You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We examine the effects of the engulfment of planets by giant stars on the evolution of late-type stars. We show that the rate at which dynamo-generated magnetic energy is being released exceeds 10% of the wind kinetic energy when the orbital angular momentum of the engulfed planet is more than ten times the angular momentum of the star as it leaves the main sequence. A significant enhancement in the mass-loss rate may be expected in this case, due to the formation of cool magnetic spots. We use the existing sample of extrasolar planets to estimate that at least 3.5% of the evolved solar-type stars will be significantly affected by the presence of planetary companions.'
author:
- Mario Livio
- Noam Soker
title: 'The Effects of Planets and Brown Dwarfs on Stellar Rotation and Mass-Loss'
---

Introduction
============

Radial velocity surveys of nearby stars have been enormously successful in discovering extrasolar planets. These surveys have demonstrated that a significant fraction of the stars in the solar neighborhood have massive planets with orbital radii that are substantially smaller than Jupiter’s [e.g. @marcy00; @udry01; @butler01; @vogt02]. The engulfing of these planets by their parent stars is an inevitable outcome of stellar evolution (either directly or due to tidal interaction) in the red giant branch (RGB) or asymptotic giant branch (AGB) phase [e.g. @livio84; @siess99a; @siess99b]. The capture of planetary-mass companions has been suggested to be the potential cause for the high rotational velocities ($V_{\mathrm{rot}}\sin i\ga10$ km s$^{-1}$) observed in some field red giants [@stefanik01] and blue horizontal branch stars [e.g. @peterson83; @soker00].

In the present work we examine specifically the effects of planet/brown dwarf “swallowing” on the rotation rates of and mass loss from giants. In particular, we estimate the expected statistics of influenced stars.

Rotation and Mass Loss in Evolved Late-Type Stars
=================================================

When a star in the RGB or AGB phase engulfs a planet/brown dwarf the latter deposits its angular momentum into the giant’s envelope. The resulting rotation frequency, as a fraction of the critical stellar rotation frequency $\omega_k\equiv(GM_*/R^3_*)^{\frac{1}{2}}$ (where $M_*$, $R_*$ are the stellar mass and radius respectively) is given by $$\frac{\omega}{\omega_k}\simeq0.10
\left( \frac{M_c}{0.01~\mathrm{M}_{\odot}} \right)
\left( \frac{M_\mathrm{env}}{\mathrm{M}_{\odot}} \right)^{-1}
\left( \frac{k^2_g}{0.1} \right)^{-1} 
\left( \frac{a}{R_*} \right)^{\frac{1}{2}}~~.$$ Here $M_c$ is the companion mass, $M_{\mathrm{env}}$ is the mass of the giant’s envelope, $Mk^2_gR^2$ is the star’s moment of inertia and $a$ is the initial separation between the giant and the companion. Since stars like the Sun rotate only at a fraction of a percent of their critical rate (on the main sequence) and this fraction is further reduced (by conservation of angular momentum) to $$\left( \frac{\omega}{\omega_k} \right)_{\mathrm{AGB}} \lesssim0.1
\left( \frac{\omega}{\omega_k} \right)_{\mathrm{MS}}
\left( \frac{R_{\mathrm{MS}}}{0.01R_{\mathrm{AGB}}} \right)^{\frac{1}{2}}~~,$$ as the stars ascend the RGB or AGB, we see from eq. (1) that even Jupiter-mass planets can have a very significant effect on the rotation rate of late-type stars. Even when the increase in central condensation is taken into account, typically, $(\omega /\omega_k)_\mathrm{AGB} \sim(0.01-0.1)(\omega /\omega_k)_\mathrm{MS}$ [e.g. @eriguchi92]. The effects on mass loss rates are more difficult to estimate because they involve the processes of generation of magnetic activity (especially in spots) and mass loss via radiation pressure on molecules and dust, both of which are rather poorly understood. Nevertheless, it has been shown semi-quantitatively that the spiraling-in process of the companion inside the giant’s envelope can result in a dynamo generation of a magnetic field [@regos95].

Here we shall adopt a simplified, more heuristic approach [similar to that of @sokerharpaz92] to estimate at what rotation rates we can expect significant effects on mass loss. Dynamo-generated magnetic fields are limited by the requirement that the amplification timescale $\tau_a$ be of the order of the buoyant rise time of magnetic flux tubes [e.g. @parker79 §8.7]. In a typical stellar $\alpha\omega$ dynamo the amplification timescale is given approximately by [e.g. @zeldovich83] $$\tau_a\simeq\pi\left( \frac{2}{\alpha R_* k\nabla\omega} \right)^{\frac{1}{2}}~~,$$ where $\alpha=\ell\omega_o/3$ is determined by convection ($\ell$ is the mixing length and $\omega_o$ the surface angular velocity), $k$ is the wave number and $\nabla\omega$ is a measure of differential rotation. The critical wave number $k_c$ below which dynamo waves are amplified is given by ($\nu_{\tau}$ is the turbulent viscosity) $$k^3_c\simeq\alpha R_* \nabla\omega/2\nu^2_{\tau}~~.$$ Adopting the same values as Zeldovich et al. (1983), $\nabla\omega=\omega_o/(0.3R_*)$, $\nu_{\tau}=\ell v_c/3$, where $v_c$ is the convective velocity, and taking the wave number to be equal to the critical one, we find $$\tau_a\simeq\frac{2\pi}{\omega_o}
\left( \frac{v_c}{7\ell\omega_o} \right)^{\frac{1}{3}}~~.$$ For typical AGB star parameters, $v_c\simeq10$ km s$^{-1}$, $<$$\ell$$>\sim R_*/4$, the obtained average amplification time is of the order of $\tau_a\simeq9\tau_{\mathrm{rot}}$, where $\tau_{\mathrm{rot}}$ is the rotation period. The local rise time of a flux tube is given approximately by [@parker79] $$\tau_r\simeq\frac{\Lambda v_c}{V^2_A}=\frac{4\pi\rho\Lambda v_c}{B^2}~~,$$ where $V_A$ is the Alfven speed, $B$ is the magnetic field strength, $\Lambda\sim r/2$ [where $r$ is the radial distance from the center; e.g. @sokerharpaz99] is the local scaleheight and $\rho$ is the density. The density in the convective zone of an AGB star can be well approximated by [@soker92] $$\rho(r)\simeq2\times10^{-8}
\left( \frac{M_{\mathrm{env}}}{0.1~\mathrm{M}_{\odot}} \right)
\left( \frac{R_*}{300~\mathrm{R}_{\odot}} \right)^{-1}
\left( \frac{r}{100~\mathrm{R}_{\odot}} \right)^{-2}\ \mathrm{g~cm}^{-3}~~.$$ Equating the flux tube rise time $\tau_r$ to the regeneration and amplification time $\tau_a$ yields a value for the magnetic field $$B\simeq1.8
\left( \frac{M_*}{\mathrm{M}_{\odot}} \right)^{\frac{1}{3}}
\left( \frac{R_*}{300~\mathrm{R}_{\odot}} \right)^{-\frac{4}{3}}
\left( \frac{M_{\mathrm{env}}}{0.1~\mathrm{M}_{\odot}} \right)^{\frac{1}{2}}
\left( \frac{v_c}{10~\mathrm{km~s}^{-1}} \right)^{\frac{1}{3}}
\left( \frac{\omega_o/\omega_k}{10^{-3}} \right)^{\frac{2}{3}}
\left( \frac{r}{100~\mathrm{R}_{\odot}} \right)^{-\frac{1}{2}}\ \mathrm{G}~~.$$ Magnetic energy is thus being released at a rate $$\begin{aligned}
\dot{E}_B&=&\frac{1}{\tau_a}\int\frac{B^2}{8\pi} 4\pi r^2 d r\nonumber\\ 
&\simeq&10^{28}
\left( \frac{M_{\mathrm{env}}}{0.1~\mathrm{M}_{\odot}} \right)
\left( \frac{v_c}{10~\mathrm{km~s}^{-1}} \right)^{\frac{1}{3}}
\left( \frac{R_*}{300~\mathrm{R}_{\odot}} \right)^{-\frac{7}{3}}
\left( \frac{M_*}{\mathrm{M}_{\odot}} \right)^{\frac{4}{3}}
\left( \frac{\omega_o/\omega_k}{10^{-3}} \right)^{\frac{8}{3}}~\mathrm{erg~s}^{-1}~~.\end{aligned}$$

The kinetic energy associated with the mass loss from an AGB star is of order $$\dot{E}_{\mathrm{wind}}\simeq3\times10^{31}
\left( \frac{\dot{M}}{10^{-6}~\mathrm{M}_{\odot}~\mathrm{yr}^{-1}} \right)
\left( \frac{V_w}{10~\mathrm{km~s}^{-1}} \right)^2~\mathrm{erg~s}^{-1}~~,$$ where $\dot{M}$ is the mass-loss rate and $V_w$ is the wind velocity. An examination of eqs. (9) and (10) reveals that for the companion to have an appreciable effect on mass loss (i.e. to have $\dot{E}_B>0.1\dot{E}_\mathrm{wind}$) it must increase the angular momentum of the envelope (and thereby $\omega_o/\omega_k$) by an order of magnitude (or more). Consequently, in our estimate of the fraction of stars that are affected significantly by planetary companions, we shall adopt as the defining condition $$J_p>10J_*~~,$$ where $J_p$ is the orbital angular momentum of the planet, and $J_*$ is the angular momentum of the star as it leaves the main sequence \[eq. (2)\]. Note that this is probably a reasonable criterion for the planet having a significant effect on stellar evolution even considering the uncertain nature of the effects on mass loss. In fact, given that the magnetic field in stellar spots can be $\sim10^3$ times stronger than the average field [e.g. @priest87], condition (11) may be too stringent, since even a field lower by an order of magnitude than that given by eq. (8) will result in spots that are significantly cooler than their surroundings (magnetic pressure larger than the photospheric pressure).

Statistics on Importance of Planets
===================================

We define the following quantities $$\begin{aligned}
\nu &\equiv&\log (a/AU)\nonumber\\
\mu &\equiv&\log (M/M_J)\end{aligned}$$ where $a$ is the orbital separation, $M$ is the minimum planetary mass, and $M_J$ is Jupiter’s mass. In Fig. 1 we show the known 75 planets (taken in January 2002 from the *Extrasolar Planets Encyclopaedia* compiled by J. Schneider, at http://www.obspm.fr/encycl/encycl.html) in the $\nu$–$\mu$ plane. In planetary systems where more than one planet is known we mark the most massive planet by an asterisk, while the others are marked by the + sign. Several authors have attempted recently to obtain unbiased distributions of planets as a function of $\nu$ and $\mu$ [e.g. @lineweaver02; @mazeh02; @tabachnik02; @armitage02]. In view of the uncertainties, we adopt the following simple form for the fraction of systems $dN$ within a box $d\nu d\mu$ (for the parameter space indicated below) $$dN=C(1+\alpha\nu)(1-\beta\mu)d\mu d\nu~~,$$ for $-1.5<\nu <0.5$ and $\mu<0.5\nu +0.75$. Here $C$, $\alpha$ and $\beta$ are parameters. This form is the same as the one used by @lineweaver02, who found (their figures 3 and 4) $\alpha \simeq 0.8$ and $\beta \simeq 0.8$. Using the well populated parallelogram [i.e. excluding areas in the $\nu-\mu$ plane with large gaps; see also @zucker02] marked by lines 1 to 4 on Figure 1, we estimate $\alpha \simeq 0.3$ and $\beta \simeq 0.5$. Examining in addition the results of @mazeh02, @tabachnik02, and @armitage02, and fitting where necessary their functional forms to ours, gives average values of $<$$\alpha$$> \simeq0.6$ and $<$$\beta$$> \simeq0.5$, which we shall now adopt.

The value of $C$ can be determined for fixed values of $\alpha$ and $\beta$ in the following way. From a sample of 1,200 stars discussed by Vogt et al. (2002; where more details can be found) 44 planets and 4 brown dwarfs have been detected. Hence 3.7% of the surveyed stars were found to have planets. The 75 known extrasolar planets (as of the end of January 2002) are distributed in 67 planetary systems. Using this ratio we find that the fraction of stars being detected to harbor planetary systems is $\sim 3.3$%. If we examine the well populated area of Figure 1 (bounded by the four straight lines $\nu = 0.5$; $\mu = 0.5 \nu +0.75$; $\nu =-1.5$; and $\mu = 0.5 \nu$), we find that out of the 67 planetary systems, 48 are inside this parallelogram. This corresponds to $\sim 2.4$% of the stars having their massive planet within this region. Considering that the right side of this region is close to the detection limit (no planets are found near $\nu \simeq 0.4$ and $\mu \sim 0.3$), we take the detection fraction within this region to be $N_s \simeq 2.5$% of all stars. For stars with a metallicity above solar, \[Fe/H\] $>0$, the detection fraction is $\sim 2.6$ times higher than that for the entire sample [@vogt02]. Hence, in the same scaling-region the fraction is $N_s \simeq 6.5$%. The large increase with metallicity in the probability of harboring a planet was studied by @gonzalez97 and @reid02. The latter author argues that most stars with a metallicity \[Fe/H\] $>0.3$ have planets around them. We therefore take $N_s=0.025$ for all stars and $N_s=0.065$ for metal rich stars, and find the value of $C$ from the expression $$N_s= C \int_{-1.5}^{0.5} d \nu \int_{0.5 \nu}^{0.5 \nu +0.75} 
d \mu   (1+ \alpha \nu)(1- \beta \mu) 
= \frac{C}{32}(48-24 \alpha -6 \beta - 5 \alpha \beta) =
0.9094C.$$ where the last equality was obtained by substituting the average values $\alpha=0.6$ and $\beta=0.5$.

In the previous section we derived as our condition for a significant effect on mass-loss from the parent star, $J_p>10J_*$. We noted, however, that it is possible that even a lower angular momentum might play some role, especially since the star loses angular momentum as it evolves [@soker01]. Soker further argues that planets of masses as low as $0.01 M_J$ are sufficient to influence the mass loss *geometry* from AGB stars. An indirect hint that even a relatively slow rotation may influence the mass loss rate comes from the distribution of stars on the horizontal branch in globular clusters (GCs). @dcruz96 show that the distribution of stars on the horizontal branch in the HR diagram can be explained by variations in the mass loss rate on the RGB, by a factor which depends on the metallicity (ranging from 1.5 in metal rich GCs to $\gtrsim 5$ in metal poor GCs). We should note that red horizontal branch stars also rotate [@stefanik01]. However, using the results of @behr00 for M13, @soker00 showed that the average angular momentum of the progenitors of hotter horizontal branch stars is larger than that of cooler ones. It is thought that rotation is the main factor that influences the mass loss rate variations (R. Rood, private communication, 2002), with the exception of very high mass loss rates which may result from interactions with stellar binary companions. Since main sequence stars in GCs are expected to rotate very slowly (since they have experienced a long period of angular momentum losses due to magnetic activity), if the rotation conjecture is correct, then very slow rotation is sufficient to influence mass loss on the RGB. Our condition, therefore, may be somewhat conservative.

The present angular momentum of the Sun is $1.7 \times 10^{48}$g cm$^2$ s$^{-1}$. A planet with a Jupiter mass at 1 AU from a solar type star has $\sim 50$ times the present angular momentum of the Sun. Taking as a typical value $J_* = J_\odot$, our condition in equation (11) reads $(M/M_J)(a/~{\rm AU})^{1/2} \gtrsim 0.2 $. For a more general form $J_p>\eta J_*= \eta J_\odot$, our condition reads $$\mu + 0.5 \nu \gtrsim  \log (\eta/10)-0.7 \equiv j_m~~.$$ In addition, we limit the planet to $a<2$ AU (or $\nu < 0.3$), in order for tidal forces to be able to bring it into the RGB or AGB star’s envelope. These conditions are represented (for $\eta=10$) as the two thick lines in Fig. 1. Planets to the left of and above these lines affect the evolution of their parent stars significantly (neglecting eccentric orbits of planets at larger orbital separations).

Not including the few systems above the line marked ‘2’ in Fig. 1, the total fraction of planetary systems that are likely to significantly influence the evolution of the central star at late stages is therefore, $$N({\rm total})= C \int_{-1.5}^{0.3} d \nu \int_{j_m-0.5 \nu}^{0.5 \nu +0.75} 
d \mu   (1+ \alpha \nu)(1- \beta \mu)~~.$$ Note that this integral is meanigful for $j_m<-0.7$. For $j_m>-0.7$ the thick diagonal line in Fig. 1 crosses the upper limit as given by line ‘2.’ Performing this integral for $\alpha=0.6$ and $\beta=0.5$, and substituting for $C$ from equation (14), we find $$N(total)=N_s(0.4-1.2j_m+0.3 j_m^2)~~.$$

Equation (17) shows that for our strong condition of $\eta =10$, we find that more than 3.5% of all solar-type stars are significantly affected (in terms of rotation rate and mass loss) by planets or brown dwarf companions. This number increases to more than 9% for metal rich (above solar) stars. Relaxing the condition to $\eta=1$, i.e. when the planet’s orbital angular momentum equals the present one for the sun, increases the numbers to be $>8$% and $>21$% respectively. All of these numbers should probably be regarded as lower limits, since we have neglected the effects of planets at 2 AU$\lesssim a\lesssim5$ AU in eccentric orbits [which could also be pulled into the giant’s envelope by tidal interaction; see also @debes02], and used only the minimum masses (as well as not having included the planets above line ‘2’ in Fig. 1).

SUMMARY
=======

We have shown that the swallowing of planetary or brown dwarf companions can have significant effects in the late stages of the evolution of late-type stars. Specifically, we considered rotation and mass loss. On the basis of the properties of the observed extrasolar planets, we estimated that $\sim4$–10% of the stars should experience enhanced mass loss due to the effects of planets. We acknowledge useful comments by an anonymous referee. N. S. was supported by a Celerity Foundation Distinguished Visiting Scholar grant at the University of Virginia, and by a grant from the US-Israel Binational Science Foundation. He also asknowledges the hospitality of ST ScI.

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