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Elson

cosmic.sample.cmc.elson.M_enclosed(r, gamma, rho_0)

Compute the mass enclosed in an Elson profile at radius r with slope gamma, central concentration rho_0, and assumed scale factor a = 1

In practice, this is only used to sample the positions of stars, so rho_0 is just picked to normalize the distribution (i.e. rho_0 s.t. M_enclosed(rmax) = 1)

cosmic.sample.cmc.elson.draw_r_vr_vt(N=100000, r_max=300, gamma=4)

Draw random velocities and positions from the Elson profile.

N = number of stars r_max = maximum number of virial radii for the farthest star gamma = steepness function of the profile

Note that gamma=4 is the Plummer profile

returns (vr,vt,r) in G=M_cluster=1 units

N.B. if you’re getting a “Expect x to not have duplicates” error with a large gamma, use a smaller r_max.

cosmic.sample.cmc.elson.find_rmax_vir(r_max, gamma): This is a little tricky: because the virial radius of the Elson profile depends on the maximum radius, if the profile is very flat (e.g. gamma~2) then you need a large maximum radius to get a large number of virial radius in there. This basically finds r such that r / rVir(r) = r_max. In other word, how far out do we need to go to have r_max number of virial radii in the sample.

cosmic.sample.cmc.elson.find_sigma_sqr(r, r_max_cluster, gamma): Find the 1D velocity dispersion at a given radius r using one of the spherial Jeans equations (and assuming velocity is isotropic)

cosmic.sample.cmc.elson.get_positions(N, r_max_cluster, gamma): This one’s easy: the mass enclosed function is just the CDF of the mass density, so just invert that, and you’ve got positions.

cosmic.sample.cmc.elson.get_velocities(r, r_max_cluster, gamma)

The correct way to generate velocities: generate the distribution function from rho, then use rejection sampling.

returns (vr,vt) with same length as r

cosmic.sample.cmc.elson.get_velocities_old(r, r_max_cluster, gamma)

Uses the spherical Jeans functions to sample the velocity dispersion for the cluster at different radii, then draws a random, isotropic velocity for each star

NOTE: this gives you correct velocity dispersion and produces a cluster in virial equilibrium, but it’s not strictly speaking correct (the distribution should have some kurtosis, in addition to getting the variance of the Gaussian correct). You can see the disagreement in the tail of the distributions when sampling a Plummer sphere. This is what mcluster does…

Use the new get_velocities function, which generates a distribution function directly from rho and samples velocities from it

This is kept around because it’s super useful to sample from when the rejection sampling gets too slow at the last X samples (e.g. gamma ~ 10)

returns (vr,vt) with same length as r

cosmic.sample.cmc.elson.get_velocities_plummer(r, r_max_cluster)

The correct way to generate velocities

this is a special case for gamma=4, which is the plummer sphere (and has an analytic distribution function)

returns (vr,vt) with same length as r

cosmic.sample.cmc.elson.normal(loc=0.0, scale=1.0, size=None)

Draw random samples from a normal (Gaussian) distribution.

The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2], is often called the bell curve because of its characteristic shape (see the example below).

The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2].

Note

New code should use the normal method of a default_rng() instance instead; please see the Quick Start.

Parameters

locfloat or array_like of floats: Mean (“centre”) of the distribution.
scalefloat or array_like of floats: Standard deviation (spread or “width”) of the distribution. Must be non-negative.
sizeint or tuple of ints, optional: Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.

Returns

outndarray or scalar: Drawn samples from the parameterized normal distribution.

See also

scipy.stats.norm: probability density function, distribution or cumulative density function, etc.
random.Generator.normal: which should be used for new code.

Notes

The probability density for the Gaussian distribution is

$p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },$

where $\mu$ is the mean and $\sigma$ the standard deviation. The square of the standard deviation, $\sigma^2$ , is called the variance.

The function has its peak at the mean, and its “spread” increases with the standard deviation (the function reaches 0.607 times its maximum at $x + \sigma$ and $x - \sigma$ [2]). This implies that normal is more likely to return samples lying close to the mean, rather than those far away.

References

1: Wikipedia, “Normal distribution”, https://en.wikipedia.org/wiki/Normal_distribution
2(1,2,3): P. R. Peebles Jr., “Central Limit Theorem” in “Probability, Random Variables and Random Signal Principles”, 4th ed., 2001, pp. 51, 51, 125.

Examples

(Source code, png, hires.png, pdf)

../_images/cosmic-sample-cmc-elson-1_00_00.png

cosmic.sample.cmc.elson.phi_r(r, gamma, rho_0)

Compute the gravitational potential of an Elson profile at radius r with slope gamma, central concentration rho_0, and assumed scale factor a = 1

Needed to compute the escape speed (for resampling stars)

cosmic.sample.cmc.elson.rho_r(r, gamma, rho_0): Compute the density of the Elson profile at radius r Best to use the same normalized rho_0 from M_enclosed

cosmic.sample.cmc.elson.scale_pos_and_vel(r, vr, vt)

Scale the positions and velocities to be in N-body units If we add binaries we’ll do this again in initialcmctable.py

takes r, vr, and vt as input

returns (r,vr,vt) scaled to Henon units

cosmic.sample.cmc.elson.uniform(low=0.0, high=1.0, size=None)

Draw samples from a uniform distribution.

Samples are uniformly distributed over the half-open interval [low, high) (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by uniform.

Note

New code should use the uniform method of a default_rng() instance instead; please see the Quick Start.

Parameters

lowfloat or array_like of floats, optional: Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0.
highfloat or array_like of floats: Upper boundary of the output interval. All values generated will be less than or equal to high. The high limit may be included in the returned array of floats due to floating-point rounding in the equation low + (high-low) * random_sample(). The default value is 1.0.
sizeint or tuple of ints, optional: Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if low and high are both scalars. Otherwise, np.broadcast(low, high).size samples are drawn.

Returns

outndarray or scalar: Drawn samples from the parameterized uniform distribution.