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structure

Functions for generating structure. This includes both cluster profiles and substructure.

a10(dx, dy, dz, theta, P0, c500, m500, gamma, alpha, beta, z, xyz)

gNFW pressure profile in 3d based on Arnaud2010. Compared to the function gnfw, this function fixes r1/r2/r3 to r500. This function does not include smoothing or declination stretch which should be applied at the end.

See the docstring for gnfw for more details.

Parameters:

Name Type Description Default
dx float

RA of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dy float

Dec of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dz float

Line of sight offset of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
theta float

Angle to rotate in xy-plane. Passed to utils.transform_grid.

 required
P0 float

Amplitude of the pressure profile

 required
c500 float

Concentration parameter at a density contrast of 500

 required
m500 float

Mass at a density contrast of 500

 required
gamma float

The central slope

 required
alpha float

The intermediate slope

 required
beta float

The outer slope

 required
z float

Redshift of cluster

 required
xyz tuple[Array, Array, Array, float, float]

Coordinte grid to calculate model on. See containers.Model.xyz for details.

 required

Returns:

Name Type Description
model Array

The gnfw model evaluated on the grid.
Source code in witch/structure.py 
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@jax.jit
def a10(
    dx: float,
    dy: float,
    dz: float,
    theta: float,
    P0: float,
    c500: float,
    m500: float,
    gamma: float,
    alpha: float,
    beta: float,
    z: float,
    xyz: tuple[jax.Array, jax.Array, jax.Array, float, float],
) -> jax.Array:
    r"""
    gNFW pressure profile in 3d based on [Arnaud2010](https://ui.adsabs.harvard.edu/abs/2010A%26A...517A..92A/).
    Compared to the function gnfw, this function fixes r1/r2/r3 to r500.
    This function does not include smoothing or declination stretch
    which should be applied at the end.

    See the docstring for `gnfw` for more details.

    Parameters
    ----------
    dx : float
        RA of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    theta : float
        Angle to rotate in xy-plane.
        Passed to `utils.transform_grid`.
    P0 : float
        Amplitude of the pressure profile
    c500 : float
        Concentration parameter at a density contrast of 500
    m500 : float
        Mass at a density contrast of 500
    gamma : float
        The central slope
    alpha : float
        The intermediate slope
    beta : float
        The outer slope
    z : float
        Redshift of cluster
    xyz : tuple[jax.Array, jax.Array, jax.Array, float, float]
        Coordinte grid to calculate model on.
        See `containers.Model.xyz` for details.

    Returns
    -------
    model : jax.Array
        The gnfw model evaluated on the grid.
    """

    nz = get_nz(z)
    hz = get_hz(z)
    da = get_da(z)  # TODO pass these arguments rather than recompute them everytime???

    r500 = (m500 / (4.00 * jnp.pi / 3.00) / 5.00e02 / nz) ** (1.00 / 3.00)
    r_1, r_2, r_3 = r500 / da, r500 / da, r500 / da

    x, y, z, *_ = transform_grid(dx, dy, dz, r_1, r_2, r_3, theta, xyz)

    r = c500 * jnp.sqrt(x**2 + y**2 + z**2)
    denominator = (r**gamma) * (1 + r**alpha) ** ((beta - gamma) / alpha)

    P500 = (
        1.65e-03
        * (m500 / (3.00e14 / h70)) ** (2.00 / 3.00 + ap)
        * hz ** (8.00 / 3.00)
        * h70**2
    )

    return P500 * P0 / denominator

add_exponential(pressure, xyz, dx, dy, dz, r_1, r_2, r_3, theta, amp, xk, x0, yk, y0, zk, z0)

Add ellipsoid with exponential structure to 3d pressure profile.

After transforming the grid the region where \(\sqrt{x^2 + y^2 + z^2} \leq 1\) will be multiplied by a factor of \(1 + P_{0} e^{x_k(x-x_0) + y_k(y-y_0) + z_k(z-z_0)}\) where \(P_{0}\) is amp.

Parameters:

Name Type Description Default
pressure Array

The pressure profile to modify with this ellipsoid. Should be evaluated on the same grid as xyz.

 required
xyz tuple[Array, Array, Array, float, float]

Coordinte grid to calculate model on. See containers.Model.xyz for details.

 required
dx float

RA of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dy float

Dec of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dz float

Line of sight offset of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
r_1 float

Amount to scale along x-axis. Passed to utils.transform_grid.

 required
r_2 float

Amount to scale along y-axis. Passed to utils.transform_grid.

 required
r_3 float

Amount to scale along z-axis. Passed to utils.transform_grid.

 required
theta float

Angle to rotate in xy-plane. Passed to utils.transform_grid.

 required
amp float

Factor by which pressure is enhanced at the peak of ellipsoid.

 required
xk float

Power of exponential in RA direction

 required
x0 float

RA offset of exponential. Note that this is in transformed coordinates so x0=1 is at xs + sr_1.

 required
yk float

Power of exponential in Dec direction

 required
y0 float

Dec offset of exponential. Note that this is in transformed coordinates so y0=1 is at ys + sr_2.

 required
zk float

Power of exponential along the line of sight

 required
z0 float

Line of sight offset of exponential. Note that this is in transformed coordinates so z0=1 is at zs + sr_3.

 required

Returns:

Name Type Description
new_pressure Array

Pressure profile with ellipsoid added.
Source code in witch/structure.py 
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@jax.jit
def add_exponential(
    pressure: jax.Array,
    xyz: tuple[jax.Array, jax.Array, jax.Array, float, float],
    dx: float,
    dy: float,
    dz: float,
    r_1: float,
    r_2: float,
    r_3: float,
    theta: float,
    amp: float,
    xk: float,
    x0: float,
    yk: float,
    y0: float,
    zk: float,
    z0: float,
) -> jax.Array:
    r"""
    Add ellipsoid with exponential structure to 3d pressure profile.

    After transforming the grid the region where $\sqrt{x^2 + y^2 + z^2} \leq 1$
    will be multiplied by a factor of $1 + P_{0} e^{x_k(x-x_0) + y_k(y-y_0) + z_k(z-z_0)}$
    where $P_{0}$ is `amp`.

    Parameters
    ----------
    pressure : jax.Array
        The pressure profile to modify with this ellipsoid.
        Should be evaluated on the same grid as `xyz`.
    xyz : tuple[jax.Array, jax.Array, jax.Array, float, float]
        Coordinte grid to calculate model on.
        See `containers.Model.xyz` for details.
    dx : float
        RA of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    r_1 : float
        Amount to scale along x-axis.
        Passed to `utils.transform_grid`.
    r_2 : float
        Amount to scale along y-axis.
        Passed to `utils.transform_grid`.
    r_3 : float
        Amount to scale along z-axis.
        Passed to `utils.transform_grid`.
    theta : float
        Angle to rotate in xy-plane.
        Passed to `utils.transform_grid`.
    amp : float
        Factor by which pressure is enhanced at the peak of ellipsoid.
    xk : float
        Power of exponential in RA direction
    x0 : float
        RA offset of exponential.
        Note that this is in transformed coordinates so x0=1 is at xs + sr_1.
    yk : float
        Power of exponential in Dec direction
    y0 : float
        Dec offset of exponential.
        Note that this is in transformed coordinates so y0=1 is at ys + sr_2.
    zk : float
        Power of exponential along the line of sight
    z0 : float
        Line of sight offset of exponential.
        Note that this is in transformed coordinates so z0=1 is at zs + sr_3.

    Returns
    -------
    new_pressure : Array
        Pressure profile with ellipsoid added.
    """
    x, y, z, *_ = transform_grid(dx, dy, dz, r_1, r_2, r_3, theta, xyz)

    exponential = amp * jnp.exp(((x - x0) * xk) + ((y - y0) * yk) + ((z - z0) * zk))

    new_pressure = jnp.where(
        jnp.sqrt(x**2 + y**2 + z**2) > 1, pressure, (1 + exponential) * pressure
    )
    return new_pressure

add_powerlaw(pressure, xyz, dx, dy, dz, r_1, r_2, r_3, theta, amp, phi0, k_r, k_phi)

Add ellipsoid with power law structure to 3d pressure profile.

After transforming the grid the region where \(\sqrt{x^2 + y^2 + z^2} \leq 1\) will be multiplied by a factor of \(1 + P_{0}(1 - {1 + r}^{-k_r})(1 - {1 + \phi}^{-k_{\phi}})\). Where \(r\) and \(\phi\) are the usual polar coordinates and \(P_{0}\) is amp.

Parameters:

Name Type Description Default
pressure Array

The pressure profile to modify with this ellipsoid. Should be evaluated on the same grid as xyz.

 required
xyz tuple[Array, Array, Array, float, float]

Coordinte grid to calculate model on. See containers.Model.xyz for details.

 required
dx float

RA of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dy float

Dec of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dz float

Line of sight offset of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
r_1 float

Amount to scale along x-axis. Passed to utils.transform_grid.

 required
r_2 float

Amount to scale along y-axis. Passed to utils.transform_grid.

 required
r_3 float

Amount to scale along z-axis. Passed to utils.transform_grid.

 required
theta float

Angle to rotate in xy-plane. Passed to utils.transform_grid.

 required
amp float

Factor by which pressure is enhanced within the ellipsoid.

 required
phi0 float

Polar angle of nose of power law.

 required
k_r float

Slope of power law in radial direction.

 required
k_phi float

Slope of power law in polar direction.

 required

Returns:

Name Type Description
new_pressure Array

Pressure profile with ellipsoid added.
Source code in witch/structure.py 
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@jax.jit
def add_powerlaw(
    pressure: jax.Array,
    xyz: tuple[jax.Array, jax.Array, jax.Array, float, float],
    dx: float,
    dy: float,
    dz: float,
    r_1: float,
    r_2: float,
    r_3: float,
    theta: float,
    amp: float,
    phi0: float,
    k_r: float,
    k_phi: float,
) -> jax.Array:
    r"""
    Add ellipsoid with power law structure to 3d pressure profile.

    After transforming the grid the region where $\sqrt{x^2 + y^2 + z^2} \leq 1$
    will be multiplied by a factor of $1 + P_{0}(1 - {1 + r}^{-k_r})(1 - {1 + \phi}^{-k_{\phi}})$.
    Where $r$ and $\phi$ are the usual polar coordinates and $P_{0}$ is `amp`.

    Parameters
    ----------
    pressure : jax.Array
        The pressure profile to modify with this ellipsoid.
        Should be evaluated on the same grid as `xyz`.
    xyz : tuple[jax.Array, jax.Array, jax.Array, float, float]
        Coordinte grid to calculate model on.
        See `containers.Model.xyz` for details.
    dx : float
        RA of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    r_1 : float
        Amount to scale along x-axis.
        Passed to `utils.transform_grid`.
    r_2 : float
        Amount to scale along y-axis.
        Passed to `utils.transform_grid`.
    r_3 : float
        Amount to scale along z-axis.
        Passed to `utils.transform_grid`.
    theta : float
        Angle to rotate in xy-plane.
        Passed to `utils.transform_grid`.
    amp : float
        Factor by which pressure is enhanced within the ellipsoid.
    phi0 : float
        Polar angle of nose of power law.
    k_r : float
        Slope of power law in radial direction.
    k_phi : float
        Slope of power law in polar direction.

    Returns
    -------
    new_pressure : Array
        Pressure profile with ellipsoid added.
    """
    x, y, z, *_ = transform_grid(dx, dy, dz, r_1, r_2, r_3, theta, xyz)
    r = jnp.sqrt(x**2 + y**2 + z**2)
    phi = abs((jnp.arctan2(y, x) - phi0) % (2 * jnp.pi) - jnp.pi) / jnp.pi

    powerlaw = (
        amp
        * (1 - jnp.float_power(1 + r, -1.0 * k_r))
        * (1 - jnp.float_power(1 + phi, -1 * k_phi))
    )
    new_pressure = jnp.where(r > 1, pressure, (1 + powerlaw) * pressure)
    return new_pressure

add_powerlaw_cos(pressure, xyz, dx, dy, dz, r_1, r_2, r_3, theta, amp, phi0, k_r, omega)

Add ellipsoid with radial power law and angular cosine dependant structure to 3d pressure profile.

After transforming the grid the region where \(\sqrt{x^2 + y^2 + z^2} \leq 1\) will be multiplied by a factor of \(1 + P_{0} ({1 + r}^{-k_r}) \left| cos(\omega\phi) \right|\). Where \(r\) and \(\phi\) are the usual polar coordinates and \(P_{0}\) is amp.

Parameters:

Name Type Description Default
pressure Array

The pressure profile to modify with this ellipsoid. Should be evaluated on the same grid as xyz.

 required
xyz tuple[Array, Array, Array, float, float]

Coordinte grid to calculate model on. See containers.Model.xyz for details.

 required
dx float

RA of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dy float

Dec of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dz float

Line of sight offset of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
r_1 float

Amount to scale along x-axis. Passed to utils.transform_grid.

 required
r_2 float

Amount to scale along y-axis. Passed to utils.transform_grid.

 required
r_3 float

Amount to scale along z-axis. Passed to utils.transform_grid.

 required
theta float

Angle to rotate in xy-plane. Passed to utils.transform_grid.

 required
amp float

Factor by which pressure is enhanced within the ellipsoid.

 required
phi0 float

Polar angle of nose of power law.

 required
k_r float

Slope of power law in radial direction.

 required
omega float

Angular freqency of the cosine term.

 required

Returns:

Name Type Description
new_pressure Array

Pressure profile with ellipsoid added.
Source code in witch/structure.py 
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@jax.jit
def add_powerlaw_cos(
    pressure: jax.Array,
    xyz: tuple[jax.Array, jax.Array, jax.Array, float, float],
    dx: float,
    dy: float,
    dz: float,
    r_1: float,
    r_2: float,
    r_3: float,
    theta: float,
    amp: float,
    phi0: float,
    k_r: float,
    omega: float,
) -> jax.Array:
    r"""
    Add ellipsoid with radial power law and angular cosine dependant structure to 3d pressure profile.

    After transforming the grid the region where $\sqrt{x^2 + y^2 + z^2} \leq 1$
    will be multiplied by a factor of $1 + P_{0} ({1 + r}^{-k_r}) \left| cos(\omega\phi) \right|$.
    Where $r$ and $\phi$ are the usual polar coordinates and $P_{0}$ is `amp`.

    Parameters
    ----------
    pressure : jax.Array
        The pressure profile to modify with this ellipsoid.
        Should be evaluated on the same grid as `xyz`.
    xyz : tuple[jax.Array, jax.Array, jax.Array, float, float]
        Coordinte grid to calculate model on.
        See `containers.Model.xyz` for details.
    dx : float
        RA of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    r_1 : float
        Amount to scale along x-axis.
        Passed to `utils.transform_grid`.
    r_2 : float
        Amount to scale along y-axis.
        Passed to `utils.transform_grid`.
    r_3 : float
        Amount to scale along z-axis.
        Passed to `utils.transform_grid`.
    theta : float
        Angle to rotate in xy-plane.
        Passed to `utils.transform_grid`.
    amp : float
        Factor by which pressure is enhanced within the ellipsoid.
    phi0 : float
        Polar angle of nose of power law.
    k_r : float
        Slope of power law in radial direction.
    omega : float
        Angular freqency of the cosine term.

    Returns
    -------
    new_pressure : Array
        Pressure profile with ellipsoid added.
    """
    x, y, z, *_ = transform_grid(dx, dy, dz, r_1, r_2, r_3, theta, xyz)
    r = jnp.sqrt(x**2 + y**2 + z**2)
    phi = (jnp.arctan2(y, x) - phi0) % (2 * jnp.pi)

    powerlaw = amp * jnp.abs(jnp.cos(omega * phi)) * jnp.float_power(r, k_r)
    new_pressure = jnp.where(r > 1, pressure, (1 + powerlaw) * pressure)
    return new_pressure

add_uniform(pressure, xyz, dx, dy, dz, r_1, r_2, r_3, theta, amp)

Add ellipsoid with uniform structure to 3d pressure profile.

After transforming the grid the region where \(\sqrt{x^2 + y^2 + z^2} \leq 1\) will be multiplied by a factor of \(1 + P_{0}\) where \(P_{0}\) is amp.

Parameters:

Name Type Description Default
pressure Array

The pressure profile to modify with this ellipsoid. Should be evaluated on the same grid as xyz.

 required
xyz tuple[Array, Array, Array, float, float]

Coordinte grid to calculate model on. See containers.Model.xyz for details.

 required
dx float

RA of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dy float

Dec of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dz float

Line of sight offset of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
r_1 float

Amount to scale along x-axis. Passed to utils.transform_grid.

 required
r_2 float

Amount to scale along y-axis. Passed to utils.transform_grid.

 required
r_3 float

Amount to scale along z-axis. Passed to utils.transform_grid.

 required
theta float

Angle to rotate in xy-plane. Passed to utils.transform_grid.

 required
amp float

Factor by which pressure is enhanced within the ellipsoid.

 required

Returns:

Name Type Description
new_pressure Array

Pressure profile with ellipsoid added.
Source code in witch/structure.py 
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@jax.jit
def add_uniform(
    pressure: jax.Array,
    xyz: tuple[jax.Array, jax.Array, jax.Array, float, float],
    dx: float,
    dy: float,
    dz: float,
    r_1: float,
    r_2: float,
    r_3: float,
    theta: float,
    amp: float,
) -> jax.Array:
    r"""
    Add ellipsoid with uniform structure to 3d pressure profile.

    After transforming the grid the region where $\sqrt{x^2 + y^2 + z^2} \leq 1$
    will be multiplied by a factor of $1 + P_{0}$ where $P_{0}$ is `amp`.

    Parameters
    ----------
    pressure : jax.Array
        The pressure profile to modify with this ellipsoid.
        Should be evaluated on the same grid as `xyz`.
    xyz : tuple[jax.Array, jax.Array, jax.Array, float, float]
        Coordinte grid to calculate model on.
        See `containers.Model.xyz` for details.
    dx : float
        RA of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    r_1 : float
        Amount to scale along x-axis.
        Passed to `utils.transform_grid`.
    r_2 : float
        Amount to scale along y-axis.
        Passed to `utils.transform_grid`.
    r_3 : float
        Amount to scale along z-axis.
        Passed to `utils.transform_grid`.
    theta : float
        Angle to rotate in xy-plane.
        Passed to `utils.transform_grid`.
    amp : float
        Factor by which pressure is enhanced within the ellipsoid.

    Returns
    -------
    new_pressure : Array
        Pressure profile with ellipsoid added.
    """
    x, y, z, *_ = transform_grid(dx, dy, dz, r_1, r_2, r_3, theta, xyz)

    new_pressure = jnp.where(
        jnp.sqrt(x**2 + y**2 + z**2) > 1, pressure, (1 + amp) * pressure
    )
    return new_pressure

cylindrical_beta(dx, dy, dz, L, theta, P0, r_c, beta, xyz)

This function does not include smoothing or declination stretch which should be applied at the end.

Once the grid is transformed the profile is computed as:

\[ P_{0}\left( 1 + \frac{y^2 + z^2}{{r_c}^2} \right)^{-1.5\beta} \]

Parameters:

Name Type Description Default
dx float

RA of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dy float

Dec of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dz float

Line of sight offset of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
L float

Length of the cylinder. Aligned with the x-axis. Note that we consider anything where \(\left| x \right| \\leq L\) to be in the profile, so the actual length is \(2L\).

 required
theta float

Angle to rotate in xy-plane. Passed to utils.transform_grid.

 required
P0 float

Amplitude of the pressure profile.

 required
r_c float

The critical radius of the cylindrical profile.

 required
beta float

Beta value of isobeta model.

 required
xyz tuple[Array, Array, Array, float, float]

Coordinte grid to calculate model on. See containers.Model.xyz for details.

 required

Returns:

Name Type Description
model Array

The cylindrical beta model evaluated on the grid.
Source code in witch/structure.py 
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@jax.jit
def cylindrical_beta(
    dx: float,
    dy: float,
    dz: float,
    L: float,
    theta: float,
    P0: float,
    r_c: float,
    beta: float,
    xyz: tuple[jax.Array, jax.Array, jax.Array, float, float],
) -> jax.Array:
    r"""

    This function does not include smoothing or declination stretch
    which should be applied at the end.

    Once the grid is transformed the profile is computed as:

    $$
    P_{0}\left( 1 + \frac{y^2 + z^2}{{r_c}^2} \right)^{-1.5\beta}
    $$

    Parameters
    ----------
    dx : float
        RA of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    L : float
        Length of the cylinder.
        Aligned with the x-axis.
        Note that we consider anything where $\left| x \right| \\leq L$
        to be in the profile, so the actual length is $2L$.
    theta : float
        Angle to rotate in xy-plane.
        Passed to `utils.transform_grid`.
    P0 : float
        Amplitude of the pressure profile.
    r_c : float
        The critical radius of the cylindrical profile.
    beta : float
        Beta value of isobeta model.
    xyz : tuple[jax.Array, jax.Array, jax.Array, float, float]
        Coordinte grid to calculate model on.
        See `containers.Model.xyz` for details.

    Returns
    -------
    model : jax.Array
        The cylindrical beta model evaluated on the grid.
    """
    x, y, z, *_ = transform_grid(dx, dy, dz, 1.0, 1.0, 1.0, theta, xyz)
    r = jnp.sqrt(y**2 + z**2)
    powerlaw = P0 / (1.0 + (r / r_c) ** 2) ** (3.0 / 2.0 * beta)

    pressure = jnp.where(jnp.abs(x) >= L / 2.0, 0, powerlaw)

    return pressure

ea10(dx, dy, dz, r_1, r_2, r_3, theta, P0, c500, m500, gamma, alpha, beta, z, xyz)

Eliptical gNFW pressure profile in 3d based on Arnaud2010. r_ell is computed in the usual way for an a10 profile, then the axes are scaled according to r_1, r_2, r_3, with a normalization applied. This function does not include smoothing or declination stretch which should be applied at the end.

See the docstring for gnfw for more details.

Parameters:

Name Type Description Default
dx float

RA of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dy float

Dec of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dz float

Line of sight offset of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
r_1 float

Amount to scale along x-axis. Units are arbitrary, only ratio of r_1/r_2, r_1/r_3, r_2/r_3 matters Passed to utils.transform_grid.

 required
r_2 float

Amount to scale along y-axis. Passed to utils.transform_grid.

 required
r_3 float

Amount to scale along z-axis. Passed to utils.transform_grid.

 required
theta float

Angle to rotate in xy-plane. Passed to utils.transform_grid.

 required
P0 float

Amplitude of the pressure profile

 required
c500 float

Concentration parameter at a density contrast of 500

 required
m500 float

Mass at a density contrast of 500

 required
gamma float

The central slope

 required
alpha float

The intermediate slope

 required
beta float

The outer slope

 required
z float

Redshift of cluster

 required
xyz tuple[Array, Array, Array, float, float]

Coordinte grid to calculate model on. See containers.Model.xyz for details.

 required

Returns:

Name Type Description
model Array

The gnfw model evaluated on the grid.
Source code in witch/structure.py 
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@jax.jit
def ea10(
    dx: float,
    dy: float,
    dz: float,
    r_1: float,
    r_2: float,
    r_3: float,
    theta: float,
    P0: float,
    c500: float,
    m500: float,
    gamma: float,
    alpha: float,
    beta: float,
    z: float,
    xyz: tuple[jax.Array, jax.Array, jax.Array, float, float],
) -> jax.Array:
    r"""
    Eliptical gNFW pressure profile in 3d based on Arnaud2010.
    r_ell is computed in the usual way for an a10 profile, then the axes are
    scaled according to r_1, r_2, r_3, with a normalization applied.
    This function does not include smoothing or declination stretch
    which should be applied at the end.

    See the docstring for `gnfw` for more details.

    Parameters
    ----------
    dx : float
        RA of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    r_1 : float
        Amount to scale along x-axis.
        Units are arbitrary, only ratio of r_1/r_2, r_1/r_3, r_2/r_3 matters
        Passed to `utils.transform_grid`.
    r_2 : float
        Amount to scale along y-axis.
        Passed to `utils.transform_grid`.
    r_3 : float
        Amount to scale along z-axis.
        Passed to `utils.transform_grid`.
    theta : float
        Angle to rotate in xy-plane.
        Passed to `utils.transform_grid`.
    P0 : float
        Amplitude of the pressure profile
    c500 : float
        Concentration parameter at a density contrast of 500
    m500 : float
        Mass at a density contrast of 500
    gamma : float
        The central slope
    alpha : float
        The intermediate slope
    beta : float
        The outer slope
    z : float
        Redshift of cluster
    xyz : tuple[jax.Array, jax.Array, jax.Array, float, float]
        Coordinte grid to calculate model on.
        See `containers.Model.xyz` for details.

    Returns
    -------
    model : jax.Array
        The gnfw model evaluated on the grid.
    """
    nz = get_nz(z)
    hz = get_hz(z)
    da = get_da(z)  # TODO pass these arguments rather than recompute them everytime???

    r500 = (m500 / (4.00 * jnp.pi / 3.00) / 5.00e02 / nz) ** (1.00 / 3.00)
    r_ell = r500 / da
    r_norm = (r_1 * r_2 * r_3) ** (1 / 3)

    r_1 *= r_ell / r_norm
    r_2 *= r_ell / r_norm
    r_3 *= r_ell / r_norm

    x, y, z, *_ = transform_grid(dx, dy, dz, r_1, r_2, r_3, theta, xyz)

    r = c500 * jnp.sqrt(x**2 + y**2 + z**2)
    denominator = (r**gamma) * (1 + r**alpha) ** ((beta - gamma) / alpha)

    P500 = (
        1.65e-03
        * (m500 / (3.00e14 / h70)) ** (2.00 / 3.00 + ap)
        * hz ** (8.00 / 3.00)
        * h70**2
    )

    return P500 * P0 / denominator

egaussian(dx, dy, dz, r_1, r_2, r_3, theta, sigma, amp, xyz)

Elliptical gaussian profile in 3d. This function does not include smoothing or declination stretch which should be applied at the end.

Once the grid is transformed the profile is computed as:

\[ P_{0} e^{-\frac{x^2 + y^2 + z^2}{2\sigma^2}} \]

where \(P_{0}\) is amp.

Parameters:

Name Type Description Default
dx float

RA of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dy float

Dec of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dz float

Line of sight offset of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
r_1 float

Amount to scale along x-axis. Passed to utils.transform_grid.

 required
r_2 float

Amount to scale along y-axis. Passed to utils.transform_grid.

 required
r_3 float

Amount to scale along z-axis. Passed to utils.transform_grid.

 required
theta float

Angle to rotate in xy-plane. Passed to utils.transform_grid.

 required
sigma float

Sigma value of gaussian model.

 required
amp float

Amplitude of gaussian model.

 required
xyz tuple[Array, Array, Array, float, float]

Coordinte grid to calculate model on. See containers.Model.xyz for details.

 required

Returns:

Name Type Description
model Array

The gaussain model evaluated on the grid.
Source code in witch/structure.py 
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@jax.jit
def egaussian(
    dx: float,
    dy: float,
    dz: float,
    r_1: float,
    r_2: float,
    r_3: float,
    theta: float,
    sigma: float,
    amp: float,
    xyz: tuple[jax.Array, jax.Array, jax.Array, float, float],
) -> jax.Array:
    r"""
    Elliptical gaussian profile in 3d.
    This function does not include smoothing or declination stretch
    which should be applied at the end.

    Once the grid is transformed the profile is computed as:

    $$
    P_{0} e^{-\frac{x^2 + y^2 + z^2}{2\sigma^2}}
    $$

    where $P_{0}$ is `amp`.

    Parameters
    ----------
    dx : float
        RA of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    r_1 : float
        Amount to scale along x-axis.
        Passed to `utils.transform_grid`.
    r_2 : float
        Amount to scale along y-axis.
        Passed to `utils.transform_grid`.
    r_3 : float
        Amount to scale along z-axis.
        Passed to `utils.transform_grid`.
    theta : float
        Angle to rotate in xy-plane.
        Passed to `utils.transform_grid`.
    sigma : float
        Sigma value of gaussian model.
    amp : float
        Amplitude of gaussian model.
    xyz : tuple[jax.Array, jax.Array, jax.Array, float, float]
        Coordinte grid to calculate model on.
        See `containers.Model.xyz` for details.

    Returns
    -------
    model : jax.Array
        The gaussain model evaluated on the grid.
    """
    x, y, z, *_ = transform_grid(dx, dy, dz, r_1, r_2, r_3, theta, xyz)

    rr = x**2 + y**2 + z**2
    power = -1 * rr / (2 * sigma**2)

    return amp * jnp.exp(power)

gaussian(dx, dy, sigma, amp, xyz)

Standard gaussian profile in 2d. This function does not include smoothing or declination stretch which should be applied at the end. The transform_grid call is awkward and can probably be removed/worked around. Function exists to match existing guassian interfaces.

Once the grid is transformed the profile is computed as:

\[ P_{0} e^{-\frac{x^2 + y^2}{2\sigma^2}} \]

where \(P_{0}\) is amp.

Parameters:

Name Type Description Default
dx float

RA of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dy float

Dec of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
sigma float

Sigma value of gaussian model.

 required
amp float

Amplitude of gaussian model.

 required
xyz tuple[Array, Array, Array, float, float]

Coordinte grid to calculate model on. We only care about x and y here. See containers.Model.xyz for details.

 required

Returns:

Name Type Description
model Array

The gaussian model evaluated on only the 2d xy grid.
Source code in witch/structure.py 
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@jax.jit
def gaussian(
    dx: float,
    dy: float,
    sigma: float,
    amp: float,
    xyz: tuple[jax.Array, jax.Array, jax.Array, float, float],
) -> jax.Array:
    r"""
    Standard gaussian profile in 2d.
    This function does not include smoothing or declination stretch
    which should be applied at the end. The transform_grid call is
    awkward and can probably be removed/worked around. Function exists
    to match existing guassian interfaces.

    Once the grid is transformed the profile is computed as:

    $$
    P_{0} e^{-\frac{x^2 + y^2}{2\sigma^2}}
    $$

    where $P_{0}$ is `amp`.

    Parameters
    ----------
    dx : float
        RA of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    sigma : float
        Sigma value of gaussian model.
    amp : float
        Amplitude of gaussian model.
    xyz : tuple[jax.Array, jax.Array, jax.Array, float, float]
        Coordinte grid to calculate model on.
        We only care about x and y here.
        See `containers.Model.xyz` for details.

    Returns
    -------
    model : jax.Array
        The gaussian model evaluated on only the 2d xy grid.
    """
    x, y, *_ = transform_grid(dx, dy, 0, 1, 1, 1, 0, xyz)
    rr = x[..., 0] ** 2 + y[..., 0] ** 2
    power = -1 * rr / (2 * sigma**2)

    return amp * jnp.exp(power)

gnfw(dx, dy, dz, r_1, r_2, r_3, theta, P0, c500, m500, gamma, alpha, beta, z, xyz)

Elliptical gNFW pressure profile in 3d. This function does not include smoothing or declination stretch which should be applied at the end. TODO: Add units to add parameters!

Once the grid is transformed the profile is computed as:

\[ \dfrac{P_{500} * P_{0}}{{\left( r^{\gamma}\left( 1 + r^{\alpha} \right) \right)}^{\dfrac{\beta - \gamma}{\alpha}}} \]

where:

\[ r = c_{500} \sqrt{x^2 + y^2 + z^2} {\frac{3m_{500}}{2000 \pi n_z}}^{-\frac{1}{3}} \]
\[ P_{500} = 1.65 \times 10^{-3} {\frac{m_{500}*h_{70}}{3 \times 10^{14}}}^{\frac{2}{3} + ap}{h_z}^{\frac{8}{3}}{h_{70}}^2 \]

\(n_z\) is the critical density at the cluster redshift and \(h_z\) is the Hubble constant at the cluster redshift.

Parameters:

Name Type Description Default
dx float

RA of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dy float

Dec of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dz float

Line of sight offset of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
r_1 float

Amount to scale along x-axis. Passed to utils.transform_grid.

 required
r_2 float

Amount to scale along y-axis. Passed to utils.transform_grid.

 required
r_3 float

Amount to scale along z-axis. Passed to utils.transform_grid.

 required
theta float

Angle to rotate in xy-plane. Passed to utils.transform_grid.

 required
P0 float

Amplitude of the pressure profile.

 required
c500 float

Concentration parameter at a density contrast of 500.

 required
m500 float

Mass at a density contrast of 500.

 required
gamma float

The central slope.

 required
alpha float

The intermediate slope.

 required
beta float

The outer slope.

 required
z float

Redshift of cluster.

 required
xyz tuple[Array, Array, Array, float, float]

Coordinte grid to calculate model on. See containers.Model.xyz for details.

 required

Returns:

Name Type Description
model Array

The gnfw model evaluated on the grid.
Source code in witch/structure.py 
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@jax.jit
def gnfw(
    dx: float,
    dy: float,
    dz: float,
    r_1: float,
    r_2: float,
    r_3: float,
    theta: float,
    P0: float,
    c500: float,
    m500: float,
    gamma: float,
    alpha: float,
    beta: float,
    z: float,
    xyz: tuple[jax.Array, jax.Array, jax.Array, float, float],
) -> jax.Array:
    r"""
    Elliptical gNFW pressure profile in 3d.
    This function does not include smoothing or declination stretch
    which should be applied at the end.
    TODO: Add units to add parameters!

    Once the grid is transformed the profile is computed as:

    $$
    \dfrac{P_{500} * P_{0}}{{\left( r^{\gamma}\left( 1 + r^{\alpha} \right) \right)}^{\dfrac{\beta - \gamma}{\alpha}}}
    $$

    where:

    $$
    r = c_{500} \sqrt{x^2 + y^2 + z^2} {\frac{3m_{500}}{2000 \pi n_z}}^{-\frac{1}{3}}
    $$

    $$
    P_{500} = 1.65 \times 10^{-3} {\frac{m_{500}*h_{70}}{3 \times 10^{14}}}^{\frac{2}{3} + ap}{h_z}^{\frac{8}{3}}{h_{70}}^2
    $$

    $n_z$ is the critical density at the cluster redshift and $h_z$ is the Hubble constant at the cluster redshift.

    Parameters
    ----------
    dx : float
        RA of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    r_1 : float
        Amount to scale along x-axis.
        Passed to `utils.transform_grid`.
    r_2 : float
        Amount to scale along y-axis.
        Passed to `utils.transform_grid`.
    r_3 : float
        Amount to scale along z-axis.
        Passed to `utils.transform_grid`.
    theta : float
        Angle to rotate in xy-plane.
        Passed to `utils.transform_grid`.
    P0 : float
        Amplitude of the pressure profile.
    c500 : float
        Concentration parameter at a density contrast of 500.
    m500 : float
        Mass at a density contrast of 500.
    gamma : float
        The central slope.
    alpha : float
        The intermediate slope.
    beta : float
        The outer slope.
    z : float
        Redshift of cluster.
    xyz : tuple[jax.Array, jax.Array, jax.Array, float, float]
        Coordinte grid to calculate model on.
        See `containers.Model.xyz` for details.

    Returns
    -------
    model : jax.Array
        The gnfw model evaluated on the grid.
    """
    nz = get_nz(z)
    hz = get_hz(z)

    x, y, z, *_ = transform_grid(dx, dy, dz, r_1, r_2, r_3, theta, xyz)

    r500 = (m500 / (4.00 * jnp.pi / 3.00) / 5.00e02 / nz) ** (1.00 / 3.00)

    r = c500 * jnp.sqrt(x**2 + y**2 + z**2) / r500
    denominator = (r**gamma) * (1 + r**alpha) ** ((beta - gamma) / alpha)

    P500 = (
        1.65e-03
        * (m500 / (3.00e14 / h70)) ** (2.00 / 3.00 + ap)
        * hz ** (8.00 / 3.00)
        * h70**2
    )

    return P500 * P0 / denominator

isobeta(dx, dy, dz, r_1, r_2, r_3, theta, beta, amp, xyz)

Elliptical isobeta pressure profile in 3d. This function does not include smoothing or declination stretch which should be applied at the end.

Once the grid is transformed the profile is computed as:

\[ P_{0}\left( 1 + x**2 + y**2 + z**2 \right)^{-1.5\beta} \]

where \(P_{0}\) is amp.

Parameters:

Name Type Description Default
dx float

RA of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dy float

Dec of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
dz float

Line of sight offset of cluster center relative to grid origin. Passed to utils.transform_grid.

 required
r_1 float

Amount to scale along x-axis. Passed to utils.transform_grid.

 required
r_2 float

Amount to scale along y-axis. Passed to utils.transform_grid.

 required
r_3 float

Amount to scale along z-axis. Passed to utils.transform_grid.

 required
theta float

Angle to rotate in xy-plane. Passed to utils.transform_grid.

 required
beta float

Beta value of isobeta model.

 required
amp float

Amplitude of isobeta model.

 required
xyz tuple[Array, Array, Array, float, float]

Coordinte grid to calculate model on. See containers.Model.xyz for details.

 required

Returns:

Name Type Description
model Array

The jax.isobeta model evaluated on the grid.
Source code in witch/structure.py 
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@jax.jit
def isobeta(
    dx: float,
    dy: float,
    dz: float,
    r_1: float,
    r_2: float,
    r_3: float,
    theta: float,
    beta: float,
    amp: float,
    xyz: tuple[jax.Array, jax.Array, jax.Array, float, float],
) -> jax.Array:
    r"""
    Elliptical isobeta pressure profile in 3d.
    This function does not include smoothing or declination stretch
    which should be applied at the end.

    Once the grid is transformed the profile is computed as:

    $$
    P_{0}\left( 1 + x**2 + y**2 + z**2 \right)^{-1.5\beta}
    $$

    where $P_{0}$ is `amp`.

    Parameters
    ----------
    dx : float
        RA of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `utils.transform_grid`.
    r_1 : float
        Amount to scale along x-axis.
        Passed to `utils.transform_grid`.
    r_2 : float
        Amount to scale along y-axis.
        Passed to `utils.transform_grid`.
    r_3 : float
        Amount to scale along z-axis.
        Passed to `utils.transform_grid`.
    theta : float
        Angle to rotate in xy-plane.
        Passed to `utils.transform_grid`.
    beta : float
        Beta value of isobeta model.
    amp : float
        Amplitude of isobeta model.
    xyz : tuple[jax.Array, jax.Array, jax.Array, float, float]
        Coordinte grid to calculate model on.
        See `containers.Model.xyz` for details.

    Returns
    -------
    model : Array
        The jax.isobeta model evaluated on the grid.
    """
    x, y, z, *_ = transform_grid(dx, dy, dz, r_1, r_2, r_3, theta, xyz)

    rr = 1 + x**2 + y**2 + z**2
    power = -1.5 * beta
    rrpow = rr**power

    return amp * rrpow