}}cos\left(\theta \right)$, and $y=R_{\mathrm{wall}}^{\mathrm{down}}sin\left(\theta \right)$. Since the protoplanetary disk is assumed to be projected on the plane of the sky $(X,Y)$, as seen in Figure 2, it is considered a third coordinate system $(X,Y,Z)$ also centered at the star, where the Z-axis is the line of sight. When the disk is face on, the coordinate systems coincide. There exists a transformation between the three coordinate systems:

$$X=x,$$

(1a)

$$Y=ycos\left(i\right)-zsin\left(i\right),$$

(1b)

$$Z=zcos\left(i\right)-ysin\left(i\right),$$

(1c)

where $0^{\circ }\le i\le {90}^{\circ }$ is the disk inclination angle, that is, the angle between the z-axis and the plane of the sky $(X,Y)$. Here, $i=0^{\circ }$ means that the disk is face on, and $i={90}^{\circ }$ means that the disk is edge on.

The amount of visible surface of the wall, projected on the plane of the sky, depends on the disk inclination angle, and there are two possibilities: (i) when the star is visible (corresponding to $\delta <1$; see Equations (4) and (9) for a definition of $\delta$), and (ii) when the star in invisible (corresponding to $\delta >1)$, as seen in Figure 2. A surface element of the visible area is $dA=dXdY=-R_{\mathrm{wall}}sin\theta d\theta dY$, with $R_{\mathrm{wall}}^{\mathrm{down}}\le R_{\mathrm{wall}}\le R_{\mathrm{wall}}^{\mathrm{up}}$.

Let $\mathcal{R}$ be the visible surface of the wall projected on the plane $(X,Y)$ for both cases, as seen in Figure 3. Then, the boundary of this region is defined by two ellipses ${\epsilon }_{\mathrm{up}}$ and ${\epsilon }_{\mathrm{down}}$ (see Appendix A.1) given by the projections of the up and down edges of the tri-dimensional conic wall. The big ellipse ${\epsilon }_{\mathrm{up}}$ is defined as $Y_{\mathrm{up}}=\mathrm{s}{Y}_{\mathrm{up}}\cup \mathrm{i}{Y}_{\mathrm{up}}$, where $\mathrm{s}{Y}_{\mathrm{up}}$ and $\mathrm{i}{Y}_{\mathrm{up}}$ are the upper and lower parts of this ellipse, respectively, such that

$${\displaystyle \frac{\mathrm{s}{Y}_{\mathrm{up}}}{R_{\mathrm{wall}}^{\mathrm{up}}}}=cos\left(i\right)\sqrt{1-{\left({\displaystyle \frac{X}{R_{\mathrm{wall}}^{\mathrm{up}}}}\right)}^2}+{\displaystyle \frac{H_{\mathrm{wall}}^{\mathrm{up}}}{R_{\mathrm{wall}}^{\mathrm{up}}}}sin\left(i\right),$$

(2a)

$${\displaystyle \frac{\mathrm{i}{Y}_{\mathrm{up}}}{R_{\mathrm{wall}}^{\mathrm{up}}}}=-cos\left(i\right)\sqrt{1-{\left({\displaystyle \frac{X}{R_{\mathrm{wall}}^{\mathrm{up}}}}\right)}^2}+{\displaystyle \frac{H_{\mathrm{wall}}^{\mathrm{up}}}{R_{\mathrm{wall}}^{\mathrm{up}}}}sin\left(i\right).$$

(2b)

Similarly, the small ellipse ${\epsilon }_{\mathrm{down}}$ is defined as $Y_{\mathrm{down}}=\mathrm{s}{Y}_{\mathrm{down}}\cup \mathrm{i}{Y}_{\mathrm{down}}$, where $\mathrm{s}{Y}_{\mathrm{down}}$ and $\mathrm{i}{Y}_{\mathrm{down}}$ are the upper and lower parts of this ellipse, respectively, such that

$${\displaystyle \frac{\mathrm{s}{Y}_{\mathrm{down}}}{R_{\mathrm{wall}}^{\mathrm{down}}}}=cos\left(i\right)\sqrt{1-{\left({\displaystyle \frac{X}{R_{\mathrm{wall}}^{\mathrm{down}}}}\right)}^2}-{\displaystyle \frac{H_{\mathrm{wall}}^{\mathrm{down}}}{R_{\mathrm{wall}}^{\mathrm{down}}}}sin\left(i\right),$$

(3a)

$${\displaystyle \frac{\mathrm{i}{Y}_{\mathrm{down}}}{R_{\mathrm{wall}}^{\mathrm{down}}}}=-cos\left(i\right)\sqrt{1-{\left({\displaystyle \frac{X}{R_{\mathrm{wall}}^{\mathrm{down}}}}\right)}^2}-{\displaystyle \frac{H_{\mathrm{wall}}^{\mathrm{down}}}{R_{\mathrm{wall}}^{\mathrm{down}}}}sin\left(i\right).$$

(3b)

Ellipses ${\epsilon }_{\mathrm{up}}$ and ${\epsilon }_{\mathrm{down}}$ intersect at critical angles ${\theta }_{\mathrm{c}}$ and $\pi -{\theta }_{\mathrm{c}}$, where ${\theta }_{\mathrm{c}}$ is given by

$$sin{\theta }_{\mathrm{c}}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{H_{\mathrm{wall}}^{\mathrm{up}}}{R_{\mathrm{wall}}^{\mathrm{up}}}}+{\displaystyle \frac{H_{\mathrm{wall}}^{\mathrm{down}}}{R_{\mathrm{wall}}^{\mathrm{down}}}}\right)tan\left(i\right)=\delta .$$

(4)

Depending on the wall inclination angle i, there exist two possibilities to know whether both ellipses can intersect: if $\delta <1$ or $\delta >1$, as seen in Figure 3.

For the case $\delta <1$, the region $\mathcal{R}$ is composed of two sub-regions, ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$:

where $0<\theta <arcsin\left(\delta \right)$:

$${\mathcal{R}}_1=\left\{(X,Y):-X_0\le X\le X_0\wedge \mathrm{s}{Y}_{\mathrm{down}}\le Y\le \mathrm{s}{Y}_{\mathrm{up}}\right\},$$

(5a)

and where $arcsin\left(\delta \right)<\theta <\frac{\pi }{2}$:

$${\mathcal{R}}_2=\left\{(X,Y):\pm X_0\le X\le \pm R_{\mathrm{wall}}^{\mathrm{up}}\wedge \mathrm{i}{Y}_{\mathrm{up}}\le Y\le \mathrm{s}{Y}_{\mathrm{up}}\right\},$$

(5b)

that means $\mathcal{R}={\mathcal{R}}_1\cup {\mathcal{R}}_2$ (see Appendix A.3 and Appendix A.4).

For the case $\delta >1$, the region $\mathcal{R}$ is defined as follows

$$\mathcal{R}=\left\{(X,Y):-R_{\mathrm{wall}}\le X\le R_{\mathrm{wall}}\wedge \mathrm{i}{Y}_{\mathrm{up}}\le Y\le \mathrm{s}{Y}_{\mathrm{up}}\right\}.$$

(6)

2.2. Vertical Walls

If in Definition 1, I set $R_{\mathrm{wall}}^{\mathrm{up}}=R_{\mathrm{wall}}^{\mathrm{down}}=R_{\mathrm{wall}}$, a two-dimensional vertical wall is obtained, as seen in Figure 4a. By rotating this line segment around the z-axis, a cylindrical ring is generated, which lies in the Euclidean space $(x,y,z)$, as seen in Figure 4b. This ring is a tri-dimensional cylindrical wall.

Following the same mathematical procedure as in the case of an inclined wall, I obtain

$$\begin{array}{cc}\hfill {\epsilon }_{\mathrm{up}}:Y_{\mathrm{up}}& =\mathrm{s}{Y}_{\mathrm{up}}\cup \mathrm{i}{Y}_{\mathrm{up}},\hfill \end{array}$$

(7a)

$$\begin{array}{cc}\hfill {\epsilon }_{\mathrm{down}}:Y_{\mathrm{down}}& =\mathrm{s}{Y}_{\mathrm{down}}\cup \mathrm{i}{Y}_{\mathrm{down}},\hfill \end{array}$$

(7b)

where

$${\displaystyle \frac{\mathrm{s}{Y}_{\mathrm{up}}}{R_{\mathrm{wall}}}}=cos\left(i\right)\sqrt{1-{\left({\displaystyle \frac{X}{R_{\mathrm{wall}}}}\right)}^2}+{\displaystyle \frac{H_{\mathrm{wall}}}{R_{\mathrm{wall}}}}sin\left(i\right),$$

(8a)

$${\displaystyle \frac{\mathrm{i}{Y}_{\mathrm{up}}}{R_{\mathrm{wall}}}}=-cos\left(i\right)\sqrt{1-{\left({\displaystyle \frac{X}{R_{\mathrm{wall}}}}\right)}^2}+{\displaystyle \frac{H_{\mathrm{wall}}}{R_{\mathrm{wall}}}}sin\left(i\right),$$

(8b)

$${\displaystyle \frac{\mathrm{s}{Y}_{\mathrm{down}}}{R_{\mathrm{wall}}}}=cos\left(i\right)\sqrt{1-{\left({\displaystyle \frac{X}{R_{\mathrm{wall}}}}\right)}^2}-{\displaystyle \frac{H_{\mathrm{wall}}}{R_{\mathrm{wall}}}}sin\left(i\right),$$

(8c)

$${\displaystyle \frac{\mathrm{i}{Y}_{\mathrm{down}}}{R_{\mathrm{wall}}}}=-cos\left(i\right)\sqrt{1-{\left({\displaystyle \frac{X}{R_{\mathrm{wall}}}}\right)}^2}-{\displaystyle \frac{H_{\mathrm{wall}}}{R_{\mathrm{wall}}}}sin\left(i\right).$$

(8d)

Both ellipses intersect at critical angles ${\theta }_{\mathrm{c}}$ and $\pi -{\theta }_{\mathrm{c}}$, where ${\theta }_{\mathrm{c}}$ is given by

$$sin{\theta }_{\mathrm{c}}={\displaystyle \frac{H_{\mathrm{wall}}}{R_{\mathrm{wall}}}}tan\left(i\right)=\delta .$$

(9)

3. The RHADaMAnTe Code

To create synthetic SEDs as arising from the inner curved wall of a gap or hole open by a planet in a protoplanetary disk, a computational code was developed, written in the fortran 90 language, called rhadamante. This code is coupled to the artemise code because the geometry of the wall is required.

As I am interested in estimating the radiation re-emitted by a tri-dimensional wall $\mathcal{W}$ projected on the plane of the sky, in this code, I firstly calculate the angle between the radial ray and the normal ray to the two-dimensional wall ${\mathcal{W}}_{{\pi }_0}$ for each incidental radial radiation ray coming from the central star, as seen in Figure 5.

Then, I construct the tri-dimensional wall as the finite union of tri-dimensional conic rings obtained by rotating inclined line segments about the z-axis at different heights (see Figure 6a and Figure 7).

Next, I calculate the surface projection on the plane of the sky of these rings, which is needed to estimate the radiation emitted by each of them. Some ideas from an algorithm developed for vertical walls [9] are implemented to carry out this task. Finally, I sum the contribution of the emission of all the projected rings to create a synthetic SED.

3.1. The Geometry of the Radiation Re-Emitted by the Wall

rhadamante, coming from Radial Geometry Algorithm for Calculating the Radiation Emitted by a Wall, is a geometrical algorithm which at first calculates the angle between the stellar radiation along a radial ray ${\mathcal{R}}_k$ and the normal to the two-dimensional wall ${\mathcal{W}}_{{\pi }_0}$, as seen in Figure 5. Secondly, this algorithm discretizes the two-dimensional wall, which is not continuous, as seen in Figure 6a.

Let ${\theta }_{\mathit{k}}$ be the angle between the normal ${\mathit{\eta }}_{\mathit{k}}$ on the point $W_{\mathit{k}}$ belonging to the wall ${\mathcal{W}}_{{\pi }_0}$, and the stellar radiation ray ${\mathcal{R}}_{\mathit{k}}$, such that

$${\theta }_{\mathit{k}}={\theta }_{{\mathit{\eta }}_{\mathit{k}}}+{\phi }_k,$$

(10)

where ${\theta }_{{\mathit{\eta }}_{\mathit{k}}}$ is the minimal angle between the normal ${\mathit{\eta }}_{\mathit{k}}$ and the mid-plane (r-axis), and ${\phi }_k$ is the angle between the ray ${\mathcal{R}}_k$ and the mid-plane, as seen in Figure 5.

The angle ${\theta }_k$ is required to calculate the re-emitted stellar radiation by the wall. Because of the wall’s curvature and the radial geometry of the stellar radiation, each parcel of the wall does not absorb the total radiation, as it does in the case of vertical walls. In this case, each parcel absorbs only a fraction of the radiation, which depends on the $cos\left({\theta }_k\right)$.

Let ${\ell }_{\mathit{k}}$ be the tangent line to the wall ${\mathcal{W}}_{{\pi }_0}$ at the point $W_{\mathit{k}}$ with a positive slope $m_{{\ell }_k}$. It follows that the inclination angle of such a line, measured from the r-axis, is ${\theta }_{{\ell }_k}={tan}^{-1}\left(m_{{\ell }_k}\right)$, where

$$m_{{\ell }_k}={\left.{\displaystyle \frac{\mathit{d}\mathcal{W}}{\mathit{d}\mathit{r}}}\right|}_{W_k}.$$

Physically, the wall ${\mathcal{W}}_{{\pi }_0}$ should be characterized by a mathematical continuous function. However, in this case, because of the numerical simulation, the wall is transformed, via the discretization process described in Section 3.2, into a discrete counterpart. So, as the points $W_k$ with $k=1,2,\cdots ,k_{max}$ defining ${\mathcal{W}}_{{\pi }_0}$ are close enough, it is possible to find an approximation of its derivative.

Consider the points $W_k$ and $W_{k+1}$ in the wall ${\mathcal{W}}_{{\pi }_0}$ to be connected along the segment line ${\mathcal{L}}_k$ (as seen in Figure 5); then, the slope $m_{{\mathcal{L}}_k}$ of this line approximates to the derivative with respect to r of ${\mathcal{W}}_{{\pi }_0}$ at the point $W_k$, that is

$${\left.{\displaystyle \frac{\mathit{d}{\mathcal{W}}_{{\pi }_0}}{\mathit{d}\mathit{r}}}\right|}_{W_k}\approx m_{{\mathcal{L}}_k}.$$

Hence, ${\theta }_{{\ell }_k}\approx {\theta }_{{\mathcal{L}}_k}={tan}^{-1}\left(m_{{\mathcal{L}}_k}\right)$.

Next, since the line ${\mathcal{L}}_k$ is almost perpendicular to the normal ${\mathit{\eta }}_{\mathit{k}}$, it follows

$${\theta }_{{\mathit{\eta }}_{\mathit{k}}}\approx \frac{\pi }{2}-{\theta }_{{\mathcal{L}}_k}.$$

(11)

Finally, since the star is located at the origin of coordinate system, it is easy to calculate the angle between the ray ${\mathcal{R}}_k$ and the mid-plane

$${\phi }_k={tan}^{-1}\left({\displaystyle \frac{W_k^{\left(z\right)}}{W_k^{\left(r\right)}}}\right),$$

(12)

where $W_k^{\left(r\right)}$ and $W_k^{\left(z\right)}$ are the r and z coordinates of the point $W_k$.

3.2. Discretization of the Two-Dimensional Wall

By applying the artemise code, a set of points $W_k=(r_k,z_k)$ with $k=1,2,\dots ,k_{max}$ is obtained, defining the two-dimensional wall ${\mathcal{W}}_{{\pi }_0}$. This means that the wall is not a continuous curve. Then, the wall is discretized as a finite union of infinitesimal inclined walls: I connect each couple of points $W_{k-1}$ and $W_k$ by inclined line segments ${\mathcal{W}}_{W_k}$ with height $2H_{\mathrm{wall}}^k$, as seen in Figure 6a.

Consider an inclined line segment ${\mathcal{W}}_{W_k}$ with boundaries $W_{k-1}$ and $W_k$ in the wall ${\mathcal{W}}_{{\pi }_0}$, and let $Pm(W_{k-1},W_k)$ be the mid-point of ${\mathcal{W}}_{W_k}$. As it is required that the vertical height of this line segment to be $2H_{\mathrm{wall}}^k$, I define

$$H_{\mathrm{wall}}^k:=W_k^{\left(z\right)}-P{m}^{\left(z\right)}(W_{k-1},W_k)=\frac{1}{2}(z_k-z_{k-1})\mathrm{if}k=2,3,\dots ,k_{max},$$

where $W_k^{\left(z\right)}$ and $P{m}^{\left(z\right)}(W_k,W_{k-1})$ are the z coordinates of $W_k$ and the mid-point between the points $W_k$ and $W_{k-1}$. See Figure 6b.

If $k=1$, a vertical line segment ${\mathcal{W}}_{W_1}$ is constructed, with boundaries $W_0=(r_1,0)$ and $W_1$, and height $z_1$, which connects to the mid-plane.

3.3. Curved Wall

Let ${\mathcal{W}}_{{\pi }_0}={\bigcup }_k^{k_{max}}{\mathcal{W}}_{W_k}$ be a two-dimensional wall discretized by infinitesimal inclined line segments ${\mathcal{W}}_{W_k}$ for $k=2,\dots ,k_{max}$, and a vertical line segment ${\mathcal{W}}_{W_1}$.

By rotating each inclined line segment around z-axis, one generates a conic ring ${\mathcal{W}}_k$ with minimum radius $R_{k-1}$, maximum radius $R_k$, and total height $2H_{\mathrm{wall}}^k$. Whereas, by rotating the vertical line segment, a cylindrical ring with radius $R_1$ and total height $2H_{\mathrm{wall}}^1$, is obtained. It follows that the tri-dimensional curved wall can be defined as the finite union of a cylindrical ring and several conic rings: $\mathcal{W}={\bigcup }_k^{k_{max}}{\mathcal{W}}_k$. See Figure 7.

Projection on the Plane of the Sky

The wall $\mathcal{W}$ has to be projected on the plane of the sky $(X,Y)$ to calculate the amount of visible surface. Therefore, I consider the coordinate system $(X,Y,Z)$, where Z is along the line of sight, such that

$$X=x,$$

(13a)

$$Y=ycos\left(i\right)-zsin\left(i\right),$$

(13b)

$$Z=zcos\left(i\right)-ysin\left(i\right).$$

(13c)

Since it is necessary to apply the same algorithms of projection described in Section 2.1 and Section 2.2, a geometric translation of each ring ${\mathcal{W}}_k$ to a secondary coordinate system $(x^{\prime },y^{\prime },z^{\prime })$ is required, such that the translated ring ${\mathcal{W}}_k^{\prime }$ is centered at the origin. It is easy to observe that there exists a translation transformation ${\Theta }_1:(x,y,z)\to (x^{\prime },y^{\prime },z^{\prime })$:

$$x^{\prime }=x$$

(14a)

$$y^{\prime }=y$$

(14b)

$$z^{\prime }=z-\delta z_k,$$

(14c)

where $\delta z_k=z_k+\frac{1}{2}\Delta z_k$ is the displacement of the ring along z-axis, to be centered at the origin of the system $(x^{\prime },y^{\prime },z^{\prime })$—see Figure 8—where $z_k$ is the z coordinate of the point $W_k$ and $\Delta z_k=2H_{\mathrm{wall}}^k$.

Applying this translation, it is possible to use the coordinate system $(X^{\prime },Y^{\prime },Z^{\prime })$ to project the ring ${\mathcal{W}}_k^{\prime }$ on the plane $(X^{\prime },Y^{\prime })$:

$$X^{\prime }=x,$$

(15a)

$$Y^{\prime }=y^{\prime }cos\left(i\right)+z^{\prime }sin\left(i\right),$$

(15b)

$$Z^{\prime }=z^{\prime }cos\left(i\right)-y^{\prime }sin\left(i\right).$$

(15c)

From Equations (13)–(15), it follows that there exists a translation transformation ${\Theta }_2:(X^{\prime },Y^{\prime },Z^{\prime })\to (X,Y,Z)$, such that:

$$X=X^{\prime },$$

(16a)

$$Y=Y^{\prime }+\delta z_{k}sin\left(i\right)$$

(16b)

$$Z=Z^{\prime }+\delta z_{k}cos\left(i\right),$$

(16c)

where $\delta z_{k}sin\left(i\right)$ is the projection on the Y-axis of the displacement of the ring along the z-axis, as seen in Figure 8.

Combining Equation (16) with Equations (2) and (3), it follows that for the k-th conic ring

$${\displaystyle \frac{\mathrm{s}{Y}_{\mathrm{up}}^{\left(k\right)}}{R_{\mathrm{wall},k}}}=cos\left(i\right)\sqrt{1-{\displaystyle \frac{X^2}{R_{\mathrm{wall},k}^2}}}+{\displaystyle \frac{H_{\mathrm{wall},k}+\delta z_k}{R_{\mathrm{wall},k}}}sin\left(i\right),$$

(17a)

$${\displaystyle \frac{\mathrm{i}{Y}_{\mathrm{up}}^{\left(k\right)}}{R_{\mathrm{wall},k}}}=-cos\left(i\right)\sqrt{1-{\displaystyle \frac{X^2}{R_{\mathrm{wall},k}^2}}}+{\displaystyle \frac{H_{\mathrm{wall},k}+\delta z_k}{R_{\mathrm{wall},k}}}sin\left(i\right).$$

(17b)

$${\displaystyle \frac{\mathrm{s}{Y}_{\mathrm{down}}^{\left(k\right)}}{R_{\mathrm{wall},k-1}}}=cos\left(i\right)\sqrt{1-{\displaystyle \frac{X^2}{R_{\mathrm{wall},k-1}^2}}}-{\displaystyle \frac{H_{\mathrm{wall},k-1}-\delta z_{k-1}}{R_{\mathrm{wall},k-1}}}sin\left(i\right),$$

(17c)

$${\displaystyle \frac{\mathrm{i}{Y}_{\mathrm{down}}^{\left(k\right)}}{R_{\mathrm{wall},k-1}}}=-cos\left(i\right)\sqrt{1-{\displaystyle \frac{X^2}{R_{\mathrm{wall},k-1}^2}}}-{\displaystyle \frac{H_{\mathrm{wall},k-1}-\delta z_{k-1}}{R_{\mathrm{wall},k-1}}}sin\left(i\right).$$

(17d)

whereas, for the cylindrical ring, by combining Equation (16) with Equation (8), it follows

$${\displaystyle \frac{\mathrm{s}{Y}_{\mathrm{up}}^{\left(1\right)}}{R_{\mathrm{wall},k}}}=cos\left(i\right)\sqrt{1-{\displaystyle \frac{X^2}{R_{\mathrm{wall},k}^2}}}+{\displaystyle \frac{H_{\mathrm{wall},k}+\delta z_k}{R_{\mathrm{wall},k}}}sin\left(i\right),$$

(18a)

$${\displaystyle \frac{\mathrm{i}{Y}_{\mathrm{up}}^{\left(1\right)}}{R_{\mathrm{wall},k}}}=-cos\left(i\right)\sqrt{1-{\displaystyle \frac{X^2}{R_{\mathrm{wall},k}^2}}}+{\displaystyle \frac{H_{\mathrm{wall},k}+\delta z_k}{R_{\mathrm{wall},k}}}sin\left(i\right).$$

(18b)

$${\displaystyle \frac{\mathrm{s}{Y}_{\mathrm{down}}^{\left(1\right)}}{R_{\mathrm{wall},k}}}=cos\left(i\right)\sqrt{1-{\displaystyle \frac{X^2}{R_{\mathrm{wall},k}^2}}}-{\displaystyle \frac{H_{\mathrm{wall},k}-\delta z_k}{R_{\mathrm{wall},k}}}sin\left(i\right),$$

(18c)

$${\displaystyle \frac{\mathrm{i}{Y}_{\mathrm{down}}^{\left(1\right)}}{R_{\mathrm{wall},k}}}=-cos\left(i\right)\sqrt{1-{\displaystyle \frac{X^2}{R_{\mathrm{wall},k}^2}}}-{\displaystyle \frac{H_{\mathrm{wall},k}-\delta z_k}{R_{\mathrm{wall},k}}}sin\left(i\right).$$

(18d)

3.4. Emission of the Wall

To calculate the emission or emergent flux $F_{\nu }$ of the visible wall projected on the plane of the sky, the total emergent intensity $I_{\nu }$ is multiplied by the solid angle ${\Omega }_{\mathrm{wall}}$ of the visible surface of the wall, whose geometry has been described in detail in Section 3.3.

For each element in the visible surface of the projected wall, the thermal emergent intensity, approximated as isotropic, is given by

$$I_{\nu }\approx {\int }_0^{\infty }{B}_{\nu }\left[T_{\mathrm{d}}\left({\tau }_{\mathrm{d}}\right)\right]exp\left(-{\tau }_{\nu }\right)d{\tau }_{\nu }$$

(19)

(see [9] for derivation), where $B_{\nu }$ is the Planck function, ${\tau }_{\mathrm{d}}$ is the total mean optical depth at the disk frequency band, and ${\tau }_{\nu }={\tau }_{\mathrm{d}}({\kappa }_{\nu }/{\chi }_{\mathrm{d}})$, with opacity ${\kappa }_{\nu }$.

The wall temperature $T_{\mathrm{d}}$ is a function of the optical depth of the disk, and it is calculated as follows (see [9] for derivation)

$$T_{\mathrm{d}}^4\left({\tau }_{\mathrm{d}}\right)=\alpha {\displaystyle \frac{F_0}{4{\sigma }_{\mathrm{R}}}}\left[C_1^{\prime }+C_2^{\prime }exp(-q{\tau }_{\mathrm{d}})+C_3^{\prime }exp(-\beta q{\tau }_{\mathrm{d}})\right],$$

(20)

where

$$C_1^{\prime }=(1+C_1)\left(2+{\displaystyle \frac{3}{q}}\right)+C_2\left(2+{\displaystyle \frac{3}{\beta q}}\right),$$

(21a)

$$C_2^{\prime }=(1+C_1)\left({\displaystyle \frac{q{\chi }_{\mathrm{d}}}{{\kappa }_{\mathrm{d}}}}-{\displaystyle \frac{3}{q}}\right),$$

(21b)

$$C_3^{\prime }=C_2\beta \left({\displaystyle \frac{q{\chi }_{\mathrm{d}}}{{\kappa }_{\mathrm{d}}}}-{\displaystyle \frac{3}{q{\beta }^2}}\right),$$

(21c)

and

$$C_1=-{\displaystyle \frac{3w}{1-{\beta }^2}},$$

(22a)

$$C_2={\displaystyle \frac{5w}{\beta \left(1+\frac{2\beta }{3}\right)(1-{\beta }^2)}},$$

(22b)

with $\alpha =1-w$, $\beta =\sqrt{3\alpha }$, and $w={\sigma }_{\mathrm{s}}/{\chi }_{\mathrm{s}}$ is the mean albedo to the stellar radiation and $F_0=L_{★}/4\pi R_{\mathrm{wall}}^2$, where $L_{★}$ is the stellar luminosity.

At a distance d from the observer, the total solid angle is given by

$${\Omega }_{\mathrm{wall}}=\{\begin{array}{c}cos\left(i\right){\left({\displaystyle \frac{R_{\mathrm{wall}}^{\mathrm{up}}+R_{\mathrm{wall}}^{\mathrm{down}}}{d}}\right)}^2\left[\delta \sqrt{1-{\delta }^2}+arcsin\left(\delta \right)\right],\mathrm{if}\delta <1,\hfill \\ \pi cos\left(i\right){\left({\displaystyle \frac{R_{\mathrm{wall}}^{\mathrm{up}}}{d}}\right)}^2,\mathrm{if}\delta \ge 1,\hfill \end{array}$$

(23)

with

$$\delta ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{H_{\mathrm{wall}}^{\mathrm{up}}}{R_{\mathrm{wall}}^{\mathrm{up}}}}+{\displaystyle \frac{H_{\mathrm{wall}}^{\mathrm{down}}}{R_{\mathrm{wall}}^{\mathrm{down}}}}\right)tan\left(i\right),$$

for conic rings, and

$${\Omega }_{\mathrm{wall}}=\{\begin{array}{c}2cos\left(i\right){\left({\displaystyle \frac{R_{\mathrm{wall}}}{d}}\right)}^2\left[\delta \sqrt{1-{\delta }^2}+arcsin\left(\delta \right)\right],\mathrm{if}\delta <1,\hfill \\ \pi cos\left(i\right){\left({\displaystyle \frac{R_{\mathrm{wall}}}{d}}\right)}^2,\mathrm{if}\delta \ge 1,\hfill \end{array}$$

(24)

with

$$\delta ={\displaystyle \frac{H_{\mathrm{wall}}}{R_{\mathrm{wall}}}}tan\left(i\right),$$

for cylindrical rings.

3.4.1. Rosseland Mean Opacity

Equation (19) requires the calculation of the opacity ${\kappa }_{\nu }$. This dominant opacity depends on the chemical composition, pressure, and temperature of the gas, as well as the frequency $\nu$ of the incident light. This is a complex endeavour. The problem can be simplified by using a mean opacity averaged over all frequencies, so that only the dependence on the physical properties of the gas remains. In the current work, the Rosseland mean opacity is used, defined as

$${\displaystyle \frac{1}{\langle {\kappa }^R\rangle }}:={\displaystyle \frac{{\displaystyle {\int }_0^{\infty }\frac{1}{{\kappa }_{\nu }}\frac{\partial B_{\nu }}{\partial T}d\nu }}{{\displaystyle {\int }_0^{\infty }\frac{\partial B_{\nu }}{\partial T}d\nu }}},$$

(25)

where $B_{\nu }\left(T\right)$ is the Planck function, and T is the disk temperature [34].

To calculate the total Rosseland mean opacity ${\kappa }^{\mathrm{R}}$, I consider that all the dust grains exist and the mixture of dust grains is made of small and big grains. Using the previous assumptions, the total Rosseland mean opacity is calculated as follows:

$${\kappa }_{\mathrm{R}}(x,z)={\zeta }_{\mathrm{small}}(x,z){\kappa }_{{\mathrm{R}}_{\mathrm{small}}}+{\zeta }_{\mathrm{big}}(x,z){\kappa }_{{\mathrm{R}}_{\mathrm{big}}},$$

(26)

where ${\kappa }_{{\mathrm{R}}_{\mathrm{small}}}$ and ${\kappa }_{{\mathrm{R}}_{\mathrm{big}}}$ are the Rosseland mean opacities associated with the small dust-grain size distribution and big dust-grain size distribution, respectively. ${\zeta }_{\mathrm{small}}$ and ${\zeta }_{\mathrm{big}}$ represent the abundances (dust-to-gas mass ratio) of the small and big grains, respectively:

$${\zeta }_{\mathrm{small}}(x,z)=\frac{1}{2}{\zeta }_{\mathrm{small},0}\left\{1-tanh\left[k\left(1-{\displaystyle \frac{z}{\delta H}}\right)\right]\right\},$$

(27a)

$${\zeta }_{\mathrm{big}}(x,z)=\frac{1}{2}{\zeta }_{\mathrm{big},0}\left\{1+tanh\left[k\left(1-{\displaystyle \frac{z}{\delta H}}\right)\right]\right\},$$

(27b)

here, $\delta H$ represents a small fraction of the scale height of the disk, and k is a factor which defines a smooth transition between small and big grain population [35].

The monochromatic opacity ${\kappa }_{\nu }$ in Equation (25) depends on the dust species in the mixture and their physical and chemical properties, such that it is calculated as the sum of the monochromatic opacity of each grain species:

$${\kappa }_{\nu }=\sum _q^{q_{max}}{\kappa }_{\nu }^q(a_{min}^q,a_{max}^q,{\sigma }^q,{\eta }^q),$$

(28)

where $a_{min}^q$ and $a_{max}^q$ are the sizes of the small and big grains, and ${\sigma }^q$ and ${\eta }^q$ are the abundance and refraction index of the species. Here, q indicates the name of the species (e.g., silicates, organics, amorphous carbon, ice and troilite) in the dust composition of the disk. The monochromatic opacities are calculated using the Mie theory by implementing some modified routines of a code developed in [9].

Summarizing, to calculate the emergent flux emitted by the curved wall, I have developed a computational code called rhadamante. This code is based on the geometry of the wall calculated by the artemise code. Figure 9 shows the flowchart of the code. For some tests, see Appendix B.

4. Results: Implementation in the Stellar System LkCa 15

In this section, I present a model of the truncated dusty disk of the T Tauri star LkCa 15 that accounts for the Spitzer Infrared Spectrograph observations. The mid-infrared spectral energy distribution has been modeled from 5 and 40$\mathsf{\mu }\mathrm{m}$, as arising from the inner curved wall of the outer disk. In this model, a planet with a mass of $10M_{♃}$ is responsible for the wall curvature. The free dust hole has a radius of ∼$53\mathrm{AU}$ along the mid-plane. The wall has a half-height of ∼$12\mathrm{AU}$ and it is illuminated at normal incidence by the central star, but it is also shadowed because of the presence of an internal optically thick disk.

4.1. Simulation: Planet–Disk Interaction

As I am interested in characterizing the geometry of the wall of the disk gap, in the LkCa 15 system, it is necessary to analyze the vertical structure of the disk. Assuming the gap was opened by an embedded planet, I use the fargo-3d hydro-dynamical code to launch two numerical simulations of the disk–planet interaction until the 500th. Simulations are scale-free units. The only difference among these simulations is the size in resolution ($N_{\mathrm{X}}\times N_{\mathrm{Y}}\times N_{\mathrm{Z}}$). The low resolution of $50\times 50\times 30$ was used to find quickly the orbit where the system reaches a quasi-stationary state. The medium resolution of $250\times 250\times 100$ was used to obtain a better approximation of the wall. The fargo-3d simulation parameters are described in

Table 1. Disk–planet simulation parameters.

	Parameter	Value
Disk	Aspect ratio H	$0.045$
	Surface density ${\Sigma }_0$	$1.44666\times {10}^{-4}$
	$\alpha$-viscosity	$0.0$
	${\Sigma }_0$ slope	1.0
	Flaring index	0.0
Planet	$m_{\mathrm{p}}$ (mass)	$10M_{♃}$
	$R_0$ (position)	1
	RocheSmoothing	0.4
	Acretion	No
Mesh	Units	unitless
	Dimension	3D
	Geometry	spherical
	$[X_{min},X_{max}]$	$[-\pi ,\pi ]$
	$[Y_{min},Y_{max}]$	$[0.1,3.666]$
	$[Z_{min},Z_{max}]$	$[1.37340076,\pi /2]$
Timing	Orbits	500

. It is worth mentioning that if such parameters were changed, the geometry and SED of the wall would also change.

In Figure 10, the tri-dimensional structure of the 100th orbit of the LkCa 15 disk simulation is presented. This figure shows the dust density of the disk in an unitless logarithmic scale. Furthermore, it is in such an orbit that the system reaches a stationary state.

Figure 11 shows the mid-plane and the vertical plane of the disk for the 100th orbit. The planet is located at the point $(x,y)=(1,0)$, as seen in Figure 11a, corresponding to $R_{\mathrm{p}}=32.3$ AU. The density isocontours for $-12.0<log\left(\rho \right)<0.0$ can be observed, where $\rho$ is in $M_{\odot }/r_{\mathrm{p}}^{-3}$). The zone with no lines is a dust-free region, that is, the gap.

More quantitative proof of the gap opening can be appreciated in Figure 12, which shows the surface-density profile of the disk for some orbits. The solid line represents the initial surface-density profile of the protoplanetary disk (i.e., the full disk or orbit number 0). This profile follows a power law $\rho \left(r\right)\propto r^{-2.0}$ because of the initial condition of the disk surface-density profile is $\Sigma \left(r\right)\propto r^{-1.0}$. As the system (disk + planet) is evolving, the initial density profile is perturbed due to density spiral waves created by the embedded planed, and all these new profiles tend to follow or approach the power law of the initial condition. It is in a neighborhood of the planet, with ∼$16\mathrm{AU}$ in radius, where the density changes drastically with respect to the initial condition.

4.2. Dust Grain

The optical depth of the disk depends on the disk material opacity, it is assumed that the disk is a mixture of grains composed of silicates with mass fraction ${\zeta }_{\mathrm{sil}}=0.0034$, organics with ${\zeta }_{\mathrm{org}}=0.0041$, and troilite with ${\zeta }_{\mathrm{troi}}=8.0\times {10}^{-4}$, consistent with the model proposed in [36]. The grains are assumed to be spheres, which obey the standard MRN grain-size distribution $n\left(a\right)\sim$$a^{-3.5}$ [10].

I consider two grain populations: small grains between minimum radius $a_{min}=0.005\mathsf{\mu }\mathrm{m}$ and maximum radius $a_{max}=0.25\mathsf{\mu }\mathrm{m}$, and big grains between minimum radius $a_{min}=0.005\mathsf{\mu }\mathrm{m}$ and maximum radius $a_{max}=1000\mathsf{\mu }\mathrm{m}$. A smooth transition between both dust populations is assumed—see Equation (27) in Section 3.4.1—with $k=20$, $\delta =0.1$, ${\zeta }_{\mathrm{small},0}=0.5$, and ${\zeta }_{\mathrm{big},0}=6.8$. The optical constants for silicates were taken from [37,38,39], with the organics taken from [36], and the troilite from [36,40]. The sublimation temperature of the grains in the mixture $T_{\mathrm{sil}}=1400\mathrm{K}$, $T_{\mathrm{org}}=425\mathrm{K}$, and $T_{\mathrm{troi}}=680\mathrm{K}$ is also taken into account.

For the composition of the silicate dust grains, amorphous Mg–Fe glassy olivine (${\mathrm{Mg}}_{2x}{\mathrm{Fe}}_{2-2x}{\mathrm{SiO}}_4$) or glassy pyroxene (${\mathrm{Mg}}_x{\mathrm{Fe}}_{1-x}{\mathrm{SiO}}_3$) is considered, as implemented in [41]. Here, the subscripts represent the fraction of Mg and Fe in the silicate grain. The optical properties of the silicate grains were taken from [38].

4.3. The Vertical Geometry of the Wall

It was found that the $10M_{♃}$ mass planet candidate, when located at $32.3\mathrm{AU}$ from the central star, opens a gap around the young transitional disk host LkCa 15. The artemise code was implemented to analyze the 3D simulation data to obtain the geometry of the wall, considering that the mixture of dust grains is made of organics, troilite, and silicates. The radii of the wall along the mid-plane of the disk and the heights of the wall have a dependence on the chemical composition of the silicate grains, as shown in

Table 2. The parameters of the wall for different chemical compositions of silicate dust grains: pyroxene and olivine. The organic and troilite grain composition is the same for all cases.

silicate	${\mathit{R}}_{\mathbf{wall}}$ [AU]	${\mathit{H}}_{\mathbf{wall}}$ [AU]
pyroxene
${\mathrm{Mg}}_{0.4}$${\mathrm{Fe}}_{0.6}$${\mathrm{SiO}}_3$	56.0	11.5
${\mathrm{Mg}}_{0.6}$${\mathrm{Fe}}_{0.4}$${\mathrm{SiO}}_3$	49.8	12.3
${\mathrm{Mg}}_{0.8}$${\mathrm{Fe}}_{0.2}$${\mathrm{SiO}}_3$	51.0	12.0
${\mathrm{Mg}}_{0.95}$${\mathrm{Fe}}_{0.025}$${\mathrm{SiO}}_3$	50.0	13.2
olivine
${\mathrm{Mg}}_{0.8}$${\mathrm{Fe}}_{0.2}$${\mathrm{SiO}}_4$	52.5	10.0
${\mathrm{Mg}}_{0.5}$${\mathrm{Fe}}_{0.5}$${\mathrm{SiO}}_4$	53.0	12.0

. Figure 13 shows the geometry of a wall where the silicate dust grains are glassy olivine with 50% Fe and 50% Mg.

The location of the planet candidate is not consistent with the observations (e.g., [27]), which suggest that the possible massive planets LkCa 15b and LkCa 15c are located at $14.7\pm 2.1\mathrm{AU}$ and $18.6\pm 2.5\mathrm{AU}$, respectively, along the semi-major axis. However, the radii of the wall along the mid-plane $R_{\mathrm{wall}}^{\mathrm{mid}-\mathrm{plane}}$ are similar to those measured in [42,43], at ∼$50\mathrm{AU}$, and [24], at ∼$56\mathrm{AU}$.

4.4. SED of the Wall

I model LkCa 15 as a central star with the properties described in Section 1.2, surrounded by an optically thick inner disk (as shown in Figure 11) and an outer disk truncated at ∼$120\mathrm{AU}$. It is considered that the gap has a curved wall at different locations and heights according to Table 2. In the models, it is assumed that LkCa 15 is at $140\mathrm{pc}$ from Earth in the Taurus–Auriga star-forming region [23] and the disk inclination is $i={50}^{\circ }$ [27]. A representation of the model is showed in Figure 14.

The inner disk casts an umbra over the wall of the outer disk, as seen in Figure 14. This umbra has to be removed from the SED of the outer disk wall. In order to find $z_{\mathrm{umb}}$, I implement some improved routines developed in [44] for curved sublimation walls. This code uses opacities to calculate the shape of the wall and assumes that the stellar rays are parallel to the mid-plane. It was found that the wall of the inner disk starts at ∼$0.04932\mathrm{AU}$, from the central star, and runs until ∼$0.18407\mathrm{AU}$, where it reaches ∼$2.3550\times {10}^{-2}\mathrm{AU}$ in height. Also, the temperature of the sublimation wall decreases with radius, ranging from $1014.3\mathrm{K}$ to $1443.7\mathrm{K}$.

In a first approximation, assuming the star as a point, the sublimation wall produces only an umbra over the wall of the outer disk (see Figure 14). To calculate the size $z_{\mathrm{umb}}$ of this umbra, the angle ${\theta }_{\mathrm{shw}}$ subtended by the height $z_{\mathrm{wall}}^{\mathrm{inner}}$ of the sublimation wall is needed. In addition, for some points $(r,z)$ in the outer wall, I calculate the angle ${\theta }_{\mathrm{shw}}^{\prime }=arctan(z/r)$ until it reaches the value ${\theta }_{\mathrm{shw}}$.

Applying the previous algorithm to the geometry of the wall (see Figure 13), it was found that the umbra produced by the sublimation wall of the inner disk onto the wall of the outer disk is $7.44\mathrm{AU}$ in height (above and below the disk mid-plane). It means the contribution to the SED of the outer wall comes from a region of the wall from ∼$58.84\mathrm{AU}$ to ∼$68.7\mathrm{AU}$ along the radial direction, and from $7.44\mathrm{AU}$ to $12.22\mathrm{AU}$ along the vertical direction. In Figure 15a, the surface area of the whole curved wall is shown, and in Figure 15b, the surface area considered the umbra cast by the inner disk is shown.

The synthetic SEDs of the wall of the outer disk, where the dust consists of grains of glassy pyroxene with different concentration of Fe and Mg are shown in Figure 16. A chi-square test was performed for each model, to examine whether the synthetic SED fits to the Spitzer IRS SED. I found that no model is able to fit the observed SED (${\chi }^2$∼300) or reproduce the silicate peak at ∼$10\mathsf{\mu }\mathrm{m}$. However, for glassy pyroxene with 60% Fe and 40% Mg (see Figure 16a), it seems the silicate peak tries to appear at ∼$10\mathsf{\mu }\mathrm{m}$; this leads us to think that a lesser concentration of Mg in the pyroxene composition would produce the silicate feature. Unfortunately, the optical constants needed to calculate the opacities for such chemical concentrations are not available in the literature.

Figure 17 shows the synthetic SEDs of the wall of the outer disk, where the dust mixture consists of grains of glassy olivine with a different concentration of Fe and Mg. I found that none of these configurations is capable of fitting the observed SED (${\chi }^2>200$). However, in both cases, a silicate peak appears at ∼$10\mathsf{\mu }\mathrm{m}$. A concentration of 50% Fe and 50% Mg produces the best fit (see Figure 17b).

Previous results lead us to say that the SED of LkCa 15 is not dominated by the contribution of the curved wall of the outer disk in mid-infrared. However, when olivine grains with a concentration of 50% Fe and 50% Mg or 80% Fe and 20% Mg are in the dust mixture, a silicate feature appears at ∼$10\mathsf{\mu }\mathrm{m}$.

4.5. SED of the System

Considering the SED contribution of the inner sublimation wall and the star, in addition to the SED contribution of the wall of the outer disk, I present a more complete model of the stellar system LkCa 15. For the stellar SED, a Kurucz atmosphere model was used with $log[Z/H]=0.5$ and $log\left[g\right]=4.0$.

Table 3. Stellar and model properties: $R_{\mathrm{wall}}^{min}$ and $H_{\mathrm{wall}}$ in the case of the outer wall are measured at the location of the umbra cast by the inner disk. The olivine silicate grain composition is 50% Fe and 50% Mg.

	parameter	value
Star	$M_{★}$	$1.0M_{\odot }$
	$R_{★}$	$1.6R_{\odot }$
	$T_{★}$	4370 K
	$d_{★}$	$120\mathrm{pc}$
	$\dot{M}$	$3.3\times {10}^{-9}{M}_{\odot }{\mathrm{yr}}^{-1}$
Disk	Inclination	${50}^{\circ }$
Inner wall	$a_{min}$	$0.005\mathsf{\mu }\mathrm{m}$
	$a_{max}$	$1.0\mathsf{\mu }\mathrm{m}$
	$R_{\mathrm{wall}}^{min}$	$0.04932\mathrm{AU}$
	$R_{\mathrm{wall}}^{max}$	$0.18407\mathrm{AU}$
	$H_{\mathrm{wall}}$	$0.02335\mathrm{AU}$
	$T_{\mathrm{wall}}^{min}$	$1014.3\mathrm{K}$
	$T_{\mathrm{wall}}^{max}$	$1443.7\mathrm{K}$
Dust	-	silicates
	-	graphite
Outer wall	$a_{min}^{\mathrm{small}}$	$0.005\mathsf{\mu }\mathrm{m}$
	$a_{min}^{\mathrm{small}}$	$0.25\mathsf{\mu }\mathrm{m}$
	$a_{min}^{\mathrm{big}}$	$0.005\mathsf{\mu }\mathrm{m}$
	$a_{min}^{\mathrm{big}}$	$1000\mathsf{\mu }\mathrm{m}$
	$R_{\mathrm{wall}}^{min}$	$58.84\mathrm{AU}$
	$R_{\mathrm{wall}}^{max}$	$68.70\mathrm{AU}$
	$z_{\mathrm{umb}}$	$7.44\mathrm{AU}$
	$H_{\mathrm{wall}}$	$4.78\mathrm{AU}$
	$T_{\mathrm{wall}}^{min}$	$59.76\mathrm{K}$
	$T_{\mathrm{wall}}^{max}$	$80.55\mathrm{K}$
Dust		olivine
	-	organics
	-	troilite

lists the parameters for this model. In Figure 18, the contributions of the star, the inner sublimation wall, the wall of the outer disk, the observed SED, and the total synthetic SED are shown.

Figure 19 shows a comparison between the observed and synthetic SEDs. It can be observed that, only for some wavelengths in the mid-infrared, ∼$15.0\mathsf{\mu }\mathrm{m}<\lambda <$∼$20.0\mathsf{\mu }\mathrm{m}$, and the synthetic SED fit to LkCa 15 observed SED; I estimated ${\chi }^2$∼$0.45$. For all the other wavelengths in the field of view of the Spitzer IRS (5.217–37.86 $\mathsf{\mu }$m), the synthetic SED is below the observed SED. For the wavelengths ∼$5.217\mathsf{\mu }\mathrm{m}<\lambda \le$∼$8.0\mathsf{\mu }\mathrm{m}$ the difference is not very high (${\chi }^2$∼$3.36$); however, for the wavelengths ∼$8.0\mathsf{\mu }\mathrm{m}<\lambda \le$∼$15.0\mathsf{\mu }\mathrm{m}$ (with ${\chi }^2$∼$17.72$) and ∼$20.0\mathsf{\mu }\mathrm{m}<\lambda \le$∼$37.86\mathsf{\mu }\mathrm{m}$ (with ${\chi }^2$∼$19.08$), this difference becomes significant.

I can suggests that the inner sublimation wall and the stellar photo-sphere cannot account for the significant near-infrared excess in LkCa 15. It means that the SED model also requires an optically thin dust region inside the gap to explain the silicate feature at $10\mathsf{\mu }\mathrm{m}$ measured by Spitzer and shown in Figure 19. In other words, the gap of LkCa 15 is not completely dust free. Similarly, as the wall of the outer disk cannot account for the excess in the mid- and long-infrared, I assume that this SED model also requires of the contribution of the outer disk.

5. Discussion: Vertical-Wall SED vs. Curved-Wall SED

Wall SED models of LkCa 15 (and many other pre-TDs and TDs) are based on vertical walls (e.g., [2,25]). The rhadamante code is able to construct wall SEDs based on this geometry. Here, I present some wall SED models of LkCa 15, considering vertical walls to compare with the best-fit curved-wall SED model (as seen in Figure 20).

In these vertical wall SED models, it is also considered that the umbra cast by the inner disk is $6.73\mathrm{AU}$ (measured from the disk mid-plane to up). The height of the wall is $12\mathrm{AU}$ (measured from the disk mid-plane to up). The size and composition of the dust grains remain the same as described for the best-fit curved-wall SED model.

The vertical walls were located at 30, $58.11$, $60.85$ and $68.7\mathrm{AU}$. rhadamante estimated the area of the visible surface projected on the plane of the sky of these vertical walls, $A_{\mathrm{wall}}^{\mathrm{vert}}$, and the curved wall, $A_{\mathrm{wall}}^{\mathrm{cur}}$. It was found that $A_{\mathrm{wall}}^{\mathrm{vert}}>A_{\mathrm{wall}}^{\mathrm{cur}}$, if the umbra of the inner disk is considered. While if the emission of the whole wall is considered, $A_{\mathrm{wall}}^{\mathrm{vert}}<A_{\mathrm{wall}}^{\mathrm{cur}}$.

Wall temperatures were also estimated for the vertical walls, via Equation (20), and it was found that such temperatures were $27.63$, $26.41$, $25.80$, and $24.29\mathrm{K}$, respectively. This means that the temperature of a vertical wall, $T_{\mathrm{wall}}$, decreases if its radius, $R_{\mathrm{wall}}$, increases. Furthermore, the radiation emitted by a vertical wall, $F_{\lambda }$, also decreases as $R_{\mathrm{wall}}$ increases, as seen in Figure 20.

Finally, I compared the radiation emitted by vertical walls, $F_{\lambda }^{\mathrm{vert}}$, with the radiation emitted by the curved wall, $F_{\lambda }^{\mathrm{curved}}$ (see Figure 20). It was found that $F_{\lambda }^{\mathrm{vert}}>F_{\lambda }^{\mathrm{curved}}$ for wavelengths between 5 and $1000\mathsf{\mu }\mathrm{m}$. The difference between fluxes becomes significant by about one order of magnitude for $9\mathsf{\mu }\mathrm{m}<\lambda <500\mathsf{\mu }\mathrm{m}$.

This infrared excess arises, in part, from the angle, $\theta$, between the radiation ray and the normal to the wall (see Section 3.1), because $F_{\lambda }\propto cos\theta$. For vertical walls, $cos\left(\theta \right)\sim 1$ for all the radiation rays hitting the wall, because the normal wall is always parallel to the disk mid-plane. Whereas for curved walls $0\le cos\left(\theta \right)\le 1$, because of the wall curvature. In addition, the highest regions of the curved wall are farther from the central star than the vertical one. That is, curved walls are cooler in the top that vertical walls.

The two facts described above induce a lower exposure of the curved wall to the host-star radiation, which derives from the lower radiative heating of the wall, and, consequently, in the significantly lower radiative infrared cooling flux. For this reason, the infrared emission in the curved wall model, which is one order of magnitude lower, is as significant, as it is correct in terms of the very basic physical considerations: the height of the curved wall and the umbra cast by the inner disk onto the outer disk.

It is worth mentioning that the geometry, location, and height of the curved wall arise from a physical mechanism considering the opacities and chemical composition of the disk, where the disk is the result of a three-dimensional hydro-dynamical simulation, whereas for the vertical walls there is not any physics involved in the selection of these parameters.

The curved-wall SED model is the best choice to fit the Spitzer IRS SED of LkCa 15, as is shown in Figure 19. However, to improve this fit, in addition to the contribution of the central star, the inner disk, and the wall of the gap, it is necessary to consider the contribution of the outer disk and an optically thin dust region inside the gap, as discussed in Section 4.4.

In the literature, it is possible to find some models of the SED of LkCa 15 that fit data from sub-millimeter [45] and millimeter [46] dust continuum observations. However, there are some differences between these models and the one presented in this work. The main difference arises on the geometry of LkCa 15.

In [45], two possible geometries are considered for LkCa 15: a model with a 1 AU optically thick inner disk that rises out of its shadow, and a second model with a 5 AU optically thin spherical dust shell and a flatter outer disk with a fully illuminated inner rim. Meanwhile, in [46], a model with a dust ring inside the gap is considered.

In this work, the model considers the curved wall of an optically thick inner disk, and the curved wall of the gap, that is, the inner wall of the outer disk. It was assumed that such a wall could explain the silicate feature at ∼$10\mathsf{\mu }$m. Models with two different silicate composition with a mixture of organics and troilite were considered. Some models implemented silicate dust grains such as pyroxene with different concentrations of Mg and Fe and, and other models implemented olivine with different concentrations of Mg and Fe.

6. Conclusions

The computational code rhadamante was developed in order to calculate synthetic SEDs of protoplanetary disks. As an initial parameter, the code requires the geometry of the wall coming from a hydro-dynamical three-dimensional simulation of the planet–disk interaction and dust-grain properties. This code is useful to explain the observed SED of young stellar systems in transition stage. It would lead to an unveiling of the structure of the system, such as, the inner disk, the gap, and the outer disk, and even the location and mass of the embedded planet responsible of the gap opening. The code is free and it is available for collaboration by contacting the author.

From the implementation of this code in the pre-transitional disk LkCa 15, it can be concluded that:

With all models of the SED consisting only of a curved wall, using different concentrations of Fe and Mg for the silicate (pyroxene and olivine) grains, suggest that LkCa 15’s Spitzer IRS SED cannot be accounted for by the emission of a curved wall. Chi-square tests indicate the models are not good (${\chi }^2$∼${10}^2$). In addition, for models with a dust mixture containing glassy amorphous olivine grains, ${\mathrm{Mg}}_{0.2}$${\mathrm{Fe}}_{0.2}$${\mathrm{SiO}}_4$ or ${\mathrm{Mg}}_{0.5}$${\mathrm{Fe}}_{0.5}$${\mathrm{SiO}}_4$, a $10\mathsf{\mu }\mathrm{m}$ silicate feature can be observed. However, the intensity of this silicate emission is very weak compared to that observed in the Spitzer IRS SED.

The total synthetic SED of LkCa 15 considers the contribution of the stellar photo-sphere, the sublimation wall of the inner disk, and the curved wall of the gap. For the dust grains in the disk, it is assumed a mixture of amorphous glassy olivine with 50% Fe and 50% Mg, and a small amount of organics and troilite grains. This synthetic SED fits the observed SED better (${\chi }^2$∼45) than synthetic SEDs, accounting for silicate grains with other concentrations of Mg and Fe. However, this SED cannot fit the Spitzer IRS SED. The fit is only good (${\chi }^2$∼$0.076$) for a small band in the mid-infrared, ∼$15.5\mathsf{\mu }\mathrm{m}<\lambda <$∼$18.0\mathsf{\mu }\mathrm{m}$.

Several limitations exist in the current model: The contribution of the inner disk was not considered, neither the contribution of an optically thin region inside the gap (e.g., [2,25]), nor the optically thick outer disk. The inner disk might contribute to the SED at wavelengths in the near infrared. The optically thin region might explain the silicate feature of the Spitzer IRS SED at $10\mathsf{\mu }\mathrm{m}$, while the outer disk might contribute to the SED at wavelengths longer than $18\mathsf{\mu }\mathrm{m}$.

Vertical-wall SED models, via rhadamante code, show a difference of one order of magnitude in the flux, $F_{\lambda }^{\mathrm{vert}}$, compared to the curved-wall SED model, $F_{\lambda }^{\mathrm{curved}}$, for wavelengths from 8 to $35\mathsf{\mu }\mathrm{m}$. This difference arises from dependency of the flux, $F_{\lambda }$, on the cosine of the angle, $\theta$, between the stellar radiation ray and the normal to the wall. This lower exposure of the curved wall to the stellar radiation results in a much lower radiative heating of the wall, and, consequently, in the significantly lower radiative infrared cooling flux.

The synthetic SED of a curved wall, estimated by the rhadamante code, includes the physical and chemical mechanisms absent in the estimation of the SED of a vertical wall. That is, a curved-wall SED is better to fit the Spitzer IRS SED of LkCa 15 or any other protoplanetary disk.