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: 'Although the gravitational wave kick velocity in the orbital plane of coalescing black holes has been understood for some time, apparently conflicting formulae have been proposed for the dominant out-of-plane kick, each a good fit to different data sets. This is important to resolve because it is only the out-of-plane kicks that can reach more than 500 and can thus eject merged remnants from galaxies. Using a different ansatz for the out-of-plane kick, we show that we can fit almost all existing data to better than 5%. This is good enough for any astrophysical calculation, and shows that the previous apparent conflict was only because the two data sets explored different aspects of the kick parameter space.'
author:
- 'James R. van Meter , M. Coleman Miller , John G. Baker , William D. Boggs , Bernard J. Kelly'
bibliography:
- 'refs.bib'
title: A General Formula for Black Hole Gravitational Wave Kicks
---

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

When two black holes spiral together and merge, the gravitational radiation they emit is usually asymmetric and thus the remnant black hole acquires a linear velocity relative to the original center of mass. The speed can reach over 3,000  [@Dain:2008ck], which would eject the remnant from any galaxy in the universe (see Figure 2 of @Merritt:2004xa). As discussed in @Merritt:2004xa, the kick magnitude and distribution are important for discussions of hierarchical merging, supermassive black hole formation, galactic nuclear dynamics, and the degree to which black holes influence galaxy formation. The community has converged on the formula for the kick speed in the original orbital plane [@Baker:2007gi; @Campanelli:2007cg; @Gonzalez:2006md], but apparently conflicting dependences for the out-of-plane kick (which dominates the total kick for most configurations) have been proposed. [@Lousto:2008dn] suggested that for a binary with component masses $m_1$ and $m_2\geq m_1$, the kick scales as $\eta^2$, where $\eta\equiv
m_1m_2/(m_1+m_2)^2$ is the *symmetric mass ratio*; however the data in [@Baker:2008md] were fit much better with an $\eta^3$ dependence. The conflict is only apparent, however, because there were no runs in common between the two data sets. This suggests that an analysis might be performed with a new ansatz that can fit all of the existing data.

Here we perform such a fit, and demonstrate that there is a single formula for the out-of-plane kick that fits almost all existing data to better than 5% accuracy. The new ansatz is similar to one recently suggested by [@Lousto:2009mf], for which, however, a fit was not attempted. Its form is based straightforwardly on the post-Newtonian (PN) approximation and includes both the aforementioned $\eta^2$ and $\eta^3$ terms, as well as a slightly more complicated spin-angle dependence. In § 2 we describe our new runs and we describe the new ansatz in § 3. In § 4 we list all 95 runs we have fit, from different numerical relativity groups, and our fitting procedure and best-fit parameters. In § 5 we discuss the implications of our results, and indicate the fraction of kicks above 500 and 1,000 for representative spin and mass ratio distributions, comparing it with previous results. The goodness of our fit to the entire usable data set of out-of-plane kicks suggests that the full three-dimensional kick is now modelled well enough that it will not limit the accuracy of any astrophysical calculation.

Numerical simulations {#sec:simulations}
=====================

We have performed new simulations representing 22 distinct physical cases. Defining $q\equiv m_1/m_2$ and $\alpha_i\equiv
S_i/m_i^2$ where $S_i$ is the spin angular momentum of the $i$th black hole, we used mass ratios of $q=0.674$ or $q=0.515$, and spins initially within the orbital plane of $\alpha_1=\alpha_1^{\perp}=0.367$ and $\alpha_2=\alpha_1^{\perp}=0.177$ or $\alpha_2=\alpha_1^{\perp}=0.236$, where the $\perp$ superscript indicates orthogonality to the orbital axis. For each mass ratio we used one of eleven different spin orientations (given in Table 1), in order to probe the spin-angle dependence of the final recoil. Initial parameters were informed by a quasicircular PN approximation and initial data constructed using the spectral solver [TwoPunctures]{} [@Ansorg:2004ds]. Evolutions were performed with the Einstein-solver [Hahndol]{} [@Imbiriba:2004tp; @vanMeter:2006vi; @Baker:2007gi], with all finite differencing and interpolation at least fifth-order-accurate in computational grid spacing. Note for the purpose of characterizing the simulations, it is convenient to use units in which $G=c=1$ and specify distance and time in terms of $M\equiv
m_1+m_2$. The radiation field represented by the Weyl scalar $\Psi_4$ was interpolated to an extraction sphere at coordinate radius $r=50M$ and integrated to obtain the radiated momentum using the standard formula [@Schnittman:2007ij]: $$P_i=\int_{-\infty}^t dt\frac{r^2}{16\pi}\int d\Omega \frac{x_i}{r}\left|\int_{-\infty}^t dt\Psi_4 \right|^2.$$ For each of the 22 cases we ran two resolutions, with fine-grid spacings of $h_f=3M/160$ and $h_f=M/64$. To give some indication of our numerical error, the total recoil from the two resolutions for each of the $q=0.674$ cases agreed to within ${\lower0.3em\hbox{$\,\buildrel <\over\sim\,$}}6\%$, and for each of the $q=0.515$ cases to within $1\%$ (the latter using a more optimal grid structure).

In these simulations, variation in the magnitude of the final recoil within the $x$-$y$ plane suggested non-negligible precession of the orbital plane. Indeed the coordinate trajectories of the black holes showed precession of up to $\sim 10^{\circ}$. To calculate the component of the recoil velocity parallel to the “final" orbital axis, $V_{||}$, we tried two different methods. In one method we simply took the dot product of the final, numerically computed recoil velocity ${\mathbf V}$ with the normalized orbital angular momentum ${\mathbf L}|L|^{-1}$ as calculated from the coordinate trajectories of the black holes, just when the common apparent horizon was found: $V_{||}\equiv {\mathbf V}\cdot{\mathbf L }{|L|}^{-1}$. In our second method we assumed that each black hole spin, which is initially orthogonal to the orbital angular momentum, remained approximately orthogonal throughout the simulation, i.e. $$\label{eq:LS0}
{\mathbf L}\cdot{\mathbf S}_1\approx{\mathbf L}\cdot{\mathbf S}_2\approx 0.$$ This assumption is consistent with PN calculations, to linear order in spin [@Racine:2008qv; @Kidder:1995zr], and is also supported numerically by the fact that the merger times in our simulations were independent of the initial spin, to within $\Delta t<1M$. In this case the in-plane recoil should depend only on the mass ratio (and not the spin), and we assume it is given by the formula found by [@Gonzalez:2006md], with coefficient values given by a previous fit [@Baker:2008md]. This implies $$\label{eq:vpar}
V_{||}\equiv \sqrt{V^2-(V_{\perp}^{pred})^2}\frac{V_z}{|V_z|},$$ where we have further assumed that the sign of $V_{||}$ should be the same as that of $V_z$, given the modest amount of precession.

These two definitions for $V_{||}$ were found to differ by a relative error of ${\lower0.3em\hbox{$\,\buildrel <\over\sim\,$}}5\%$ for all points except one for which $V_{||} \ll V_{\perp}$. Relative to the maximum $V_{||}$ per mass ratio, they were found to differ only by $<2\%$. Note this lends further support to our assumption that the spins are orthogonal to the orbital angular momentum to good approximation throughout these simulations (Eq. (\[eq:LS0\])). Even $V_z$ was found to differ from the above definitions for $V_{||}$ by only ${\lower0.3em\hbox{$\,\buildrel <\over\sim\,$}}2\%$, relative to the maximum. We will use Eq. (\[eq:vpar\]) to give our canonical $V_{||}$, for the purpose of analytic fits to the data.

An additional assumption we will make about our data, important for fitting purposes, is that the amount of precession undergone by the spins in the orbital plane, i.e. the difference in spin-angles, between the initial data and the merger, is independent of the initial spin orientation. This assumption, valid to linear order in spin according to the PN approximation (Eq. (2.4) of [@Kidder:1995zr]), was previously found to be the case through explicit computation of the spins in the simulations presented in [@Baker:2008md]. We have not explicitly computed the spins in the new simulations presented here but the independence of the merger time with respect to initial spin orientation is consistent with the assumption of similar independence of spin precession.

For the purpose of constructing an accurate phenomenological model, we added to this data set previously published data representing out-of-plane kicks. Our criteria for selecting relevant data are that the component of the final recoil parallel to the orbital plane is given and at least three different in-plane spin orientations are used. We arrived at a total of 95 data points: the 22 new ones presented here plus 73 drawn from [@Baker:2008md; @Campanelli:2007cg; @Dain:2008ck; @Lousto:2008dn]. Note that for the cases in [@Lousto:2008dn] we use the second version of their values listed in their Table III, resulting from what they consider to be their best calculation of the final orbital plane.

Ansatz
======

An ansatz for the total recoil that was found to be very consistent with numerical results has the form [@Baker:2007gi; @Campanelli:2007cg; @Gonzalez:2006md] $$\begin{aligned}
\vec{V} &=& V_{\perp m} \, {\mathbf e}_1 + V_{\perp s} (\cos\xi \, {\mathbf e}_1 + \sin\xi \, {\mathbf e}_2) + V_{\parallel} \, {\mathbf e}_3, \label{eq:V_total}\\
      V_{\perp m}    &=& A \eta^2 \sqrt{1 - 4 \eta} (1 + B \eta), \label{eq:V_mass_Gonzalez}\\
V_{\perp s}     &=& H \frac{\eta^2}{(1+q)} \left( \alpha_2^{\parallel} - q \alpha_1^{\parallel} \right), \label{eq:V_perp}\end{aligned}$$ where ${\mathbf e}_1$ and ${\mathbf e}_3$ are unit vectors in the directions of separation and the orbital axis just before merger, respectively, ${\mathbf e}_2\equiv{\mathbf e}_1\times{\mathbf e}_3$, $\alpha_1^{\parallel}$ and $\alpha_2^{\parallel}$ represent the components of spins parallel to the orbital axis, $\xi, A, B$, and $H$ are constant fitting parameters, and $V_{\parallel}$ is to be discussed.

Every component of Eq. (\[eq:V\_total\]) has been modeled to some degree after PN expressions. The use of such PN-based formulae has proven very successful. For example, Eq. (\[eq:V\_mass\_Gonzalez\]) for the in-plane, mass-ratio-determined recoil can be obtained from the corresponding PN expression (Eq. (23) of [@Blanchet:2005rj]) simply by taking the leading-order terms and replacing instances of the PN expansion parameter (in this case frequency) with constant fitting parameters. Using this, [@Gonzalez:2006md] obtained very good agreement with a large set of numerical results. [@Tiec:2009yg], who calculated the recoil using a combination of the PN method with a perturbative approximation of the ringdown, also found that Eq. (\[eq:V\_mass\_Gonzalez\]) gave a good phenomenological fit to their analytic results.

Why such a prescription for generating an ansatz from the PN approximation should apply so well to merger dynamics is not perfectly understood. The effective replacement of powers of the frequency with constants may be defensable because the majority of the recoil is generated within a narrow time-window near merger, perhaps within a narrow range of frequencies. This is particularly evident for the out-of-plane recoil speed, which rapidly and monotonically increases up to a constant value around merger. However, values of the constant coefficients that appear in the PN expansion cannot be expected to remain unchanged because, as merger is approached, high-order PN terms with the same functional dependence on mass and spin as leading order terms can become comparable in magnitude. So, it is just for this functional dependence that we look to the PN approximation for guidance.

Following the prescription implied above, from the PN expression for out-of-plane recoil given by [@Racine:2008kj], Eqs. (4.40-4.42) (neglecting spin-spin interaction), the following can be straightforwardly obtained: $$\begin{aligned}
V_{||} &=& \frac{K_2\eta^2+K_3\eta^3}{q+1}\left[q \alpha^{\perp}_1
\cos(\phi_1-\Phi_1)-\alpha^{\perp}_2\cos(\phi_2-\Phi_2)\right] \nonumber\\
  &+&\frac{K_S(q-1)\eta^2}{(q+1)^3}\left[q^2\alpha^{\perp}_1
\cos(\phi_1-\Phi_1)+\alpha^{\perp}_2\cos(\phi_2-\Phi_2)\right],\end{aligned}$$ where $K_2$, $K_3$, and $K_S$ are constants, $\alpha^{\perp}_i$ represents the magnitude of the projection of the $i$th black hole’s spin (divided by the square of the black hole’s mass) into the orbital plane, $\phi_i$ represents the angle of the same projection, as measured at some point before merger, with respect to a reference angle representing the direction of separation of the black holes, and $\Phi_i$ represents the amount by which this angle precesses before merger, which depends on the mass ratio and the initial separation. We have ignored terms quadratic in the spin because we assume they are subleading. Note that this ansatz is equivalent to one suggested by [@Lousto:2009mf], provided the angular parameters are suitably interpreted. In terms of the notation of [@Racine:2008kj], $$\begin{aligned}
(q+1)^{-1}\left[q \alpha^{\perp}_1 \cos(\phi_1-\Phi_1)-\alpha^{\perp}_2\cos(\phi_2-\Phi_2)\right]   =& -M^{-1}{\mathbf \Delta}\cdot\mathbf{n} &=  -M^{-1}\Delta^{\perp}\cos(\Theta), \\
(q+1)^{-2}\left[q^2\alpha^{\perp}_1 \cos(\phi_1-\Phi_1)+\alpha^{\perp}_2\cos(\phi_2-\Phi_2)\right]  =&  M^{-2}{\mathbf S}\cdot\mathbf{n}      &=  M^{-2}S^{\perp}\cos({\Psi}),\end{aligned}$$ where ${\mathbf{n}}$ is a unit separation vector, $\Delta\equiv \mathbf{S}_2/m_2-\mathbf{S}_1/m_1$, $\mathbf{S}\equiv \mathbf{S}_1+\mathbf{S}_2$, $\Delta^{\perp}\equiv M (q+1)^{-1}|\alpha^{\perp}_2-q\alpha^{\perp}_1|$, $S^{\perp}\equiv M^2 (q+1)^{-2}|\alpha^{\perp}_2+q^2\alpha^{\perp}_1|$, $\Theta$ is the angle between $\Delta$ and ${\mathbf{n}}$, and $\Psi$ is the angle between $\mathbf{S}$ and ${\mathbf{n}}$, all measured, for our purposes, at some point arbitrarily close to merger. It may be interesting to note that the expression multiplying $K_S$ takes into account effects of orbital precession (because it is non-vanishing if and only if the orbit precesses) and therefore represents physical phenomena neglected by previous fits.

Fitting procedure and results
=============================

The fitting of the out-of-plane recoil is in principle complicated because in addition to the overall factors $K_2$, $K_3$, and $K_S$ (which are the same for any mass ratio or spins), any particular set of runs with the same initial separation and mass ratio (which we will term a “block") has idiosyncratic values of $\Phi_1$ and $\Phi_2$ that, although not fundamentally interesting, need to be fit to the data. Therefore, in the 17 blocks of data we fit, there are formally $3+2\times 17=37$ fit parameters. In the eight data blocks for which $\alpha_1^{\perp}=0$, $\Phi_1$ never enters, and in the two for which $\alpha_1^{\perp}=\alpha_2^{\perp}$, $\Phi_1=\Phi_2$. Therefore the actual number of fitting parameters is 27, but this is still large enough that a multiparameter fit would be challenging. Fortunately, each $(\Phi_1,\Phi_2)$ pair only affects a single data block. We can therefore speed up the fitting, and incidentally concentrate on only the interesting parameters, if we (1) pick some values of $K_2$, $K_3$, and $K_S$ that apply to all data blocks, then (2) for each data block, find the values of $\Phi_2$ and possibly $\Phi_1$ that optimize the fit, repeating this using new values of $\Phi_1$ and $\Phi_2$ for each block. This gives an overall fit for the assumed values of $K_2$, $K_3$, and $K_S$, having optimized over the uninteresting $\Phi$ parameters.

The fit itself needs to be performed assuming uncertainties on each of the numerical measurements of the kick. Each such calculation is computationally expensive and systematic errors are usually difficult to quantify, hence we do not have enough information to do a true fit. As a substitute, we assume that for each block of data, the uncertainty $\sigma$ in each kick is either equal to a fraction (fixed for all blocks) of the maximum magnitude kick in the block, or to a fixed fraction of the individual kick itself. The former may be justified because some sources of error will be independent of the phase of the angles when the holes merge, but we note that the fit performed with the latter assumption (that the uncertainty equals a fractional error of each kick) yields very similar values for the fitting factors. As we do not know what the actual fractional error is, in either case we adjust it so that for our best fit we get a reduced $\chi^2$ of roughly unity (given our 95 data points and 27 fitting variables, this means we need a total $\chi^2$ of about 68). We then evaluate every $(K_2,K_3,K_S)$ triplet using $\chi^2=\sum ({\rm pred-kick})^2/\sigma^2$.

Minimizing $\chi^2$ as calculated with respect to the maximum kick per block, we find that $\sigma^2=0.0005V^2_{\rm \parallel,max}({\rm block})$ gives $\chi^2/{\rm dof}= 1.0$. We note that this value for $\sigma$ is comparable to the numerical error as measured by the difference in kicks computed at different resolutions, when available (e.g. for the new simulations presented here, or the $q=0.25$ case presented in Table IV of [@Lousto:2008dn]). It is also worth noting that for the data from the new simulations, the uncertainty due to orbital precession, discussed in § \[sec:simulations\], is less than $0.02V_{\rm \parallel,max}<\sigma$, and therefore cannot significantly affect the fit. Our best fit is $K_2=30,540$ , $K_3=115,800$ , and $K_S=17,560$ , with $1\sigma$ ranges $28,900-32,550$ , $107,300-121,900$ , and $15,900-19,000$ , respectively. We emphasize that all three coefficients are indispensable in obtaining a good fit; e.g. using the same definition of $\chi^2$ as above, if $K_3=0$ then the best fit gives $\chi^2/{\rm dof}=2.0$, or if $K_S=0$ then the best fit gives $\chi^2/{\rm dof}=1.6$.

Minimizing $\chi^2$ as calculated with respect to each individual kick, we obtain similar results. In this case, $\sigma^2=0.0016V^2_{\rm \parallel}$ gives $\chi^2/{\rm dof}= 1.0$. Our best fit becomes $K_2=32,092$ , $K_3=108,897$ , and $K_S=15,375$ , in agreement with the above fit to within $\sim 5\%$ for $K_2$ and $K_3$ and ${\lower0.3em\hbox{$\,\buildrel <\over\sim\,$}}12\%$ for $K_S$.

In Table \[table:data\] we compare the predicted out-of-plane kicks with the measured ones for the entire data set. For fit \#1 minimizing the error with respect to the maximum kick per block, of the 95 points, only 4 agree to worse than 10%, and all of those occur for kicks with magnitudes much less than the maximum in their data block. Therefore these could represent small phase errors rather than relatively large fractional velocity errors. Only 15 of the 95 points agree to worse than 5%. Fit \#2 minimizing the error with respect to individual kicks performs even better in this regard, with only 10 points differing by more than 5% and only 2 points differing by more than 10%. In either case, the fits with the new ansatz are significantly better than either the simpler $\eta^2$ fit proposed by [@Lousto:2008dn] or the $\eta^3$ fit proposed by [@Baker:2008md].

[c|r|r|r|r|r|r|r|r|r|r]{} \[table:data\] A &0.125&0.000&0.751&0.000& 1.571& -113.3& -118.5& 0.046& -118.8& 0.049\
A &0.125&0.000&0.756&0.000& 1.640& -101.9& -95.9& 0.058& -96.5& 0.053\
A &0.125&0.000&0.761&0.000& 0.223& -349.1& -354.5& 0.016& -350.1& 0.003\
A &0.125&0.000&0.747&0.000& 4.665& 131.6& 133.4& 0.014& 133.4& 0.014\
A &0.125&0.000&0.772&0.000& 3.128& 338.7& 339.6& 0.003& 335.0& 0.011\
A &0.167&0.000&0.777&0.000& 1.571& 530.6& 524.7& 0.011& 517.0& 0.026\
A &0.167&0.000&0.739&0.000& 2.688& 348.8& 369.1& 0.058& 368.6& 0.057\
A &0.167&0.000&0.786&0.000& 0.947& 320.3& 327.1& 0.021& 318.9& 0.004\
A &0.167&0.000&0.773&0.000&-1.509& -529.0& -532.0& 0.006& -524.6& 0.008\
A &0.167&0.000&0.778&0.000& 3.721& -130.8& -140.4& 0.074& -133.5& 0.021\
A &0.250&0.000&0.779&0.000& 1.571& -909.9& -908.1& 0.002& -898.0& 0.013\
A &0.250&0.000&0.788&0.000& 1.203& -815.9& -811.8& 0.005& -801.1& 0.018\
A &0.250&0.000&0.760&0.000& 0.080& 56.3& 52.2& 0.074& 56.1& 0.004\
A &0.250&0.000&0.787&0.000& 4.463& 866.0& 858.4& 0.009& 847.7& 0.021\
A &0.250&0.000&0.751&0.000& 3.041& -209.9& -209.2& 0.003& -211.3& 0.007\
A &0.333&0.000&0.794&0.000& 1.571&-1145.2&-1123.8& 0.019&-1109.6& 0.031\
A &0.333&0.000&0.794&0.000& 1.164& -832.5& -833.6& 0.001& -819.3& 0.016\
A &0.333&0.000&0.754&0.000& 0.082& 397.7& 387.9& 0.025& 391.8& 0.015\
A &0.333&0.000&0.795&0.000& 4.466& 981.7& 968.3& 0.014& 953.8& 0.028\
A &0.333&0.000&0.751&0.000& 3.017& -611.4& -603.8& 0.012& -604.9& 0.011\
A &0.400&0.000&0.793&0.000& 1.571&-1414.7&-1399.6& 0.011&-1386.6& 0.020\
A &0.400&0.000&0.798&0.000& 1.054&-1180.6&-1170.4& 0.009&-1155.3& 0.021\
A &0.400&0.000&0.767&0.000& 0.017& 91.7& 83.9& 0.084& 91.3& 0.004\
A &0.400&0.000&0.798&0.000& 4.426& 1338.6& 1319.5& 0.014& 1304.8& 0.025\
A &0.400&0.000&0.760&0.000& 2.980& -328.3& -321.0& 0.022& -326.0& 0.007\
A &0.500&0.000&0.771&0.000& 1.571& 34.0& 26.6& 0.217& 34.0& 0.000\
A &0.500&0.000&0.777&0.000& 0.642& 1261.7& 1249.7& 0.010& 1244.4& 0.014\
A &0.500&0.000&0.800&0.000&-0.323& 1528.2& 1493.9& 0.022& 1479.7& 0.032\
A &0.500&0.000&0.764&0.000& 4.320& -622.6& -603.4& 0.031& -605.6& 0.027\
A &0.500&0.000&0.800&0.000& 2.674&-1439.4&-1401.8& 0.026&-1387.2& 0.036\
A &0.666&0.000&0.772&0.000& 1.571& 895.2& 872.5& 0.025& 865.9& 0.033\
A &0.666&0.000&0.802&0.000& 0.506& 1699.3& 1672.6& 0.016& 1662.6& 0.022\
A &0.666&0.000&0.798&0.000&-0.356& 1025.0& 1000.5& 0.024& 995.7& 0.029\
A &0.666&0.000&0.784&0.000& 4.325&-1362.7&-1339.9& 0.017&-1330.8& 0.023\
A &0.666&0.000&0.795&0.000& 2.654& -823.9& -813.3& 0.013& -809.8& 0.017\
A &1.000&0.000&0.800&0.000& 1.571& 1422.3& 1422.6& 0.000& 1421.0& 0.001\
A &1.000&0.000&0.803&0.000& 0.428& 1009.3& 990.5& 0.019& 980.9& 0.028\
A &1.000&0.000&0.785&0.000&-0.461& -246.0& -237.4& 0.035& -245.3& 0.003\
A &1.000&0.000&0.805&0.000& 4.222&-1479.9&-1468.0& 0.008&-1461.9& 0.012\
A &1.000&0.000&0.786&0.000& 2.581& 390.5& 380.5& 0.026& 387.8& 0.007\
B &0.333&0.200&0.022&0.000& 3.142& 49.0& 50.5& 0.031& 52.0& 0.062\
B &0.333&0.200&0.022&5.498& 2.356& 48.0& 48.1& 0.003& 48.9& 0.018\
B &0.333&0.200&0.022&4.712& 1.571& 17.0& 17.6& 0.034& 17.2& 0.012\
B &0.333&0.200&0.022&0.000& 0.000& 114.0& 114.3& 0.003& 115.2& 0.011\
B &0.500&0.200&0.050&0.000& 3.142& -37.0& -38.2& 0.033& -36.7& 0.008\
B &0.500&0.200&0.050&5.498& 2.356& 111.0& 111.2& 0.002& 114.1& 0.028\
B &0.500&0.200&0.050&4.712& 1.571& 193.0& 195.5& 0.013& 198.0& 0.026\
B &0.500&0.200&0.050&5.498& 1.571& 75.0& 73.2& 0.023& 76.4& 0.019\
B &0.500&0.200&0.050&0.000& 1.571& -55.0& -56.7& 0.031& -55.0& 0.000\
B &0.666&0.200&0.089&1.047& 4.189& -381.0& -382.9& 0.005& -384.5& 0.009\
B &0.666&0.200&0.089&0.000& 3.142& -135.0& -132.1& 0.022& -133.4& 0.012\
B &0.666&0.200&0.089&5.498& 2.356& 168.0& 165.3& 0.016& 165.2& 0.017\
B &0.666&0.200&0.089&4.712& 1.571& 364.0& 365.9& 0.005& 367.0& 0.008\
B &0.769&0.200&0.118&0.000& 3.142& -386.0& -387.2& 0.003& -388.1& 0.005\
B &0.769&0.200&0.118&5.498& 2.356& -525.0& -521.7& 0.006& -521.2& 0.007\
B &0.769&0.200&0.118&4.712& 1.571& -348.0& -350.6& 0.008& -349.0& 0.003\
B &0.909&0.200&0.165&0.000& 3.142& -542.0& -542.8& 0.001& -542.4& 0.001\
B &0.909&0.200&0.165&5.498& 2.356& -657.0& -656.3& 0.001& -655.2& 0.003\
B &0.909&0.200&0.165&4.712& 1.571& -384.0& -385.3& 0.003& -384.1& 0.000\
C &0.674&0.367&0.236&2.513& 5.655& -973.7& -944.4& 0.030& -941.0& 0.034\
C &0.674&0.367&0.236&2.094& 4.189& -442.9& -435.8& 0.016& -431.8& 0.025\
C &0.674&0.367&0.236&2.094& 0.000& -988.5& -934.6& 0.055& -934.4& 0.055\
C &0.674&0.367&0.236&1.257& 4.398& -361.7& -344.3& 0.048& -339.4& 0.062\
C &0.674&0.367&0.236&0.000& 4.189& 374.2& 331.6& 0.114& 336.5& 0.101\
C &0.674&0.367&0.236&0.000& 3.142& 749.0& 731.6& 0.023& 731.2& 0.024\
C &0.674&0.367&0.236&0.000& 2.094& 672.4& 682.2& 0.015& 677.3& 0.007\
C &0.674&0.367&0.236&5.027& 1.885& 826.1& 796.5& 0.036& 791.4& 0.042\
C &0.674&0.367&0.236&4.189& 2.094& 619.7& 603.0& 0.027& 597.9& 0.035\
C &0.674&0.367&0.236&4.189& 0.000& -245.6& -246.4& 0.003& -245.5& 0.000\
C &0.674&0.367&0.236&3.770& 0.628& -237.7& -239.4& 0.007& -242.1& 0.019\
C &0.515&0.367&0.177&2.513& 5.655& -584.3& -578.0& 0.011& -587.0& 0.005\
C &0.515&0.367&0.177&2.094& 4.189& -123.0& -103.3& 0.160& -118.1& 0.040\
C &0.515&0.367&0.177&2.094& 0.000& -684.8& -683.1& 0.002& -685.0& 0.000\
C &0.515&0.367&0.177&1.257& 4.398& -16.9& 1.0& 1.059& -16.5& 0.024\
C &0.515&0.367&0.177&0.000& 4.189& 483.9& 462.6& 0.044& 448.5& 0.073\
C &0.515&0.367&0.177&0.000& 3.142& 579.8& 578.6& 0.002& 576.8& 0.005\
C &0.515&0.367&0.177&0.000& 2.094& 345.2& 346.7& 0.004& 357.5& 0.036\
C &0.515&0.367&0.177&5.027& 1.885& 362.4& 356.6& 0.016& 373.0& 0.030\
C &0.515&0.367&0.177&4.189& 2.094& 202.1& 220.5& 0.091& 236.6& 0.171\
C &0.515&0.367&0.177&4.189& 0.000& -231.1& -243.4& 0.053& -239.4& 0.036\
C &0.515&0.367&0.177&3.770& 0.628& -354.8& -358.2& 0.010& -346.3& 0.024\
D &1.000&0.930&0.930&1.571& 4.712& 2372.0& 2413.6& 0.018& 2423.7& 0.022\
D &1.000&0.930&0.930&1.833& 4.974& 2887.0& 2972.2& 0.030& 2975.8& 0.031\
D &1.000&0.930&0.930&2.269& 5.411& 3254.0& 3440.6& 0.057& 3432.8& 0.055\
D &1.000&0.930&0.930&3.176& 0.035& 2226.0& 2390.4& 0.074& 2366.0& 0.063\
D &1.000&0.930&0.930&4.817& 1.676&-2563.0&-2659.2& 0.038&-2666.8& 0.041\
D &1.000&0.930&0.930&4.974& 1.833&-2873.0&-2972.2& 0.035&-2975.8& 0.036\
D &1.000&0.930&0.930&5.149& 2.007&-3193.0&-3233.9& 0.013&-3232.9& 0.013\
D &1.000&0.930&0.930&5.934& 2.793&-2910.0&-3152.3& 0.083&-3133.1& 0.077\
E &1.000&0.515&0.515&1.571& 4.712& 1833.0& 1881.1& 0.026& 1875.6& 0.023\
E &1.000&0.515&0.515&0.785& 3.927& 1093.0& 1079.6& 0.012& 1076.4& 0.015\
E &1.000&0.515&0.515&3.142& 0.000& 352.0& 354.4& 0.007& 353.3& 0.004\
E &1.000&0.515&0.515&4.712& 1.571&-1834.0&-1881.1& 0.026&-1875.6& 0.023\
E &1.000&0.515&0.515&3.304& 0.162& 47.0& 46.3& 0.014& 46.2& 0.017\
E &1.000&0.515&0.515&0.000& 3.142& -351.0& -354.4& 0.010& -353.3& 0.007

Ejection probabilities and discussion
=====================================

One of the most important outputs of kick calculations and fits is the probability distribution of kicks given assumptions about the mass ratio, spin magnitudes, and spin directions. This distribution is critical to studies of hierarchical merging in the early universe (e.g., @Volonteri:2007et) as well as to the gas within galaxies [@Devecchi:2008qy] and an evaluation of the prospects for growth of intermediate-mass black holes in globular clusters [@HolleyBockelmann:2007eh]. In Table 2 we show the results of our work (from fit \#1), compared with the proposed fit formula of [@Campanelli:2007ew]. It is clear that our work gives distributions very close to those of [@Campanelli:2007ew], with perhaps slightly smaller kicks because of the $\eta^3$ term we include.

[lrrr]{} $1/10 \leq q \leq 1$, $\alpha_1=\alpha_2=0.9$ & $v>500 {\ifmmode{{\rm km~s^{-1} }}\else{km s$^{-1}$}\fi}$ & 0.364$\pm$0.0048 & 0.342526$\pm$0.00019\
& $v>1000 {\ifmmode{{\rm km~s^{-1} }}\else{km s$^{-1}$}\fi}$ & 0.127$\pm$0.0034 & 0.120974$\pm$0.00011\
$1/4 \leq q \leq 1$, $\alpha_1=\alpha_2=0.9$ & $v>500 {\ifmmode{{\rm km~s^{-1} }}\else{km s$^{-1}$}\fi}$ & $
0.699 \pm 0.0045$ & 0.697818$\pm$0.00026\
& $v>1000 {\ifmmode{{\rm km~s^{-1} }}\else{km s$^{-1}$}\fi}$ & 0.364$\pm$0.0046 & 0.353393$\pm$0.00019\
$1/4\leq q \leq 1$, $0\leq \alpha_1,\alpha_2\leq 1$ & $v>500 {\ifmmode{{\rm km~s^{-1} }}\else{km s$^{-1}$}\fi}$ & 0.428$\pm$0.0045 & 0.415915$\pm$0.00020\
& $v>1000 {\ifmmode{{\rm km~s^{-1} }}\else{km s$^{-1}$}\fi}$ & 0.142$\pm$0.0034 & 0.134615$\pm$0.00012\

In summary, we have demonstrated that a modified formula fits all available out-of-plane kicks extremely well. The wide range of mass ratios, spin magnitudes, and angles explores all the major aspects of parameter space for the out-of-plane kicks, and thus we do not expect new results to deviate significantly from our formula. The excellence of these fits suggests that the kick distribution is known to an accuracy that is sufficient for any astrophysical purpose.

New simulations used for this work were performed on Jaguar at Oak Ridge National Laboratory. MCM acknowledges partial support from the National Science Foundation under grant AST 06-07428 and NASA ATP grant NNX08AH29G. The work at Goddard was supported in part by NASA grant 06-BEFS06-19. We also wish to think S. McWilliams and A. Buonanno for helpful discussions.