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N-body simulations (gravitational)

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Michele Trenti and Piet Hut (2008), Scholarpedia, 3(5):3930. doi:10.4249/scholarpedia.3930 revision #91544 [link to/cite this article]
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Curator: Piet Hut

Gravitational N-body simulations, that is numerical solutions of the equations of motions for N particles interacting gravitationally, are widely used tools in astrophysics, with applications from few body or solar system like systems all the way up to galactic and cosmological scales. In this article we present a summary review of the field highlighting the main methods for N-body simulations and the astrophysical context in which they are usually applied.

The underlying dynamics relevant in the astrophysical context for of a system of N particles interacting gravitationally is typically Newton's law plus, in case, an external potential field (see however below for a discussion of N-body simulations in general relativity). The force \(\vec{F}_i\) acting on particle \(i\) of mass \(m_i\) is:

\[\tag{1} \vec{F}_i = - \sum_{j \ne i} G \frac{m_i m_j (\vec{r}_i-\vec{r}_j)}{|\vec{r_i}-\vec{r_j}|^3 } - \vec{\nabla} \cdot \phi_{ext}(\vec{r}_i), \]

where \(G=6.67300 \cdot 10^{-11}\) \(m^{3}\) \(kg^{-1}\) \(s^{-2}\) is the gravitational constant, and \(\phi_{ext}\) is the external potential. The problem is thus a set of non-linear second order ordinary differential equations relating the acceleration \(\partial^2 \vec{r_i} / \partial t^2 = \vec{F}_i /m_i\) with the position of all the particles in the system.

Once a set of initial condition is specified (for example the initial positions \(\vec{r}_i\) and velocities \(\vec{v}_i \equiv \partial \vec{r}_i / \partial t \) of all particles) it exists a unique solution, analytical only for up to two bodies, while larger N require numerical integration (e.g. see Press et al. 2007). However special care must be employed to ensure both accuracy and efficiency. In fact, the gravitational force (eq. (1) presents a singularity when the distance of two particles approaches 0, which can lead to arbitrarily large relative velocities. In addition, given the non-linear nature of the equations, the singularities are movable, that is they depend on the specific choice of initial conditions. In contrast, all singularities in linear ordinary differential equations are independent of initial conditions and thus easier to treat. Therefore constant timestep methods are unable to guarantee a given accuracy in the case of gravitational dynamics and lead to unphysical accelerations during close encounters, which in turn may create unbound stars. A shared, adaptive timestep scheme can correctly follow a close encounter, but the price is paid in terms of efficiency as all the other particles of the system are evolved on the timescale of the encounter, which may be several orders of magnitude smaller than the global timescale, resulting essentially in a freezing of the system.

The singularity may be avoided by introducing a softening length in Eq. (1) (e.g. see Aarseth 1963), that is by modifying the gravitational interaction at small scales. For example:

\[ \vec{F}_i = - \sum_{j \neq i} \frac{G m_i m_j (\vec{r}_i - \vec{r}_j)}{(|\vec{r}_i - \vec{r}_j|^2 + \epsilon^2)^{3/2}} , \]

where \(\epsilon > 0\) is the softening length, that is a typical distance below which the gravitational interaction is suppressed. To minimize the force errors and the global impact of the softening for distances larger than \(\epsilon\ ,\) finite size kernels that ensure continuous derivatives of the force may be employed (e.g., see Dehnen 2001). This strategy effectively suppresses binary formation and strong gravitational interactions, but at the price of altering the dynamics of the system.

The computational complexity of the numerical solution of a N-body system for a fixed number of timesteps scales as \(N^2\ ,\) as the evaluation of the force on each particle requires to take into account contributions from all other members of the system. For example, considering a single state of the art cpu core (speed \(\approx 5\) GFlops), a single force evaluation through a direct method would require about 1 second for a system with \(N=10^4\) particles (assuming 10 floating point operations per pair of particles) and more than a week for \(N=10^7\ .\)

The arbitrarily large dynamic range in the unsoftened dynamics and the expensive evaluation of the force have led to the development of a wide number of numerical techniques aimed at obtaining a reliable numerical solution with the minimum amount of computational resources, depending on the astrophysical problem of interest. Here we start by discussing the different astrophysical contexts where N-body simulations are routinely employed and we then present the state of the art techniques for these problems.

Astrophysical domains and timescales

N-body simulations are applied to a wide range of different astrophysical problems so that the most appropriate technique to use depends on the specific context, and in particular on the timescale and collisionality of the problem.

Timescales, Equilibrium and Collisionality

A system of N particles interacting gravitationally with total mass M and a reference dimension R (for example the radius containing half of the total mass) reaches a dynamic equilibrium state on a timescale comparable to a few times the typical time \(T_{cr}\) needed for a particle to cross the system \((T_{cr} \approx 1/\sqrt{GM/R^3})\ .\) This is the response time needed to settle down to virial equilibrium, that is \(2K/|W|=1\ ,\) where \(K\) is the kinetic energy of the system\[K=1/2 \sum_{i=1,N} m_i |\vec{v}_i|^2 \ ,\] and W is its potential energy\[W = - 1/2 \sum_{i \ne j} G m_i m_j /|\vec{r}_i-\vec{r}_j|\] (assuming no external field). If the system is initially out of equilibrium, this is reached through mixing in phase space due to fluctuations of the gravitational potential, a process called violent relaxation (Lynden-Bell 1967).

Once the system is in dynamic equilibrium a long term evolution is possible, driven by two-body relaxation. Energy is slowly exchanged between particles and the system tends to evolve toward thermodynamic equilibrium and energy equipartition. The timescale \(T_{rel}\) for this process depends on the number of particles and on the geometry of the system\[T_{rel} \propto N/log(0.11 N) T_{cr}\] (e.g. see Spitzer 1987). N-body systems such as galaxies and dark matter halos have a relaxation time much longer than the life of the Universe and are thus considered collisionless systems. Smaller systems, such as globular and open clusters, are instead collisional, as the relaxation time is shorter than their age. Two body relaxation is also suppressed when one particle in the system dominates the gravitational potential, such as in the case of solar system dynamics, where planets are essentially quasi-test particles.

Close encounters between three or more particles not only contribute to energy exchange, but can also lead to the formation of bound subsystems (mainly binaries). The formation and evolution of a binary population is best followed through direct, unsoftened, N-body techniques.

A self-gravitating N-body system made of single particles has a negative specific heat, that is it increases its kinetic energy as a result of energy losses (Lynden-Bell & Wood 1968). This is a consequence of the virial theorem and qualitatively it is analogous to the acceleration of a Earth artificial satellite in presence of atmospheric drag. A negative specific heat system is thermodynamically unstable and over the two body relaxation timescale it evolves toward a gravothermal collapse, creating a core-halo structure, where the core progressively increases its concentration, fueling an overall halo expansion. The collapse is eventually halted once three body interactions lead to the formation of binaries. The so called "core collapsed globular clusters" are considered to be formed as a result of this mechanism.

Mean field approach: the Boltzmann equation

A system of N particles interacting gravitationally defines a 6N+1 dimensional phase space given by the N position and velocity vectors associated to each particle at each time t. The solution of the N-body problem defines a trajectory in this phase space. If the number of particles is large enough, that is if the two body relaxation time is long compared to the time-frame one is interested in, then a statistical description of the problem is possible. This allows us to pass from a 6N+1 to a 6+1 dimension phase space. The idea is to construct a mean field description of the dynamical system in terms of a single particle distribution function \(f(\vec{x},\vec{v},t) \ ,\) where \(f(\vec{x},\vec{v},t) d^3x d^3v \) is proportional to the probability of finding a particle in a 6D element of volume \( d^3x d^3v \) centered around position \(\vec{r}\) and velocity \(\vec{v}\) at time t. Within this simplified framework the knowledge of the distribution function uniquely defines all the properties of the system. The dynamic is described by the collisionless Boltzmann equation, which derives essentially from the Liouville theorem:

\[ \frac{D f}{D t} = \frac{\partial f}{\partial t} + \vec{v} \cdot \frac{\partial f}{\partial \vec{x}} - \frac{\partial \phi_T}{\partial \vec{x}} \cdot \frac{\partial f}{\partial \vec{v}}= 0, \]

where the total potential field \(\phi_T = \phi_{ext}(\vec{x},t)+ \phi(\vec{x},t)\) is the sum of an external potential plus the self-consistent field \(\phi(\vec{x},t)\) defined from the distribution function itself through the solution of the Poisson equation:

\[ \nabla^2 \phi(\vec{x},t) = 4 \pi G \rho(\vec{r},t), \]

where \( \rho(\vec{r},t) = \int f(\vec{x},\vec{v},t) d^3v \ .\)

Given its high dimensionality (6+1), the collisionless Boltzmann equation is usually solved by sampling the initial distribution function and then by evolving the resulting N-body system by means of a numerical method that suppresses two body interactions at small scales. The interaction is softened not only for computational convenience to limit the maximum relative velocity during close encounters but especially to prevent artificial formation of binaries. This is because a simulation particle in a collisionless run represents in reality an ensemble of real particles (e.g. galaxies contain \(10^{11}\) stars but simulations typically use only \(N \in [10^6:10^9]\)). Note however that two body relaxation is driven by close as well as by distant encounters, so softening does not suppress it. In principle any numerical method that has a small scale softening is appropriate for following collisionless dynamics.

A mean field description for an N-body system is possible also for collisional systems, that is when the relaxation time is comparable to or shorter than the timeframe of interest. In this case the collisionless Boltzmann equation is modified by the introduction of a collision operator \(C[f]\) on its right side:

\[ \frac{D f}{D t} = \frac{\partial f}{\partial t} + \vec{v} \cdot \frac{\partial f}{\partial \vec{x}} - \frac{\partial \phi_T}{\partial \vec{x}} \cdot \frac{\partial f}{\partial \vec{v}}= C[f]. \]

In this framework the operator \(C[f]\) describes the probability for particles to enter/leave a phase space element as a result of gravitational encounters. The collision operator C is generally constructed assuming that encounters are (i) Markov processes, that is C depends only on the present state of the system, (ii) local, that is only the velocity of the particles are changed and not their positions and (iii) weak, that is the typical velocity change is much smaller than the velocity itself. Under these assumptions Monte Carlo methods are available to solve the dynamics of the system (see next section). Applications of the collision operator include dynamics of globular clusters and of self-interacting dark matter.

Mean Field Approach: analogies and differences with fluid dynamics

The velocity moments of the Boltzmann Equation define a set of equations known as the Jeans Equations (e.g. Binney & Tremaine 2007). The first three equations of the set are formally identical to the Navier-Stokes equations for a self-gravitating gas and, like in the fluid-dynamics analogy, express the conservation of mass, momentum and energy. Therefore the numerical algorithms developed to follow the dynamics of N-body systems find a wide application also in the context of fluid-dynamics, with one important example being the Smoothed Particle Hydrodynamics (SPH) method (Gingold & Monaghan 1977). The fundamental difference between the two cases is that the Jeans equations are derived in the limit of a collisionless system, while the Navier-Stokes equations assume a highly collisional system, with the mean free path of a particle approaching zero. For fluids, this leads to the definition of an equation of state, which closes the Navier-Stokes equations. The Jeans Equations are instead an infinite open set, where the n-th velocity moment depends on the n-th+1 moment.

Astrophysical domains

Based on the previous considerations about collisionality and timescales, four main astrophysical domains for N-body simulations can be identified, each requiring a different numerical technique to guarantee maximum performance and accuracy:

Celestial mechanics (solar and extrasolar planetary systems). Here a single body dominates the gravitational field and all the other objects move almost like test particles, subject to reciprocal perturbations. In this framework very high accuracy is required to correctly evaluate the perturbative terms and to avoid being dominated by numerical noise such as time discretization and round-offs errors.

Dense Stellar systems, such as open clusters and globular clusters. These collisional systems made of components of roughly equal mass present a rich dynamics, with multiple close encounters of stars. The evolution requires to be followed on a relaxation timescale with a correct description of the short range interactions.

Sphere of influence of a massive BH at the center of a stellar system. The sphere of influence of a BH is the volume within which the gravity of the BH dominates over that of the other particles. The situation resembles that of solar system dynamics, but here given the very high density of stars two body encounters are frequent, making the problem a difficult hybrid between the two previous cases. In addition, Post Newtonian physics may need to be included if high accuracy is required in the proximity of the BH.

Galaxy dynamics and cosmology. Galaxies, and especially dark matter halos, are constituted by a very large number of particles, so that their dynamics can be well described in terms of a mean field. Close encounters are not important and softening is usually employed in these N-body simulations to avoid the unphysical formation of binaries. Within this class, Self-Interacting Dark Matter Particles need a special mention: if dark matter halos are made of Weakly Interacting Massive Particles, then their dynamics can be modified by non-gravitational self-interactions, especially effective at the center of cuspy dark halos. The dynamics of such a system is described by the Collisional Boltzmann Equation, which can be approximately solved using Fokker-Plank methods.

Figure 1: Visualization of a simulation of galaxy collision using stars (blue) and SPH gas (yellow) particles. Credit: Frank Summers - STScI, reproduced with permission. Original images and movies available at http://terpsichore.stsci.edu/~summers/viz

Newtonian gravity: methods

The history of N-body simulations starts with a pioneering attempt by Holmberg (1941), who followed the evolution of a 37 particle system, where the force was calculated using lightbulbs and galvanometers (taking advantage of the same \(r^{-2}\) scaling of electromagnetic and gravitational interactions). Computer simulations started in the early sixties using up to 100 particles (e.g. see von Hoerner 1960 and Aarseth 1963) and had their full bloom in the eighties with the development of fast and efficient algorithms to deal with collisionless systems, such as particle-mesh codes (see Hockney & Eastwood 1988 and references therein) and the tree method (Barnes & Hut 1986). At the same time regularization techniques were developed to deal with close encounters and binary dynamics in the case of direct simulations of a collisional system (e.g. see Aarseth's NBODY-X code series based on KS and chain regularization - Aarseth 2003 and references therein). These algorithm advancements were coupled with tremendous progresses in the hardware, with the cpu speed growing exponentially. In addition to parallelization of serial codes, the field advanced also thanks to special purpose hardware, such as the GRAPE (Makino et al. 1997). Today's (2008) N-body simulations are performed with up to \(N=10^5\) (e.g. see Baumgardt & Makino 2003) for direct integration codes over a two-body relaxation timescale and up to \(N=10^{10}\) for collisionless dynamics/cosmology (e.g. see the Millennium Run - Springel et al. 2005). In the context of planetary dynamics, self-gravitating systems of disk/ring particles with \(N\approx 10^6 \) can be followed over hundreds of dynamical times (e.g. Richardson et al. 2000). Major breakthroughs are also expected in the near future thanks both to the next generation GRAPE-DR and to double precision graphic processing units, which provide extremely cost competitive high performance computing capabilities.

Figure 2: Visualization from the Millennium Run Simulation, the largest published cosmological simulation to date. The figure shows the projected density field for a 15 Mpc/h thick slice of the redshift z=0 output. The overlaid panels zoom in by factors of 4 in each case, enlarging the regions indicated by the white squares. Credit: Volker Springel and the Virgo Consortium team, reproduced with permission. Original image available at http://www.mpa-garching.mpg.de/galform/virgo/millennium/index.shtml

Direct methods

Direct methods do not introduce approximations in the solution of the equations of motions and thus deliver the highest accuracy at the price of the longest computation time, of order \(O(N^2)\) per timestep. Integration is performed using adaptive (individual) timesteps and commonly a fourth order Hermite integrator. Close encounters and bound subsystems are treated exactly in terms of Kustaanheimo-Steifel transformations. These essentially consist in transformations of coordinates using a perturbative approach over the analytical two body solution. If more than two particles are strongly interacting between each other, a chain regularization strategy (Mikkola 1990) can be used, which consists in recasting the problem in terms of a series of separate Kustaanheimo-Steifel interactions. A state of the art, publicly available, serial direct N-body integrator is Aarseth's NBODY6. Even with this specialized software, the number of particles that can be effectively followed for timescales comparable to the Hubble time is limited. For example, if one is interested in the dynamical evolution of globular clusters, currently about \(N=20000\) is the practical limit for a serial run, as such a run takes about \(1000\) cpu hours. A run with \(10^6\) particles carried out for a similar number of relaxation times \(T_{rel}\) would require about \(10^8\) cpu hours. The algorithm can be parallelized, but in practice load imbalances may saturate the gain in efficiency, so some of the most cpu demanding simulations have been carried out on special purpose hardware, such as the GRAPE, where the chip architecture has been optimized to compute gravitational interactions, thus delivering Teraflops performance.

Tree codes

The tree code method (Barnes & Hut 1986) provides a fast, general integrator for collisionless systems, when close encounters are not important and where the force contributions from very distant particles does not need to be computed at very high accuracy. In fact, with a tree code, small scale, strong interactions are typically softened (but see McMillan & Aarseth 1993), while the potentials due to distant groups of particles are approximated by multipole expansions about the group centers of mass. The resulting computation time scales as \(O(N log(N))\) but the approximations introduces small force errors. The long-range force errors are controlled by a single parameter (the opening angle) that determines how small and distant a group of particles must be to use the approximation. This strategy works well to keep the average force error low, but a worst case scenario analysis highlights that unbound errors can arise for rare, but astrophysically reasonable configurations, such as that of the classic "exploding galaxy" (Salmon & Warren 1994). In addition, force errors from the tree code may lead to violation of momentum conservation. Typical implementations of the tree code expand the potentials to quadrupole order and construct a tree hierarchy of particles using a recursive binary splitting algorithm. The tree does not need to be recomputed from scratch at every timestep, saving significant cpu time. Systems with several hundred thousands of collisionless particles can be easily simulated on a GFlops workstation for a Hubble time using this method.

Fast Multipole Methods

A standard tree code implementation does not take advantage of the fact that nearby particles will be subject to a similar acceleration due to distant groups of particles. The Fast Multipole Method (Greengard & Rokhlin 1987) exploit this idea and uses a multipole expansion to compute the force from a distant source cell within a sink cell. This additional approximation of the gravitational interaction was claimed to reduce the complexity from \(O(N log(N))\) to \(O(N)\ ,\) but the exact scaling seems implementation dependent and has been debated in the literature (e.g. see Dehnen 2000 and references therein). One advantage of the fast multipole method is that the symmetry in the treatment of sink and source cells with respect to the multipole expansion can guarantee an exact conservation of the momentum. Recent successful implementations of fast multipole codes or hybrids with tree code scheme, include Dehnen's Cartesian expansion scheme (the GyrfalcON code- Dehnen 2000) and PKDGRAV (Stadel 2001).

Particle-mesh codes

The particle mesh method represents another route to speed up direct force evaluation for collisionless systems. In this case the gravitational potential of the system is constructed over a grid starting from the density field and by solving the associated Poisson equation. Particles do not interact directly between each other but only through a mean field. The method essentially softens the gravitational interactions at small scales, that is below the cell length. The density field is constructed using a kernel to split the mass of the particles to the grid cells around the particle position. The simplest choice is to assign all the mass to a single cell, but this leads to significant force fluctuations, which can be reduced using a cloud in cell (8 points) or a triangular shaped cloud (27 points) kernel. The Poisson equation is typically solved using a Fast Fourier Transform, but other grid methods such as successive overrelaxation can also be used - e.g. see Bodenheimer et al. (2007). The deriving force, defined on the grid, is then assigned back to the particles using the same kernel employed for the density field construction, in order to avoid spurious self forces. The method achieves a linear complexity in the number of particles and (\(O(N_g log(N_g)\)) in the number of grid cells (this latter scaling is that of the FFT method). The price to pay is in terms of short range accuracy as the force is a poor approximation of Newton's law up to several grid spacing of distance.

Adaptive Mesh Refinement method

The dynamic range of particle-mesh codes can be increased by using an adaptive rather than a static grid to solve the Poisson Equation. In the Adaptive Mesh Refinement (AMR) method the grid elements are concentrated where a higher resolution is needed, for example around the highest density regions. One possibility to obtain an adaptive resolution is to first construct a low-resolution solution of the Poisson Equation and then to progressively refine regions where the local truncation error (estimated through the Richardson extrapolation) is highest. A multigrid structure needs to take into account issues such as matching the solution at the grid interfaces. AMR codes are well suited for cosmological simulations (e.g. see the ENZO code, Bryan & Norman 1998).

Self consistent field methods

A variant over the Particle Mesh code is the expansion of the density and potential of the system in terms of a basis of orthogonal eigenfunctions. Clutton-Brock (1972) was one of the first to apply this idea in stellar dynamics, while a modern implementation is that of Hernquist & Ostriker (1992). This method guarantees at fixed computational resources a higher accuracy than the tree code and the particle mesh algorithms, provided that the set of basis function is appropriately selected. This limits in practice a general application of the method, which remains however very competitive for the study of the dynamical stability of collisionless systems constructed from distributions functions models.

P3M and PM-Tree codes

In order to increase the force resolution of particle mesh codes it has been proposed to couple a mean field description on large scales with a direct, softened, treatment of the gravitational interactions on distances of the order of or below a few grid spacing (e.g. see Springel 2005). This method is called \(P^3M\ :\) Particle-Particle-Particle-Mesh and efficiently increases the dynamic range of the parent PM algorithm. However in presence of strong clustering a large number of particles will interact directly between each other, slowing down significantly the computation to \(O(N^2)\ .\) This problem can be resolved by using adaptive meshes, so that the spatial resolution is refined in regions of high density. Adaptive \(P^3M\) codes have a computational cost which scales as \(O(N log(N))\ ,\) like in a tree code. Finally another possibility is to resort to a tree code for the short range force evaluation leading to a hybrid PM-Tree scheme. These methods are generally extremely well suited for cosmological simulations, for example see Gadget2 (Springel 2005).

Celestial mechanics codes

Computational Celestial Mechanics refers to a series of methods targeted at studying the dynamics of small N systems (\( N \lesssim 20 \)). The smallest non trivial N is N=3, that is the three body problem, which has many applications ranging from space flight to planets satellite motions and to binary-single stars encounters. Celestial mechanics requires extremely high precision given the chaotic nature of the N-body problem. Numerical methods are based on the use of local system of coordinates, to fight round-off errors in systems with a wide dynamic range, such as in the study of star-planet-satellite problems, as well as on the variational equations formalism and on perturbation theory to take advantage of the analytical, unperturbed motion of planets in the gravitation field of their star (e.g. see Beutler 2005). In this context symplectic integrators are widely used (e.g. see Wisdom & Holman 1991; Leimkuhler & Reich 2005).

Mean Field Methods

As an alternative to particle based N-body methods, the dynamics of a system of particles interacting gravitationally can be followed by solving the time dependent Boltzmann Equation coupled with the self-consistent Poisson equation.

Grid based solvers for the Collisionless Boltzmann Equation

This approach can take advantage of standard computational methods developed to solve partial differential equations, such as successive over-relaxation and conjugate gradient methods. However it requires to solve a highly dimensional (6D+time) non-linear system of partial differential equations. In general, the bottleneck is thus the very large amount of memory needed (for example, Terabites just to have a moderate resolution grid with 100 elements in each dimension). However this method is competitive if the astrophysical problem of interest presents symmetries that reduce the number of dimensions needed in the model. For example, in the case of globular cluster dynamics a very good approximation can be obtained via a 3 dimensional model by assuming spherical symmetry in the position space (1D) and radial anisotropy in the velocity space (2D).

Fokker-Planck and Monte Carlo methods

These methods solve the collisional Boltzmann equation starting from a given distribution function and by following test particles in the six dimensional position-velocity phase space. At each timestep the velocity of the particles is perturbed by random fluctuations accordingly to the assumed form for the collision operator \( C[f]\ ,\) which depends on computed cross sections for two, three and four body encounters. The complexity of Monte Carlo codes is linear with the number of particles and thus a realistic number of particles can be used for simulations of collisional systems with \( N>10^5 \) with a serial code. The method is ideal for exploring grids of initial conditions, after proper validation through comparison with direct integration (e.g. see Heggie et al. 2006).

Beyond Newton: strong gravitational fields

In presence of a strong gravitational field, such as that in the proximity of the event horizon of a black hole, N-body simulations cannot be based on Newtonian physics, but must take into account a general relativity framework. As a numerical solution of the Einstein equation is extremely challenging, Post-Newtonian approximations are used when the gravitational field does not deviate too much from the Newtonian case. Post-Newtonian corrections are typically good enough to treat most astrophysical problems of the dynamics of stars around a black hole. A full general relativity framework is only required to study the merging and gravitational waves emission of two black-holes (e.g. see Baker et al. 2006).

An alternative approach to increase the efficiency of numerical solution of the N-body problem is to optimize the hardware. For direct simulations this approach can be very effective, thanks to the fact that the bottle neck of computation is just the evaluation of the gravitational force, which has a very simple expression. Along this route the GRAPE (GRavityPipE) concept has been extremely effective. The basic idea is to optimize a hardware pipeline to compute \( (\vec{r_i}-\vec{r_j})/| \vec{r_i}-\vec{r_j}|^3 \ .\) This special purpose hardware can then be interfaced with a general purpose computer, which takes care of all the other numerical operations required to solve the equations of motions. With the GRAPE-6, the largest simulation on a collisional timescale published to date has N=131028 (Baumgardt & Makino 2003).

Another recent promising hardware development is the possibility to use Graphic Cards (GPUs) to carry out the cpu intensive force evaluation. The performance of current generation of GPUs appears to be superior (in terms of Flops/$ ratio) to that of the GRAPE6 series (Portegies-Zwart et al. 2007).

Simulation environments

In addition to the availability of stand-alone codes, several software environments have been created that contain various tools to set up initial conditions, run simulations, and analyze and visualize their results. Some examples are NEMO, Starlab, ACS, MUSE.

  • Aarseth, S. 1963, MNRAS, 126, 223
  • Aarseth, S. 2003, "Gravitational N-Body Simulations: Tools and Algorithms", Cambridge University Press
  • Baker, J.G. et al. 2006, ApJ, 653, 93
  • Barnes, J.E. and Hut, P. 1986, Nature, 324, 466
  • Baumgardt, H. and Makino, J. 2003, MNRAS, 340, 227
  • Beutler G. 2005, "Methods of Celestial Mechanics", Springer
  • Binney J. & Tremaine S. 1987, "Galactic Dynamics", Princeton University Press
  • Bodenheimer, P., Laughlin, G.P., Rozyczka, M. and Yorke, H.W. 2007, "Numerical Methods in Astrophysics: An Introduction", Taylor & Francis
  • Bryan G.L. and Norman, M.L. 1998, ApJ, 495, 80
  • Clutton-Brock, M. 1972, Ap&SS, 16, 101
  • Dehnen, W. 2001, MNRAS, 324, 273
  • Dehnen, W. 2000, ApJL, 536, 39
  • Gingold, R.A. and Monaghan, J.J. 1977, MNRAS, 181, 375
  • Greengard, L. & Rokhlin, V. 1987, J. comput. Phys., 73, 325
  • Heggie, D.C., Trenti, M. and Hut, P. 2006, MNRAS, 368, 677
  • Hernquist, L and Barnes, J.E. 1990, ApJ, 349, 562
  • Hockney, R.W. and Eastwood, J.W. 1988, "Computer Simulation Using Particles", Taylor & Francis
  • Holmberg, E. 1941, ApJ, 94, 385
  • Leimkuhler, B. and Sebastian R. 2005, "Simulating Hamiltonian Dynamics", Cambridge University Press
  • Lynden-Bell, D. 1967, MNRAS, 136, 101
  • Lynden-Bell, D. and Wood, R. 1968, MNRAS, 138, 495
  • Makino, J., Fukushige, T., Koga, M. and Namura, K. 2003, PASJ, 55, 1163
  • Portegies-Zwart, S.F., Belleman, R.G. and Geldof, P.M. 2007, New Astronomy, 12, 641
  • Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery B.P. 2007, "Numerical Recipes", Cambridge University Press
  • Richardson, D.C., Quinn, T., Stadel, J. and Lake, G. 2000, Icarus, 143, 45
  • Salmon J.K. and Warren M.S. 1994, J. Comp. Phys., 111, 136.
  • Spitzer, L. 1987, "Dynamical Evolution of Globular Clusters", Princeton University Press
  • Springel, V. 2005, MNRAS, 364, 1105
  • Springel, V. et al. 2005, Nature, 435, 629
  • Stadel J. 2001, PhD. Thesis, University of Washington
  • von Hoerner, S. 1960, Z. Astrophys. 50, 184
  • Wisdom, J. and Holman, M. 1991, AJ, 102, 1528

Internal references

  • Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
  • Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
  • David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.

Recommended reading

  • "Computer Simulation Using Particles" Hockney, R.W. and Eastwood, J.W. 1988
  • "Gravitational N-Body Simulations: Tools and Algorithms" Aarseth, S. 2003
  • "The Gravitational MillionBody Problem" Heggie, D.C. and Hut, P. 2003
  • "Methods of Celestial Mechanics" Beutler, G. 2005
  • "Numerical Recipes" Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery B.P. 2007
  • "Numerical Methods in Astrophysics: An Introduction" Bodenheimer, P., Laughlin, G.P., Rozyczka, M. and Yorke, H.W. 2007

Review articles

  • "Simulations of Structure Formation in the Universe" Bertschinger, E. 1998, ARA&A, 36, 599

Web Material

External links

Open source codes

  • MUSE, a software framework for simulations of dense stellar systems: http://muse.li/

Pictures and movies from N-body simulations

See also

Computational Celestial Mechanics, KAM Theory, Three Body Problem, Computational Astrophysics, Computational Cosmology

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Parents | Raising readers & learners.

Home of Parent & Child Magazine

What Makes a Good Kindergarten?

See how to evaluate a program — and how to advocate for your child if his classroom doesn't measure up.

Learning Benefits

Hover over each Learning Benefit below for a detailed explanation.
Problem Solving
Independent Thinking

Ideally, kindergarten will be a smooth, sunny introduction to real school for your child, since it sets the stage for the rest of his education. While no program is perfect, some are better than others. Find out what sets them apart and how you can get the best possible start for your child — no matter what your options are.

Why Kindergarten?
First, consider the goal of a good kindergarten program. Why do most children attend, even though it is not mandated in many states? Kindergarten provides your child with an opportunity to learn and practice the essential social, emotional, problem-solving, and study skills that he will use throughout his schooling.

  • The development of self-esteem is one of the important goals of kindergarten. This is the process of helping your child feel good about who she is and confident in her ability to tackle the challenges of learning.
  • Kindergarten teaches cooperation: the ability to work, learn, and get along with others. A year in kindergarten provides your child with the opportunity to learn patience, as well as the ability to take turns, share, and listen to others — all skills that he will use through his school years and beyond.
  • Most children are naturally curious, but some do not know how to focus or use this curiosity. Kindergarten is a time for sparking and directing your child’s curiosity and natural love of learning.

What Does an Ideal Kindergarten Look Like?
Ask any number of educators and parents, and you will get many different descriptions of the ideal kindergarten. But there are certain basic agreements among educators as to what makes a good program. It should:

  • Expand your child’s ability to learn about (and from) the world, organize information, and solve problems. This increases his feelings of self-worth and confidence, his ability to work with others, and his interest in challenging tasks.
  • Provide a combination of formal (teacher-initiated) and informal (child-initiated) activities. Investigations and projects allow your child to work both on her own and in small groups.
  • Minimize use of large group activities that require sitting. Instead, most activities feature play-based, hands-on learning in small groups. As the year progresses, large group activities become a bit longer in preparation for 1st grade.
  • Foster a love of books, reading, and writing. There are books, words, and kids’ own writing all over the classroom.

When looking at programs, keep these elements in mind — as well as the specific needs of your child and family. Not every program is perfect for every child. Some children thrive in a program with more direction, some with less. Talk to your child’s preschool teacher, visit a few schools, and talk to the principal or a kindergarten teacher before deciding.

What if the Program Is Less Than Ideal?
Perhaps you have little or no choice about where to send your child to kindergarten but are concerned about its quality. First, give the program and teacher some time to get the year going. If you observed the class in the spring and it seems different when your child starts in the fall, there may be a good reason. Many programs start slowly, taking time to help children separate from their families and feel confident in school before adding learning demands.

If after a few weeks you still have concerns, talk to the teacher. Ask her about her goals and share your expectations. Sometimes an apparent mismatch can be just a difference in approach. Keep the dialogue going. Ask for information, but also be willing to hear the “whys” of the teacher’s philosophy.

Still, there are times when a teacher or his approach is not the right fit for your child. Then it is time to talk with the principal. Come prepared with clear points you want to make. This will help the principal see what the problem is and make suggestions to help your child.

Sometimes (but rarely) children need to switch to a different teacher or school. This can be the result of many classroom observations of your child by the teacher, principal, and/or another professional. It is important to have group consensus on this decision.

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Notre Dame School

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Academic Results

Parents Forum

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About Notre Dame School

Notre Dame School is a Voluntary Aided school located in London.

Contact Information

Notre Dame School 118 St George's Road, London, Greater London, SE1 6EX
020 7261 1121
020 7620 2922

Other Info

  • Type: Voluntary Aided
  • OFSTED: Outstanding.
  • Rating: -
  • Gender: Girls
  • Total on roll: 602

School Catchment Area

Show my postcode on the map:

New to School Reviewer is our easy to follow catchment area heat map, which accurately illustrates the current catchment area for those schools where catchment is relevant, all compiled using data supplied by the Department of Education. Simply type in your postcode and see where your address is in relation to the heatmap. Invaluable if you are looking for properties in your area or are planning to move somewhere new in the UK.

All attending pupils: National School Census Data 2015
Heat Map Key: ManySomeFew
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Academic Results

OFSTED Results

Detail Result
Last Inspection Date 22/11/2012
Overall Effectiveness Outstanding
Behaviour and Safety of Pupils Outstanding
Achievement of Pupils Outstanding
Quality of Teaching Outstanding
Leadership and Management Outstanding

The table below shows the % of students achieving A* - C Grades in each subject

Subject name 2016 2015 2014
English 73% 83% 78%
Maths 71% 60% 61%
Science (Double Award) 84% 77% 81%
Humanities 85% 71% 70%
Languages 78% 69% 75%
% achieving 5 or more GCSEs 60% NE 55%

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Teaching Your Child To Rhyme - Worksheets and Ideas

Learning to identify rhymes is a fundamental phonemic awareness skill for young children.  Not only does your child need to listen carefully to each word but he needs to listen for sounds in specific positions in words. To identify a set of rhyming words, your child will need to listen for sounds in the ending position of each word.

When a child can identify rhymes, it is evident he is able to isolate sounds in words. This will be important as your child learns to write, as writing involves your child first hearing the word he wants to write, then isolating individual sounds within the word, then applying letter(s) to each sound he hears.

The beginning rhyming worksheets are an excellent way to introduce your child to the concept of rhyming.  With four pictures on the page, your young child can consider a limited number of options while he practices.  Just be sure to to glance quickly at all of the pictures to make sure that you know what to call each one. For example, if the first picture is a bat, you would know to call the kitten a “cat” when labeling the pictures for your child.

Kindergarten worksheets - Finding rhyming words beginning - Phonemic Awareness

For young children first learning to rhyme, explain to your child that he is searching for words that end with the same sound.  Then, give your child a simple example of a rhyming pair of words, such as “mom” and “tom” or “hat” and “cat.”  Be sure to engage your child’s eyes and ears by asking him to point to each picture as you say aloud the name.

If your child is struggling to identify the rhyming pair on each beginning worksheet, try focusing only on the two words that you have identified as rhyming and ignore the other two pictures on the page. Say aloud the name of the first picture then immediately say aloud the name of the picture that completes the rhyming set.  Ask your child if he hears how they sound similar and repeat the two words again.  With practice and exposure, your child will learn to identify the rhyming pairs without such strong guidance.

The intermediate rhyming worksheets provide additional, and slightly more challenging, rhyming practice. Each page contains six pictures instead of only four, but there is still only one rhyme per worksheet.

Kindergarten worksheets - Finding rhyming words intermediate - Phonemic Awareness

To reinforce the concept of rhyming words for a child who already understands the basic concept of rhyming, I suggest playing a modified version of “I Spy.” Rather than simply giving a clue of what you spy, give your child rhyming clues such as, “I spy something blue that rhymes with hall.” (ball)

The advanced rhyming worksheets require your child to remember the words in the first column as he searches for a rhyming word in the second column. By adding more pictures to a page, your child will need to pay extra attention to each individual picture and not become distracted. Also, since each page contains four rhymes instead of one rhyme, these worksheets pose an advanced challenge for young children.

Kindergarten worksheets - Rhyming word pairs - Phonemic Awareness

Is your child ready for an advanced rhyming game?  Check back tomorrow for two Memory-style rhyming games to play with your child at home.

Questions For A Kindergarten Teacher?

Do you have questions about how to explain rhyming to your child? Or are you looking for extra suggestions for simple activities you can do at home to get your child comfortable with rhyming?

I would love to hear from you, so please leave a comment, below.

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Smart kids listen to heavy metal

Research funded by Metallica?

Gifted students who feel the pressure of their ability could be using Heavy Metal music to get rid of negative emotions.

This is the conclusion of Stuart Cadwallader and Professor Jim Campbell of The National Academy for Gifted and Talented Youth at the University of Warwick. They will discuss their findings at the British Psychological Society’s Annual Conference at the University of York on Wednesday 21 March 2007.

1, 057 students aged between 11 and 18 years old completed a survey which asked them about family, school attitudes, leisure time pursuits and media preferences. They also asked them to rank favoured genres of music.

They found that rock was the most popular form of music, closely followed by pop. But there were also differences between the type of music the young people liked and their attitudes – with those who liked Heavy Metal having lower self-esteem and ideas about themselves.

To find out why this was, the researchers then quizzed 19 gifted students via an online group interview to find out their views on Heavy Metal.

These pupils said they did not consider themselves to be ‘Metalheads’ but identified with specific aspects of this youth culture.

They spoke specifically about using Heavy Metal for catharsis, literally using the loud and often aggressive music to jump out frustrations and anger. Although the more ardent fans stated that ‘there’s Metal out there for every occasion’, many also stated they listen to the music when they are in a bad mood.

Mr Cadwallader said: “Perhaps the pressures associated with being gifted and talented can be temporarily forgotten with the aid of music. As one student suggests, perhaps gifted people may experience more pressure than their peers and they use the music to purge this negativity.”

Written from a news release by University of Warwick.


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Scientists in the Field
2005 Sibert Honor Medalist

The Tarantula Scientist

By Sy Montgomery | Photographs by Nic Bishop

Yellow blood, silk of steel, skeletons on the outside! These amazing attributes don’t belong to comic book characters or alien life forms, but to Earth’s biggest and hairiest spiders: tarantulas. Here you are invited to follow Sam Marshall, spider scientist extraordinaire (he’s never been bitten), as he explores the dense rainforest of French Guiana, knocking on the doors of tarantula burrows, trying to get a closer look at these incredible creatures. You’ll also visit the largest comparative spider laboratory in America—where close to five hundred live tarantulas sit in towers of stacked shoeboxes and plastic containers, waiting for their turn to dazzle and astound the scientists who study them.

Tarantula Scientist
Look inside the book!
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Copyright 2018 Houghton Mifflin Harcourt Books for Young Readers. All rights reserved.


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Sunday 2nd January 2011 IoS Bird Reports

The first boat of the year to Tresco today, allowing the islands birders to catch up with:

A Black-necked Grebe in The Roads, Common Scoter (7) in The Roads and an Eider off Samson. At least 2 male Pintails, TWO Buzzards over NE end of Tresco, 3 Jackdaw, 4 Pink-footed Geese, Long-tailed Duck in Pentle Bay, Red-breasted Merg off South Beach and 2 Black-throated Diver, Whimbrel, 3 Slav Grebe, 18 Pochard, Scaup and Mistle Thrush at Old Grimsby. A Spoonbill was in Old Town Bay, St Mary’s early Morning and the Hooded Crow was on Bryher viewed from Tresco.

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Home >> Listings Directory >>

palo verde Real Estate Listings

Use the following links to browse homes in palo verde from the California Regional Multiple Listing Service, Inc. (CRMLS). To search listings using criteria such as location, price, size, bedrooms, bathrooms and more, visit The Alden Team at http://RealtyOneGroup12.proiwebsites.com

Real Estate Listings in ZIP code 92266


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Sean Beaudoin

Enough excellent writing to fill a large tube sock

My Books:

"Some Tears May Be Crocodile, But My Salt Is True."

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Max City Blues

Well, just sent "Maximum City Blues" off last night. We'll see if anyone wants to buy it, but it was a lot of fun to write, a noir/detective thing, basically Lee Marvin meets Royale With Cheese. Even though a lot prep was already done, it's weird to have finished two novels in one calendar year. I sent "This Unlovely Monster" in a few months ago, but before that almost two years went by with almost nothing to show for it (a sprawling now-abandoned manuscript, angry essays, learning how to cook, etc.) It was a low period career-wise, the feeling of deep inertia and self-pity, a time to guzzle Merlot and question the purpose of literature as an ongoing concern, let alone vocation. Not to mention crumbling attention spans, shrinking budgets, and the rise of stories comprised of 140 characters. Yet somehow this year has all come together. I have no idea why. Possibly 6 straight months of rain? In any case, for the first time I like both books unequivocally. Pretty much everything before has felt to some degree fraudulent, like I knew the work could be better, or I might have been more emotionally honest and vulnerable, or any minute some guy in a beret would pop out from behind a velvet curtain and expose me in public for having zero clue what I was doing. And he would have been right. It's always possible neither book will ever actually hit shelves, but they do exist, wrenched out of sloth and pessimism and the vastly easier route of posting record covers on Facebook all day. So there's that. Anyway, 297 pages, 14 characters, 8 epilogues, 3 sex scenes, 2 Ike & Tina references, and one exploding cow.

On The Pod

Hey, I had the good fortune to appear on Brent Jensen's mighty "No Sleep Til Sudbury" podcast. It was a ton of fun. Brent's a Canadian author, musician, DJ, and all-around music industry wheel. In this episode I drop a list of the 10 most formative songs of my early years. Check it out. Hint: Lots of Winger talk.

You can find the episode on iTunes or the pod website:

Literary Happy Hour

Hey, tomorrow night I'll be reading at Capitol Cider's Literary Happy Hour with the mighty Joshua Mohr and Anca Szilagyi. Swing by for an early double Jager and crudite. I'm going to bust out a chapter of the forthcoming very adult novel "This Unlovely Monster". You've been warned. 5-7pm.

American Junkie Release Party

Hey, tonight's the launch party for the re-release of Tom Hansen's excellent American Junkie. I'll read before Tom does an encore, and then we'll break down all of music and literature and old school Seattle in an Q&A with Jarret Middleton. Definitely the literary event of the year: Lazarus emerges from the cave, shambles into Elliott Bay Book Company to drink beer and yell rude stuff at the podium. Yeah, you want to be there. 7pm.

Bookfest for Charity! Today!

Hey, this great annual event is happening again tomorrow. An absolutely killer lineup of authors, readings, food, and music. Excellent chance to buy some Christmas presents that matter, with a big chunk of the money going to charity, including PNA and the The Greater Seattle Bureau of Fearless Ideas. I'll be reading at 4pm and signing the rest of the while, but mostly standing there feeling good about something for once. You should definitely swing on by.

Election 2016

Here's my final column for The Weeklings, "25 Ways Hillary Can Fix America By Wednesday". It's one that I've been thinking about for six months, but finally got around to writing. Like it or not, Hillary will be elected tomorrow. After which I am going to drink a bowl of ayahuasca and lie in a sensory deprivation tank for two weeks. In the meantime, here are 25 non-partisan, commonsense ways that President Clinton can immediately make this country great again. In a sense, Trump has already won, because he has changed discourse forever. But not my discourse. Article includes at least three bonus Chuck Norris jokes. Plus, the first person who tattoos this entire thing on their back in a barely-legible calligraphy font wins a free T-shirt. Pics required. Selah.

Powell's Books Portland!

Sent from a friend in Portland. Always nice to be a staff pick at the great Powell's Books, Inc.

Arty Interview

So I was interviewed/photographed for this film/lit project recently, which might be amusing to check out. The cafe the photos were taken in was packed, and I was keenly aware of various people watching over their laptops and omelettes, trying to decide if I was someone famous, or an unbelievable tool. Of course they're not mutually exclusive, but I think the unspoken consensus was definitely on Phillips Head. Also, you try wearing better socks.

Amazon Pick of the Week!

For some reason, Welcome Thieves is a Kindle 25 Select all this week. Lots of imposing company on the list, and a few I'm pretty sure I could wrestle to the mat. Hey, go ahead and be cynical, but it's kinda cool to be selected as an "exciting summer read." I was born to be a summer read. And if everyone seeing this post breaks down and buys two copies, it'll sell 30 more copies than it would have otherwise. So, yea, gonna feel special all day, walk around in my boxers and drink mint tea and sing Elton John to the plants

Dock Street Salon

Hey, Saturday night I will be reading with the mighty Ben Tanzer, who's touring with his latest "Be Cool". It's part of the Dock Street Salon at Phinney Books. There will be free pudding, and rude lit spoken aloud in overly serious voices. Then a roundtable discussion between me and Ben moderated (refereed?) by Joshua Mohr. Yeah, you should definitely come. 7pm.

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X-Seal & Go is a range of collapsible food containers with airtight seals. The threaded lids are simple and easy to operate, and provide peace of mind against leaks. When not in use each container collapses to a minimal height, taking up significantly less space when stored. Available in four different sizes, the X-Seal & Go containers are perfect for culinary adventures on the trail, as well as taking lunch to school or the office.

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Syndicate contentJeff Slye

Jeff Slye

CEO (Chief Evolution Officer)
Business Evolution Consulting

Jeff Slye is considered a specialist in shaping and guiding organizations to initiate and adopt environmental programs. He has created and led â€˜greeningâ€™ programs for over 70 hotels and restaurants across the United States and Canada and was the strategic advisor to the National Restaurant Associationâ€™s effort to undertake a sustainability agenda for the restaurant industry. Mr. Slye emphasizes a â€˜business friendlyâ€™ approach and recognizes organizations have different brand, image, quality, and financial sensitivities and requirements when embracing an environmental program.

Implementing Sustainability Initiatives: Lessons from Kimpton Hotels

May 4, 2:15 PM

Kimpton Hotels is a San Francisco-based hotel chain with 42 boutique hotels across the United States. In 2005 Kimpton launched EarthCare, a major environmental sustainability program. Two years down the road, EarthCare has proven very successful, both in terms of its environmental and business benefits. Conceiving and launching a sustainability vision or program, however, is only the beginning of a long and rewarding journey. Creating a supportive organizational culture and managing the implementation of such programs is a complex process.

This workshop explores the process used to implement and institutionalize EarthCare and the lessons learned. Presenters will offer a conceptual roadmap of the steps required to successfully implement sustainable business programs.

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Google Reframes Size Debate

Google removed their you are searching 8 billion + pages and reframes the size debate, encouraging people to use Google more and search more specifically. :)

Framing is huge in marketing. If you control the language then you win the battle.

Danny wrote a great article about the history of the search database size wars, going back to the days of AltaVista & Excite page counting.

Also interesting to note that Danny loves the Lego Star Wars video game, and searched for the same piece I did. Finding the missing Star Wars lego pieces via search is no doubt a hard task, especially using UK spellings.

Danny recently turned 40. Happy Bday Danny.

Astute SEW fans may find the hidden Annie Williams paintings present suggestion in his latest article. :)

Published: September 27, 2005 by Aaron Wall in marketing

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More Free SEO Tools

I added a bunch more SEO tools to the free SEO Tools page. Most of the new tools are geared toward beginners and are perhaps a bit boring, but
I like some of them, including:

Some of the tools could be quite a bit better, but I bought a bunch of them for cheap, almost just for the heck of it because they were cheap. Let me know if you find any of them useful. I already know some tweaks I want to make when time becomes available.

The next build of Backlink Analyzer is coming along slowly but surely. Am hoping to have a cross platform version of it with a few more features before the month is out.

Published: January 4, 2006 by Aaron Wall in seo tools

January 29, 2006 - 5:41am

hey yo...yo got the typo generator working...good!

but you never wrote me back you punk ;-)

January 29, 2006 - 6:23am

I swore I wrote you back.

Notice I did not open source that one since I got it from you and did not know if it was ok.

thanks for writing it Ninja :)

Hows stuff going your way?

January 4, 2006 - 5:01pm

These tools are AWESOME! I'm training my first employee this week, and guess what we're going to be working on this morning?

Seriously, maybe this qualifies me as a raw beginner, but I think these tools rule. Thanks, Aaron. Definitely bookmarking this one.

January 4, 2006 - 5:03pm

Aaron, I just launched a new free SEO Tool too. It's focused more on web standards, and how people code their pages (semantic structure, modern web design techniques, etc...). It also has a Lynx text browser built into every report, so you can check your page for semantic structure and accessibility. The main purpose of the tool is to help people build websites that are built on modern web standards and are also optimized for search engines. It's still in beta, and any feedback would be appreciated.

January 9, 2006 - 3:49am

Aaron, seems like your notification not working....

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Google Outdistancing Yahoo! in Monetizing Branded SERPs

Yahoo! tends to be a bit more cautious than Google when it comes to allowing trademark related ads. That significantly suppresses their earnings because brand related search queries are often some of the most targeted, most commercial, highest converting, and most expensive keywords. Here is a snapshot of the current search results for SEO Book. Notice that the top 8 organic results link to SEO By loosening up on subdomains (displaying them more frequently) and providing a mini site map (called Sitelinks) in the SERPs it makes it much harder for affiliates or merchants reselling a brand to get exposure through the organic search results for the core brand name.

There are two pieces to that, as well.

  1. If the top 5 or 6 search results point at the official site searchers have to scroll down quite a bit to find commercial search results outside of the core brand. Rather than scrolling they may be more likely to click an ad.

  2. If the search results look highly informational in nature (by being harder to manipulate via commercial bias, and Google over-representing the official site) then merchants are more likely to buy ads.

The brutal part with buying the ads is that sometimes Google shows 0, 1, 2, or 3 AdWords ads above the organic search results. If the organic search results are somewhat irrelevant to the commercial intent of a searcher, those few ads at the top of the search results are going to get a high clickthrough rate. If you are a merchant who is not featured at the top of the results the right rail ads will bring you relatively little exposure. Thus a bidding war occurs for those top couple ad spots, and Google can control how many ads to show above the results to maximize earnings.

If you sell an ebook like I do, it is no big deal if one day your sales are high and the next day they are garbage, but if you run a business with fixed costs, a fixed marketing budget, and many employees then you may be stuck paying whatever it takes to be at the top of the results for brands you carry. The more dependant your business is on search the more they can bleed you dry!

While I used my brand as an example in this post, this brand factor recently played a big roll for a client, who even outranks his manufacturer for their official name in some major search engines, but is forced to pay much higher AdWords rates to get any exposure on Google. If he falls out of the top couple ad slots the right rail sends us like 10% the traffic that the top ad positions do. And I am probably going to have to bid bleed about $5 a click for a while to work that client back into the rotation of the ads at the top of the search results. And then if we knock a competing site out of position that may spur on a budget bleeding bidding war. Companies with a fixed marketing budget (my client has a variable budget based on market conditions) will suffer even worse by having to overpay for traffic until they kill their budget, and then not show up the rest of the time.

If the core brand term is priced out of reach it is important to

My client dominates the long tail, but for that particular brand he carries, most of the queries are for the official brand name. As a Google share owner and owner of a strong brand, I think they are brilliant. As a marketer who has a client getting screwed by their setup, I think a bit less of them. ;)

Also worth noting that Yahoo!'s search results are moving toward a more authority based algorithm, and they are starting to double list many brand sites, so this issue will likely rise again soon enough.

Published: October 14, 2006 by Aaron Wall in google pay per click search engines

October 19, 2006 - 11:47am

You are just right. I opened a PPC account two days ago on Yahoo following their $50 bonus. This is what i recieved back:

Account ID: XXXX
Account Name: ZZZZ

Approved Search Listings: 12
Declined Search Listings: 26
Pending Search Listings: 0

From 40 Key Words entered on the account opening, they declined 28 (the best ones). It took 2 days. All apparently due to trademark concerns. These KW perform on the other PPC vendors without problems.

It is not easy to increase reveinue with such policies. Or they think everyone will bid $5 per KW???

October 15, 2006 - 4:40am

Talk about not ranking for your own brand name. Aaron: did you ever here of the case: this guy (named coby) owns the domain name and coby inc. tried to sue him for the rights of the name. The results... coby got to keep his domain name.

This comes to mind because when you searc for coby in google: guess what... coby is literally right behind the giant electronic manufacturer's Web site-
Another Web site of the electronic manufacturer's is ecoby- and that does not rank at all for the the keyword "coby."

October 15, 2006 - 5:02am

Then I see you are correct about yahoo being less devoted towards brand names in the organic results because cobyusa does not rank in the firts spot of the organic results in yahoo, it ranks in the 2nd spot. Also in yahoo there are many more reseller/merchant related Web sites such as newegg showing and less coby related names shownig. In google on the other hand- there are definitely more "coby" related sites showing. I think that this proves that if one of the keywords of your site has the brand name that you are looking to rank for in it, then google will bias towards you- more than would yahoo.

October 15, 2006 - 7:50am

So is this an example of the strong getting stronger while for those new to the game it's going to be even more and more difficult to get noticed? Are new sites and businesses going to be forced to spend more in order to build their brand or get traffic from Google and now Yahoo?

October 15, 2006 - 7:58am

Well costs constantly increase so long as a market is underpriced and new competition keeps jumping in.

As much as anything, I think this post is just another example of why weakness in Yahoo!'s ad market may not equate to a bad quarter by Google, and a reminder to some marketers of how their marketing strategy may need to change if their search marketing is heavily reliant upon core brand related searches.

I get multi link sitemaps for searches like

  • seobook
  • seo book
  • aaron wall seo
  • aaron wall's seo

The branded multi link love will hit more queries over time.

October 16, 2006 - 12:41am

And in addition Aaron... you would most likely be dominating in more keywords if you would have had more anchor text variations earlier on in the game.

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Wednesday, 16 December 2015

John Howie Five Cheese Twice Baked Potato

"Recently voted Readers' Choice "Best Steak" in Seattle Magazine". Chef/restaurateur John Howie's definitive NW steak house, serving custom-aged USDA Prime steaks, American Wagyu Beef, Australian "A5" Full blood Waygu beef, Japanese "A5" 100% Wagyu beef, fresh seasonal Pacific fish and seafood's, wild foraged mushrooms, and locally sourced fruits and vegetables. John Howie Steak features two wood fired grills, mesquite charcoal and apple wood that add subtle flavors to the foods. The wine list features a "Best of Award of Excellence" from Wine Spectator. You can enjoy classic and seasonal specialty cocktails, or the world’s finest single-malt scotch, bourbons and fine cognacs in the lounge. There are two beautifully appointed private dining rooms that are available for meetings or special occasions.

One of my favorite sides at John Howie is the "Five Cheese Twice Baked Potato"! It's very flavorful without being too rich and cheesy.that was accompanied by oven roasted vegetables. Today, I am giving you this delicious recipe, which I got directly from John Howie! Yum!!

John Howie Five Cheese Twice Baked Potato

Makes 8 servings (1/2 potato per serving)

  • 4 large Washington Russet potatoes
  • Oil (John Howie used bacon fat)
  • ½ cup (1 stick) butter
  • 1/3 c. shredded Asiago cheese
  • 1/3 c. shredded Monterey Jack cheese
  • 1/3 c.  Rondelé cheese
  • 1/3 c.  cheddar cheese
  • 1/3 c. chopped  green onion
  • 4 slices bacon, crisp-cooked and crumbled
  • 1 teaspoon sea salt
  • ½ teaspoon freshly ground black pepper
  • ¼ teaspoon dry onion granules
  • 1/8 teaspoon dry garlic granules
  • 1/8 teaspoon white pepper (White pepper tastes hotter than black but is less complex, with fewer flavor notes)

  • 1/2 c. shaved Parmigiano-Reggiano cheese
  • 2 teaspoons minced chives

Preheat oven to 350 F degrees.
Scrub potatoes with a vegetable brush under cold running water. Brush or rub each potato with about 1 teaspoon of oil. Bake until tender when pierced with fork, about 1 hour and 30 minutes, depending on size of potato.
Remove from oven and cool just slightly. (Cooling too long will result in a lumpy potato) Cut each potato in half lengthwise and scoop out the center, leaving a shell about 1/4 inch thick. Arrange shells on baking sheet.
In large mixing bowl, with whisk attachment added to electric mixer, or use a potato masher, mix scooped pulp with butter until blended. Add, the remainder of the ingredients and beat or mash until smooth.

Refill the potato skins with all the mashed potato. Put stuffed potato halves on baking sheet and bake until tops are golden brown about 15- 20 minutes. To serve, sprinkle with Parmesan Reggiano cheese and chives.

This is a great choice with Perfect Prime Rib, Horseradish Mashed Potatoes!

Another great choice Mashed potatoes with three onion butter!

This is a delicious side with all the holiday color, Roasted Brussel Sprouts!

Are you putting together your DIY holiday gifts? Here is the entire tutorial on baking and decorating your Christmas cookies!! Part 1 - Baking the Roll out Sugar Cookies!
Part 2- Decorating and boxing the Christmas Cookies!

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This is the second chapter in the series. Hope you enjoy it
I am sorry for the errors in the first story. Hope you can read past them and enjoy the story

Lake House

My name is Dean and I am 18 years old. I have dirty blonde hair that reaches just over my ears. I have deep brown eyes that like to wonder from time to time. Last week my life changed for the better when I met my neighbour Lucy. If you read “A Dream of a Night” you will be caught up with the events that happened.
“Dean Honey time to wake up or we will be late” Mom called from the kitchen. I woke up and groggily sat up and watched the clock. 7 am was early for me on a Saturday morning. The thing we were going to be late for was our trip to the lake house that we owned. It was far out and it was a long drive to get there so Mom always wanted to get an early start so that we would have more time to spend there. I got up and opened my closet and took my bag out and started packing some clothes for the time there; swimming trunks included.

I walked into the kitchen and saw Mom was making her signature road sandwiches; ham, cheese, tomato and lettuce. My mouth watered as I looked at the sandwiches. I saw Mom’s bag already near the front door. I looked outside and saw that Lucy was carrying a bag to a car. “Mom do you mind if I said goodbye to Lucy” I asked staring out of the front window. “Of course not dear go right ahead” Mom replied packing the last of the sandwiched and drinks. I opened the front door and walked next door. Lucy turned around and looked at me. She looked so beautiful in her daisy duke jeans and white tank top. Her breasts strained at the material. I felt my dick stirring in my pants. “Hay Lucy! Where are you off too?” I asked as I walked over to give her a hug. She hugged me tightly mashing her tits into my chest. I hope she didn’t feel my growing hard-on. “I don’t know, mom just said to pack and to bring my bikini” Lucy replied letting go of our embrace. “Well we are also going away to our favourite lake house” I said and Lucy smiled. “You better not have too much fun while I am gone” Lucy said winking at me when she said ‘fun’.
We left at around 7:30 am and I was all too eager to see the lake house again. I haven’t been there in a long time. We have neglected to go there with the death of my dad which I haven’t mentioned up until now. He died in a car accident and we were the only two survivors. I sometimes blame myself for not being able to do something but Mom reassured me that Dad wouldn’t want me to beat myself up about it.
Like I said it was a long drive to the lake house and I was right. I was bored out of my mind the whole drive. The only thing that kept me busy was thinking about Lucy and munching away at the sandwiches Mom made. I wondered what Lucy might be up to and our time together. I wondered if she would have met a guy and something would happen between them. I must admit that I had a slight crush on her. I quickly pushed all the negative thoughts out of my head and thought of the positive.

We arrived at the Lake House in record time. The car stopped in front of the house at 10 am. I got out and stretched a bit and looked around. It was still like I remember. A big lake with small wood houses surrounding it with dense forest around it. I walked over to the lake and stared into the distance until Mom’s voice brought me back to reality. “Hay Honey will you help me carry the bags inside” Mom called over and I walked over to the car and started helping her carry the bags inside.
The inside of the house was pretty cosy. It had a kitchen with a sitting area and a bathroom and then stairs up to two rooms. One for me and one for Mom. I carried both our bags to their separate rooms. My room had a single bed in it with a dresser for my clothes and a little table with a chair. Mom’s room was the same but just a bit bigger with a queen size bed instead of the single.
I decided to go for a walk around to see the surrounding area. I actually needed to clear my head of this whole Lucy thing. I walked around the house and checked to see if we still had some wood for the fire. There was still more than enough left for a fire or two. I continued walking down the side of the lake throwing a rock or two into the lake every now and again. I got back at the house around 12 am. I smelled something nice coming from somewhere. I decided to follow my nose. It lead me straight to Mom cooking us some bacon and eggs. A late brunch she would call it.
We sat at the dining room table and munched away at our food. “Where was Lucy going” Mom asked. “She doesn’t know because her Mom wouldn’t tell her” I answered eating my last piece of bacon. I took both our plates to the kitchen. “You two were really getting a long last time” Mom continued. “I guess” I said not really wanting Mom to find out what really happened between the two of us.
That night I decided to make a fire in the fireplace. I lay out on the couch in front of the fire catching some heat when my phone vibrated. Opened it and saw it was from Lucy. How did she get my number?

Hay you! Just thinking about you. This place is amazing. Lucy

I replied to her message

How did you get my number? Thinking about you too.

A reply came through

That is my little secret. Dinner is almost ready. Got to go bye!

I sent her a last reply and closed my phone

Awww darn! Sleep tight. Don’t let the bed bugs bite

I woke up on the couch with Mom standing over me with a big cup of coffee. I took the cup from her and she proceeded into the kitchen. “You forgot to climb into bed last night” She stated and I sat up. “Yes I guess I must have fallen asleep on the couch” I replied taking a sip. “What are your plans for today?” Mom asked as she poured milk into two bowls and filled them up with rice crispies. “I was thinking of taking the old Jet Ski for a spin” I replied walking over to the dining room table. “That sounds like fun” Mom added as she brought the bowls over. I quickly munched my breakfast and headed to the bathroom for a quick shower. I put on my swimming trunks and headed outside.

Luckily for me the Jet Ski was in the water already and tied to the jetty. I walked over and untied the knot holding it there and climbed on. It still had a full tank of gas. I started it up and rode out into the lake. It took it easy for the first 5 minutes getting used to it again. Like it said earlier we have been neglecting to come here after Dad died. I rode past some of the other houses to see that all had occupants. Some of them had some pretty hot girls. Then one house caught my attention. It was always empty but this time I saw some lights on and a girl outside. Her back was turned to me but she had on daisy duke jeans and a white tank top. The same as Lucy. She lifted her tank top over her head and threw it on the law chair that stood on their jetty. She bent down and pulled down her jean pants. What I saw next made my heart skip a beat. I saw the same polka dot bikini that Lucy had on. The girl had the same hair as her. “Lucy is that you” I called out and the girl turned around. Sure enough it was Lucy with her blue eyes and her bright smile. “Dean! Is it really you?” She asked squinting in my direction. I rode closer and stopped at her jetty and she walked over and jumped onto the Jet Ski into my arms. I held her and she kissed me on the lips. For a moment our tongues played with each other and I held her tightly against me. All my worries about her being with another guy had disappeared that instance.

“I was wondering when Mom pulled into this lake house place” Lucy said after breaking the kiss after what seemed like ages. “When I saw your polka dot bikini I knew exactly that it was you” I said looking into her deep blue eyes. “Do you want to go for a ride” I asked and she nodded and turned around. “I want to drive” She said with a giggle so I showed her what is what and she learned very quickly and we were off at a quickened pace. She drove us in circles and made waves and started jumping them and every time we would land she would push up against my swimming trunks. Having this hot vixen in front of me was enough but when she started doing that I couldn’t take it. I held on for dear life as she kept on hopping over the waves mashing into my already hard dick. “Are those your house keys are you just happy to see me” She said looking over her shoulder with an evil grin. She didn’t wait for a reply and rode back to her jetty
We stopped at her jetty and she climbed off and I made no attempt to cover my pants. “Do you want to come over for dinner?” I asked. “I would very much like that” Lucy said waving at me then heading inside. I rode a bit more Jet Ski and decided to head on back home.

It was around 7 pm when Lucy showed up at our house and I invited her in. She looked around the room and saw only a fire burning in the fireplace. “Where is your mom” Lucy asked as she put her coat over one of the dining room chairs. “She went out to your house when she heard that you guys were also here” I said. Lucy was wearing black jean pants and a red top that shoved her cleavage nicely and a pair of flat shoes. I walked and we sat down in front of the fire on a blanket. I was dressed in shorts and a white t-shirt. There was a bowl with marshmallows and a pile of fire poker sticks lay next to it. “You first” I said as she took a marshmallow a put it on the stick and started turning it on the fire. I also took one and did the same. After about 30 seconds we took it out and I brought it up to my lips but Lucy stole my marshmallow and I grinned and I stole hers. We took more marshmallows and roasted them over the fire.
There was one marshmallow left and I put it on my stick and turned it in the fire and we both looked at it in anticipation. I held it out to my mouth but then turned to her when I saw the disappointment and before she took a bite I also took out and we kissed. Be both broke the marshmallow off and swallowed it before we kissed again. Her tongue wormed its way into my mouth and my tongue played with hers in turn. Her hands went around my neck and mine went to her hips and I slowly moved her t-shirt over her boobs. She looked down and then at me and whispered “I have waited for this for so long”. We continued kissing and my hands moved under her black lacy bra and I cupped her perfect boobs in my hands.
Her hands moved down and she lifted my shirt over my head and pushed me down onto the blanket and kissed my chest and kissed down and unbuttoned my pants. I lay there and looked into her eyes and she pulled my pants down and my boxers with it too. She took in my hard dick and licked her lips and started jacking me up and down. Her hand moved slowly at first and then faster and faster. She wrapped her mouth around my dick and I let out a sigh of pleasure and let her work her head up and down the length of my dick. She rolled her tongue around it and licked the underside. I knew I was getting close and warned her and she stopped her oral assault. She unbuttoned her pants and pulled it down revealing her lacy black panties.

She turned around and shook her ass and she pulled down her lacy panties. I was so turned on that I wanted to pull her down and do her with all I had in me. She had other ideas and she climbed onto me and straddled my legs and took my dick into her hand. She pointed it at her dripping pussy and slowly sat down onto it. “oeh that nice” She moaned and I put my hands on her hips to help her.
She slowly started rocking back and forth on my dick. “Ah yes Dean that feels so good” She moaned leaning down to bite my lip playfully. I kissed her full on and she continued rocking. I knew I was close but I wanted to cum with her so I continued to hold out. She hopped on my dick faster and faster and was moaning loud enough for the people across the lake to hear us. I felt her body start to tense up and her pussy squeezed me like a vice. I couldn’t hold out much longer and let loose a stream of cum into her pussy. She fell onto me panting like a wild animal. As we lay there in each other’s arms I stroked her back and she purred softly. “I really enjoyed that Dean” She whispered and I replied “That was amazing”. We lay there for what seemed like ages until we heard a knock at the door.

The End

Anonymous readerReport

2014-08-06 07:43:02
WVNOVc wow, awesome blog article.Really looking forward to read more. Keep writing.

anonymous readerReport

2013-04-23 03:17:55
Love your story but chapter 1 is missing, and nowhere to be found.

anonymous readerReport

2013-04-22 00:05:38
A little short but good job fixing grammar mistakes that hurt the last one

2013-04-17 02:17:42
Thanx for all the support from my readers I am really glad you are enjoying my stories.

anonymous readerReport

2013-04-15 14:50:07
Very fine story. I will watch for your writings in the future. (countrycadillac)

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Evans Farms is closing 27 under- performing locations and laying off 1,100 workers in attempt to boost profits.

Of the 27 restaurants, 21 were company-owned and closed last week. The remaining six are leased and expected to close in fiscal 2017. The company says it expects to get $20 million for the owned properties, translating to a $1 million gain in annual operating income. The New Albany, Ohio-company estimates expenses near $8 million related to the closures, most of which it expects to be non-cash charges. It expects to offer laid-off workers jobs in nearby restaurants when possible.

The company said it will update the estimated expenses related to the sale when it reports fourth-quarter results in June.

Bob Evans Farms Inc. operates more than 500 Bob Evans restaurants in 18 states across the Midwest, Mid-Atlantic and Southeast U.S. regions.

Shares in Bob Evans fell less than half of one percent in midday trading Monday, but are up more than 20 percent in 2016 and more than three percent during the past year.


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Share The Road

Past Events

Since our launch in 2008, Share the Road has coordinated events in partnership with stakeholders across Ontario. Our regional summits and community rides help to build capacity, relationships and support for cycling.

Regional Summits

Share the Road's Regional Summits bring together City Councillors and employees, tourism professionals, police, cycling advocates. bicycle retailers, area businesses and community stakeholders to learn about:

The objective of these summits is to help communities work towards a Bicycle-Friendly Community designation, or to move up to the next designation level. Recently, we've worked with partners in York Region, Quinte, Niagara, Peel Region and Kingston to host regional summits.

Community Rides

In addition to coordinating Greg's Ride each year, Share the Road staff have supported community-based rides coordinated by local partners. Past rides have included:

Share the Road is proud to support the work of local organizations as part of these community rides.

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