TRANSPORT OF THE FIRST ROCKS OF THE SOLAR SYSTEM BY X-WINDS

In this section, we provide an introduction on the X-wind configurations based on Shu et al. (1994a, 1996) and identify several stellar and disk parameters that govern the X-wind.

An accreting protoplanetary disk that interacts with the rotating magnetosphere of a protostar with a pure dipole is truncated at a radius of

$$\begin{equation} R_{\rm X}=C_1\bigg (\frac{\mu _*^4}{{\it GM}_*\dot{M}_{\rm D}^2}\bigg)^{1/7} {,} \end{equation} \tag{ 1 }$$

where μ_* is the stellar dipole moment, M_* is the stellar mass, $\dot{M}_{\rm D}$ is the inflow rate of the disk, G is the gravitational constant, and C₁ is a dimensionless coefficient that we assume to be unity in the following (e.g., Shu et al. 1996). In steady state, the inner disk edge corotates with the star at the Keplerian angular velocity of

$$\begin{equation} \Omega _*=\Omega _{\rm X}=\bigg (\frac{{\it GM}_*}{R_{\rm X}^3}\bigg)^{1/2}\ . \end{equation} \tag{ 2 }$$

The wind has a terminal velocity $\bar{v}_w$ when leaving the X region. The terminal velocity can be written as

$$\begin{equation} \bar{v}_w=(2J_g-3)^{1/3}R_{\rm X}\Omega _{\rm X}\,, \end{equation} \tag{ 3 }$$

where J_g is the dimensionless specific angular momentum of the gas (Shu et al. 1994a).

The formation of CAIs requires the thermal treatment at 1800–2200 K. CAIs and chondrules entrained in the X-wind experience the peak temperature when they are lifted into direct sunlight. The peak temperature is given by Shu et al. (1996) as

$$\begin{eqnarray} T_{\rm peak}&=& \bigg \lbrace \frac{L_*}{16\pi \sigma R_{\rm X}^2}+\frac{L_*}{8\pi ^2\sigma R_*^2}\bigg [\arcsin \bigg (\frac{R_*}{R_{\rm X}}\bigg)\nonumber\\ &&-\bigg (\frac{R_*}{R_{\rm X}}\bigg)\bigg (1-\frac{R_*^2}{R_{\rm X}^2}\bigg)^{1/2}\bigg ]\bigg \rbrace ^{1/4}\,, \end{eqnarray} \tag{ 4 }$$

where L_* is the stellar luminosity, σ is the Stefan–Boltzmann constant, and R_* is the radius of the star.

From above, we see that the physical properties of X-winds are determined by three parameters of a planetary system: M_*, μ_*, and $\dot{M}_{\rm D}$. In theory, the X-wind can exist in either the "embedded" stage, when the protostar is still embedded in its natal envelope of gas and dust, or the "revealed" stage, when the outflowing wind has reversed the infall of the envelope and revealed the central star and the accretion disk (Shu et al. 1996). At different stages, the star has different masses, the stellar magnetic dipole moment varies in different ranges, and the accretion rate has different values. As long as the lifetime of these two stages is constrained fairly well by the observations of YSOs, we use typical values of M_* and $\dot{M}_{\rm D}$ for each stages, and let μ_* vary in different ranges, respectively. Following Shu et al. (1997), we define a high state, an average state, and a low state according to the value of μ_* for each stage, as shown in Table 1. Key X-wind properties R_X, Ω_X, $\bar{v}_w$, and T_peak of each state are also given in Table 1. As evident from Table 1, a proto-Sun with a stronger magnetic field truncates the protoplanetary disk at a larger radius, and therefore has lower T_peak and $\bar{v}_w$. Notably, CAIs can only form during the EA and EL cases.

Table 1. Definition and X-wind Properties of All States Considered in This Paper

Statesa	M*	$\dot{M}_{\rm D}$	μ*	RX	ΩX	$\bar{v}_w$	Tpeakb
	(M☉)	(M☉ yr−1)	(G cm−3)	(cm)	(s−1)	(cm s−1)	(K)
EH	0.5	2 × 10−6	6	1.43 × 1012	4.76 × 10−6	1.43 × 107	1325
EA	0.5	2 × 10−6	2	7.64 × 1011	1.22 × 10−5	1.96 × 107	1837
EL	0.5	2 × 10−6	1	5.14 × 1011	2.21 × 10−5	2.39 × 107	2270
RH	0.8	1 × 10−7	3	2.12 × 1012	3.34 × 10−6	1.49 × 107	941
RA	0.8	1 × 10−7	1	1.13 × 1012	8.57 × 10−6	2.04 × 107	1299
RL	0.8	1 × 10−7	0.5	7.61 × 1011	1.55 × 10−5	2.48 × 107	1598

Notes. Each state is defined by a set of parameters (M_*, $\dot{M}_{\rm D}$, μ_*). For each case, key X-wind properties R_X, Ω_X, $\bar{v}_w$, and T_peak are calculated based on Equations (1)–(4). Full names of cases are given following the table, but simplified names, given in the first column of the table, are used in the text. ^aEH: embedded stage, high state; EA: embedded stage, average state; EL: embedded stage, low state; RH: revealed stage, high state; RA: revealed stage, average state; RL: revealed stage, low state. ^bFor EH, EA, and EL cases, L_* = 4.4 L_☉ and R_* = 3 R_☉ are used in the calculation of T_peak, where L_☉ and R_☉ are the current luminosity and radius of the Sun. Similarly, for RH, RA, and RL cases, we assume L_* = 2.5 L_☉ and R_* = 3 R_☉.

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For each state, the gas density and flow velocity of the X-wind can be determined numerically, from a set of equations involving the conservation of mass, specific momentum, and specific energy (see Shang 1998 for the detail formulation). In the following, we formulate the equation of motion in the inertial frame, with the cylindrical coordinates (l, ϕ, z) centered at the star and with z = 0 defining the midplane of the disk. The asymptotic behavior of the X-wind dynamics sufficiently far from the X-region (i.e., l > 50R_X) is (1) the poloidal velocity reaches the terminal velocity, (2) the toroidal velocity decreases as l⁻¹, and (3) the density of the wind falls off as l⁻². Assuming the streamlines to be straight lines originating from the X-region and the open angle of the bipolar wind to be θ_w = 45°, we follow the procedure of Shang (1998) to construct the X-wind as the background for the motion of the solid body.

In this section, we set up the framework to calculate the trajectories of solid bodies entrained in the X-wind. A solid body with size R_s and density ρ_s entrained in an X-wind is subject to three forces: the gravity of the star, the gravity of the protoplanetary disk, and the aerodynamic drag of the X-wind. The total force exerted on the solid body is

$$\begin{equation} {\bf F}=-\frac{{\it GM}_*m_s}{r^3}{\bf r}-A(l,z)m_s{\bf e}_z-B(l,z)m_s{\bf e}_l+{\bf F}_{\rm drag}\,, \end{equation} \tag{ 5 }$$

where m_s is the mass of the solid body and r is the distance from the solid body to the center of star. The second and the third terms on the right-hand side of Equation (5) represent the protoplanetary disk gravity, which has not been included in previous trajectory calculations. A(l, z) and B(l, z) can be calculated by the superposition of gravitational contributions from surface elements throughout the disk, namely,

$$\begin{eqnarray} &&A(l,z)=-\frac{\partial }{\partial z}\int _{R_{\rm in}}^{R_{\rm out}}\int _0^{\pi } \frac{2G\Sigma (l^{\prime })l^{\prime }d\phi ^{\prime } dl^{\prime }}{\sqrt{l^{\prime 2}+l^2+z^2-2l^{\prime }l\cos \phi ^{\prime }}} \,, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &&B(l,z)=-\frac{\partial }{\partial l}\int _{R_{\rm in}}^{R_{\rm out}}\int _0^{\pi } \frac{2G\Sigma (l^{\prime })l^{\prime }d\phi ^{\prime } dl^{\prime }}{\sqrt{l^{\prime 2}+l^2+z^2-2l^{\prime }l\cos \phi ^{\prime }}} \,, \end{eqnarray} \tag{ 7 }$$

where l' and ϕ' are dummy variables of integration spanning the whole disk, R_in and R_out are the inner and outer edges of the protoplanetary disk, and Σ(l) is the surface density of the disk. In the calculation of the disk's gravity, we assume the geometry of the disk to be a thin plane at z = 0, and use R_in = R_X and R_out = 40 AU. The integrals can be evaluated numerically once we know the surface density profile of the protoplanetary disk.

We adopt that the disk has the mass distribution as the standard minimum mass solar nebula (MMSN) given by Hayashi (1981). However, the density of the protoplanetary disk might be much higher. For example, Desch (2007) estimated a denser MMSN based on the NICE model. The disk's surface density in our model is parameterized as

$$\begin{equation} \Sigma (l)=f\times 1700 \bigg (\frac{l}{1\ {\rm AU}}\bigg)^{-3/2}\ {\rm g}\ {\rm cm^{-2}}\,, \end{equation} \tag{ 8 }$$

where f is a multiplying factor specifying the mass of the disk and the rest is the MMSN profile given by Hayashi (1981). Note that Desch (2007) suggested the surface density exponent to be −2.2 instead of −1.5, but in this work we adopt the profile by Hayashi (1981). The acceleration due to the disk gravity A(l, z) and B(l, z) can then be computed numerically.

The protoplanetary disk has the vertical extension that may intercept the solid body trajectory and influence the radius where the solid body re-enters the disk. Assuming that the disk is supported by thermal pressure, the scale height of the disk is

$$\begin{equation} H(l)=\sqrt{\frac{2k_{\rm B}T(l)l^3}{{\it GM}_{*}\mu M_{\rm H}}}\,, \end{equation} \tag{ 9 }$$

where k_B is the Boltzmann constant, μ is the mean molecular mass in the atomic mass unit (for the solar abundance μ = 2.3), M_H is the mass of hydrogen atom, and T(l) is assumed to be the temperature of the MMSN disk as T(l) = 280 l^−1/2 K (see Hartmann 2009). In the following, we regard the scale height in Equation (9) as the "boundary" of the disk. Once the vertical position of the solid body becomes smaller than the scale height of the disk, z < H, the solid body re-enters the disk, and then the trajectory calculation is terminated. We do not intend to follow the trajectories of the solid bodies inside the disk. Once the horizontal position of the solid body becomes outside the disk, or l > 40 AU, the solid body is ejected from the planetary system. In this work, we assume a thin disk when calculating the gravitational force from the disk. Still, we do consider the vertical geometry of the disk using a realistic "flared disk" model when determining the radius where the solids re-enter the disk. We will demonstrate that the vertical structure of the disk has a significant effect on the re-entering radius.

The Epstein gas drag applies when the relative velocity of the solid body and the wind are less than the sound speed of the wind, and the radius of the solid body is smaller than the mean free path of the gas. For example, the mean free path of the wind is 10–10² cm during the embedded stage, and 10²–10³ cm during the revealed stage. Therefore, when calculating the trajectories of chondrule- or CAI-size solid bodies entrained in X-winds, the Epstein gas drag law should be used, namely,

$$\begin{equation} {\bf F}_{\rm drag}=\frac{4}{3} \pi R_s^2 \rho _w \big[({\bf v}_w-{\bf v}_s)^2+c_w^2\big]^{1/2} ({\bf v}_w-{\bf v}_s)\,, \end{equation} \tag{ 10 }$$

where ρ_w, c_w, and v_w are the density, sound speed, and bulk velocity of the wind, respectively, and v_s is the velocity of the solid body (Weidenschilling 1977; Shang 1998; Youdin & Chiang 2004). The gas drag depends on the density and velocity of the wind. The acceleration due to the gas drag is

$$\begin{equation} {\bf a}_{\rm drag}\equiv \frac{{\bf F}_{\rm drag}}{m_s}=\frac{\rho _w }{\eta }\big[({\bf v}_w-{\bf v}_s)^2+c_w^2\big]^{1/2} ({\bf v}_w-{\bf v}_s)\,, \end{equation} \tag{ 11 }$$

where we define the drag efficiency parameter η ≡ R_sρ_s. The gas drag is more effective for a solid body with smaller η, meaning smaller or less dense.

Assuming that a solid body is launched at the initial angle of θ₀ with respect to the midplane of the disk from the reconnection ring at the speed of the local X-wind, we integrate the equations of motion (Equation (5)) numerically to find the trajectory of the solid body. The allowing range of θ₀ is [0,  θ_w]. In summary, the model calls six parameters: three parameters to specify the X-wind, M_*, $\dot{M}_{\rm D},$ and μ_*; f to specify the mass of the protoplanetary disk; η to specify the efficiency of the gas drag; and θ₀ to specify the initial launching angle of the solid body. For any particular state in Table 1, the former three wind-related parameters are fixed, and the latter three parameters are free to be explored.

The trajectories of solid bodies of different size and density (or η) in the EA and RA states are plotted in Figure 1. Here, we assume f = 5 and only explore the situations of maximum lift (θ₀ = θ_w) and intermediate lift (θ₀ = θ_w/2). It is evident from Figure 1 that smaller and less dense objects can be launched to a larger orbit, in agreement with previous works (e.g., Shu et al. 1996). Note that although the left and the right panels of Figure 1 appear to be very similar, they show trajectories of solid bodies with very different η (see the caption for details). Based on Figure 1, the X-wind in the EA state can lift much larger and denser (or larger η) solid bodies to a significantly larger radius than that in the RA state does.

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The vertical extent of the disk intercepts some trajectories. From Figure 1, the trajectories approach straight lines in the l–z plot at large l and z, in agreement with previous works of Shu et al. (1996), Shang (1998), and Liffman (2005). Without the consideration of disk's flared vertical geometry, these straight-line trajectories may easily be interpreted as "ejection." For example, in the EA case, f = 5 and θ₀ = θ_w, the solid body with η = 1.3 cgs units re-enters the disk at l = 8.5 AU (see the left panel of Figure 1). If the disk was assumed to be a thin surface, such an object would be ejected from the system. Therefore, one has to consider the disk's vertical geometry in order to yield realistic diagnostics on whether the solid bodies re-enter the disk or at what radius the solid bodies re-enter the disk.

The initial launching angle θ₀ has a large effect on the trajectories of the solid bodies. From Figure 1, a solid body is launched to a much larger orbit when θ₀ = θ_w. For example, in the EA case and f = 5, a solid body with η = 1 cgs unit is ejected from the system if θ₀ = θ_w, but re-enters the disk at ∼2 AU if θ₀ = θ_w/2 (see Figure 1). For another example, in the EA case and f = 5, a solid body with η = 1.3 cgs unit re-enters the disk at ∼8.5 AU if θ₀ = θ_w, but re-enters the disk at much smaller radius (∼0.7 AU) if θ₀ = θ_w/2. Therefore, if numbers of solid bodies of such η are initially launched from various angles in [0,  θ_w], the X-wind spreads them to the disk's various annuli up to ∼8.5 AU. In general, under the same condition, a solid body can be lifted to a larger orbit and thus re-enters the disk at a larger radius if initially launched from a larger angle.

There exists a threshold value of η of solid bodies for any particular X-wind configuration, disk's mass and initial launching angle, denoted as η_t, below which the solid bodies will be expelled from the planetary system (see Figure 1). This result is expected because the aerodynamic drag on the light particles by the wind is strong, i.e., these particles are more strongly coupled to the gas. For the heavy (η>η_t) particles, on the other hand, drag forces are much less important compared to the gravitational forces from the central star and the disk, and as a result, they fall back into the disk.

In this work, we include the gravity of the protoplanetary disk into the calculation of trajectories of solid bodies entrained in X-winds for the first time. Here, we explore the effect of the disk gravity and demonstrate that the disk mass modifies the trajectories and significantly lowers the re-entering radius.

In Figure 2, we plot the trajectories of the same solid body with a different disk mass factor f. If f = 0, the disk's gravity is not considered and we return to the situation of Shu et al. (1996). From Figure 2, we find that the disk's gravity has a negligible effect on the trajectory at l < 1 AU. Detail analysis on the forces reveals that at l < 1 AU, the gravity of the central star provides the major force in the −e_z direction. When l > 1 AU, the trajectories under different f become different. At such a distance, the gravity of the protoplanetary disk becomes comparable with the gravity of the central star in the −e_z direction. As a result, the solid body flies closer to the disk (smaller z at the same l) when the disk is more massive (larger f). Therefore, the re-entering radius becomes smaller also when the disk is more massive. Comparing the MMSN disk case to the non-disk case, the re-entering radius decreases by ∼10%. If we still consider the fact that an actual protoplanetary disk is very likely to be much more massive than the MMSN disk, for example f = 5, the re-entering radius can be lowered by up to ∼30%.

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Here, we find that the protoplanetary disk provides a non-negligible force in the −e_z direction and modifies the trajectories at l > 1 AU. The previous calculations of Shu et al. (1996) that did not consider the disk gravity are therefore imperfect in estimating the re-entering radius.

In this section, we study the sensitivity of η_t on the disk's mass f and the initial launching angle θ₀. We show how η_t depends on f and θ₀ in Figure 3. By definition, solid bodies of η < η_t entrained in X-winds are expelled from the solar system. As mentioned in Section 2.2, we consider a solid body to have been expelled if l > 40 AU in the trajectory calculation. Note that the value of η_t will change if this cutoff changes.

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From Figure 3, the threshold η_t depends most sensitively on the initial launching angle θ₀. Typically, when f < 10, η_t at the maximum launching angle θ₀ = θ_w is one order of magnitude greater than that at a lower launching angle of θ₀ = 0.3θ_w. For example, during the EA state and f = 5, η_t ∼ 0.98 cgs units at θ₀ = θ_w; and η_t ∼ 0.09 cgs units at θ₀ = 0.3θ_w. Similarly, during the RA state and f = 5, η_t ∼ 0.039 cgs units at θ₀ = θ_w; and η_t ∼ 0.006 cgs units at θ₀ = 0.3θ_w.

When the disk becomes more massive, the solids need to be smaller and lighter in order to be expelled by the X-winds. As seen in Figure 3, η_t decreases by ∼10% when f increases from 0 to 10, and η_t decreases by ∼30% when f increases from 0 to 100. The dependence of η_t on f is less sensitive, but should also be taken into account when considering the retention of solids in the protoplanetary disk.

The X-wind is stronger during the embedded stage, as a result, η_t during the embedded stage is much larger than that during the revealed stage, as shown in Figure 3. It is interesting to note that η_t during the revealed stage is compatible with the typical chondrule size range in meteorites, if the density is assumed to be 2–4 g cm⁻³. Finally, an important observation that we can make from Figures 1 and 3 is that the horizontal distance that a solid body can travel depends on its size and density (η) very sensitively.

In this section, we investigate how the re-entering radius depends on the stellar magnetic dipole moment and size and density of the solid bodies. We calculate the trajectories of solid bodies of different η in the states listed in Table 1 and plot the relations between η and the re-entering radius for each state in Figure 4. There are still flexibilities in choosing the value of parameter f and θ₀. In a particular state, a solid body can be launched to the maximum re-entering radius when f = 0 and θ₀ = θ_w, as plotted in Figure 4. There is no theoretical minimum of the re-entering radius for any particular state, since θ₀ can be as low as 0. However, we still plot the re-entering radius with f = 5 and θ₀ = θ_w/2 in Figure 4 to indicate how small the re-entering radius can be for the intermediate lift of any particular state. Again, it is evident that smaller bodies re-enter the disk at larger radii.

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The main result that one can draw from Figure 4 is that only the solid bodies whose η are larger than but very close to η_t of a particular state can be launched to the annuli of AB or KB. For example, during the EH state and f = 0,  θ₀ = θ_w, η_t∼ 0.7 cgs units, and the solid bodies with η∼ 1.1–1.3 cgs units re-enter the disk at the AB radius (see the solid curve of the top panel of Figure 4). For another example, during the RH state and f = 0,  θ₀ = θ_w, η_t∼ 0.03 cgs units, and the solid bodies with η∼ 0.04–0.05 cgs units re-enter the disk at the AB radius (see the solid curve of the bottom panel of Figure 4). The densities of chondrules and CAIs are in a narrow range around 2–4 g cm⁻³ (e.g., Hudges 1980). As a result, η_t is in general equivalent to the threshold size. Of course, meteoritic parent bodies may have a wider feeding zone than the current AB location during their formation. However, the result does not change even if we consider a wider feeding zone, because in order to be lifted to a radius larger than 1 AU, the solid body must already have a size close to the threshold size (see Figure 4).

In any particular state, the location of the AB, where we envisage the parent bodies of carbonaceous chondrites, can only receive solid bodies with sizes larger than but very close to the threshold size. However, the variation of initial launching angle θ₀ affects η_t significantly (see Figure 3) and enlarges the acceptable range of η significantly. In reality, we know very little about the distribution of θ₀. The X-wind model (Shu et al. 2001) suggests that the magnetorotational instability (MRI) occurs at the X-point to transport the rocks from the reconnection ring to the launching height. Based on the turbulent nature of the process, it is reasonable to assume that the distribution of θ₀ is very wide, or nearly uniform. Imagine the CAIs and chondrules form in the X-region and are launched in X-wind from a wide range of θ₀. Due to this variation of θ₀, in any particular state, the annuli of the AB can receive a wide spectrum of solid bodies with different η, in other words, different sizes. For example, during the EH state, if the condition (f = 0,  θ₀ = θ_w) is specified, the range for η to re-enter the AB radius is 1.1–1.3 cgs unit (see the solid curves of the top panel of Figure 4). However, once the variation of θ₀ from θ_w to θ_w/2 is considered, the range for η to re-enter the AB radius becomes as wide as 0.6–1.3 cgs unit. In the previous section, we find that η_t depends most sensitively on θ₀, so here the enlargement of the η range is mainly due to the variation in θ₀.

Liffman (2005) predicts very effective size sorting and very sharp size distributions in a constant outflow. This is because in Liffman (2005), the initial launching angle of particles is fixed, corresponding to a single curve in our Figure 4. Here, we step forward to suggest that the variation of θ₀ may make the size distribution of CAIs or chondrules in chondrites wider.

The second result is that an annulus of the protoplanetary disk receives one order of magnitude smaller solid bodies (assuming the density is generally the same) during the revealed stage than during the embedded stage. For example the location of AB receives solid bodies of η∼ 0.8–1.7 cgs units during the EA state, but solid bodies of η∼ 0.04–0.07 cgs units during the RA state, with the uncertainty of f and θ₀ (∼[θ_w/2, θ_w]) considered. Table 1 shows that only the EA and EL states can reach the required temperature T_peak > 1800 K and produce CAIs. In contrary, chondrules need lower temperature and are younger than CAIs (e.g., Scott 2007). It is then likely that chondrules mainly formed during the revealed stage. If this is the case, the chondrules should be smaller than CAIs in any particular chondritic group because of the X-wind transport. However, such difference could not be found in chondrites (e.g., Scott 2007). We will address this discrepancy in the following section.

Last but not the least, the variations of re-entering radius according to μ_* are different for small and large bodies. As the star becomes less magnetized, the small solid body re-enters the disk at a larger radius, but the large solid body re-enters the disk at a smaller radius. This feature may counter the physical intuition. Actually, as μ_* decreases, or the star becomes less magnetized, the accretion disk is truncated at a smaller radius. However, the star rotates much faster according to Equation (2) so that the terminal velocity of the X-wind becomes faster. For small bodies, the wind terminal velocity determines how much momentum it can gain during the aerodynamic launching phase; therefore, a less magnetized star produces faster winds, and then solid bodies re-enter the disk at larger radii. For large bodies, their aerodynamic launching phase is very short and their re-entering radii are close to the truncation radius R_X.