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 grid.transform_grid. Units: arcsec	required
dy	float	Dec of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
dz	float	Line of sight offset of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
theta	float	Angle to rotate in xy-plane. Passed to grid.transform_grid. Units: radians	required
P0	float	Amplitude of the pressure profile Units: unitless	required
c500	float	Concentration parameter at a density contrast of 500 Units: unitless	required
m500	float	Mass at a density contrast of 500 Units: M_solar	required
gamma	float	The central slope Units: unitless	required
alpha	float	The intermediate slope Units: unitless	required
beta	float	The outer slope Units: unitless	required
z	float	Redshift of cluster Units: redshift	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

@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 `grid.transform_grid`.
        Units: arcsec
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    theta : float
        Angle to rotate in xy-plane.
        Passed to `grid.transform_grid`.
        Units: radians
    P0 : float
        Amplitude of the pressure profile
        Units: unitless
    c500 : float
        Concentration parameter at a density contrast of 500
        Units: unitless
    m500 : float
        Mass at a density contrast of 500
        Units: M_solar
    gamma : float
        The central slope
        Units: unitless
    alpha : float
        The intermediate slope
        Units: unitless
    beta : float
        The outer slope
        Units: unitless
    z : float
        Redshift of cluster
        Units: redshift
    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 grid.transform_grid. Units: arcsec	required
dy	float	Dec of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
dz	float	Line of sight offset of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
r_1	float	Amount to scale along x-axis. The absolute value of these is degenerate with sigma. Units are arbitrary, only ratio of r_1/r_2, r_1/r_3, r_2/r_3 matters Passed to grid.transform_grid. Units: arbitrary	required
r_2	float	Amount to scale along y-axis. Passed to grid.transform_grid. Units: arbitrary	required
r_3	float	Amount to scale along z-axis. Passed to grid.transform_grid. Units: arbitrary	required
theta	float	Angle to rotate in xy-plane. Passed to grid.transform_grid. Units: radians	required
amp	float	Factor by which pressure is enhanced at the peak of ellipsoid. Units: unitless	required
xk	float	Power of exponential in RA direction Units: unitless	required
x0	float	RA offset of exponential. Note that this is in transformed coordinates so x0=1 is at xs + sr_1. Units: arcsec	required
yk	float	Power of exponential in Dec direction Units: unitless	required
y0	float	Dec offset of exponential. Note that this is in transformed coordinates so y0=1 is at ys + sr_2. Units: arcsec	required
zk	float	Power of exponential along the line of sight Units: unitless	required
z0	float	Line of sight offset of exponential. Note that this is in transformed coordinates so z0=1 is at zs + sr_3. Units: arcsec	required

Returns:

Name	Type	Description
new_pressure	Array	Pressure profile with ellipsoid added.

Source code in witch/structure.py

@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 `grid.transform_grid`.
        Units: arcsec
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    r_1 : float
        Amount to scale along x-axis. The absolute value of these is degenerate with sigma.
        Units are arbitrary, only ratio of r_1/r_2, r_1/r_3, r_2/r_3 matters
        Passed to `grid.transform_grid`.
        Units: arbitrary
    r_2 : float
        Amount to scale along y-axis.
        Passed to `grid.transform_grid`.
        Units: arbitrary
    r_3 : float
        Amount to scale along z-axis.
        Passed to `grid.transform_grid`.
        Units: arbitrary
    theta : float
        Angle to rotate in xy-plane.
        Passed to `grid.transform_grid`.
        Units: radians
    amp : float
        Factor by which pressure is enhanced at the peak of ellipsoid.
        Units: unitless
    xk : float
        Power of exponential in RA direction
        Units: unitless
    x0 : float
        RA offset of exponential.
        Note that this is in transformed coordinates so x0=1 is at xs + sr_1.
        Units: arcsec
    yk : float
        Power of exponential in Dec direction
        Units: unitless
    y0 : float
        Dec offset of exponential.
        Note that this is in transformed coordinates so y0=1 is at ys + sr_2.
        Units: arcsec
    zk : float
        Power of exponential along the line of sight
        Units: unitless
    z0 : float
        Line of sight offset of exponential.
        Note that this is in transformed coordinates so z0=1 is at zs + sr_3.
        Units: arcsec

    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 grid.transform_grid. Units: arcsec	required
dy	float	Dec of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
dz	float	Line of sight offset of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
r_1	float	Amount to scale along x-axis. The absolute value of these is degenerate with sigma. Units are arbitrary, only ratio of r_1/r_2, r_1/r_3, r_2/r_3 matters Passed to grid.transform_grid. Units: arbitrary	required
r_2	float	Amount to scale along y-axis. Passed to grid.transform_grid. Units: arbitrary	required
r_3	float	Amount to scale along z-axis. Passed to grid.transform_grid. Units: arbitrary	required
theta	float	Angle to rotate in xy-plane. Passed to grid.transform_grid. Units: radians	required
amp	float	Factor by which pressure is enhanced within the ellipsoid. Units: unitless	required
phi0	float	Polar angle of nose of power law. This is CCW from the x-axis, after the grid rotation. See arctan2 documentation Units: radians	required
k_r	float	Slope of power law in radial direction. Units: unitless	required
k_phi	float	Slope of power law in polar direction. Units: unitless	required

Returns:

Name	Type	Description
new_pressure	Array	Pressure profile with ellipsoid added.

Source code in witch/structure.py

@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 `grid.transform_grid`.
        Units: arcsec
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    r_1 : float
        Amount to scale along x-axis. The absolute value of these is degenerate with sigma.
        Units are arbitrary, only ratio of r_1/r_2, r_1/r_3, r_2/r_3 matters
        Passed to `grid.transform_grid`.
        Units: arbitrary
    r_2 : float
        Amount to scale along y-axis.
        Passed to `grid.transform_grid`.
        Units: arbitrary
    r_3 : float
        Amount to scale along z-axis.
        Passed to `grid.transform_grid`.
        Units: arbitrary
    theta : float
        Angle to rotate in xy-plane.
        Passed to `grid.transform_grid`.
        Units: radians
    amp : float
        Factor by which pressure is enhanced within the ellipsoid.
        Units: unitless
    phi0 : float
        Polar angle of nose of power law. This is CCW from the x-axis,
        after the grid rotation. See arctan2 documentation
        Units: radians
    k_r : float
        Slope of power law in radial direction.
        Units: unitless
    k_phi : float
        Slope of power law in polar direction.
        Units: unitless

    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 grid.transform_grid. Units: arcsec	required
dy	float	Dec of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
dz	float	Line of sight offset of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
r_1	float	Amount to scale along x-axis. The absolute value of these is degenerate with sigma. Units are arbitrary, only ratio of r_1/r_2, r_1/r_3, r_2/r_3 matters Passed to grid.transform_grid. Units: arbitrary	required
r_2	float	Amount to scale along y-axis. Passed to grid.transform_grid. Units: unitless	required
r_3	float	Amount to scale along z-axis. Passed to grid.transform_grid. Units: unitless	required
theta	float	Angle to rotate in xy-plane. Passed to grid.transform_grid. Units: radians	required
amp	float	Factor by which pressure is enhanced within the ellipsoid. Units: unitless	required
phi0	float	Polar angle of nose of power law. This is CCW from the x-axis, after the grid rotation. See arctan2 documentation Units: radians	required
k_r	float	Slope of power law in radial direction. Units: unitless	required
omega	float	Angular freqency of the cosine term. Units: unitless	required

Returns:

Name	Type	Description
new_pressure	Array	Pressure profile with ellipsoid added.

Source code in witch/structure.py

@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 `grid.transform_grid`.
        Units: arcsec
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    r_1 : float
        Amount to scale along x-axis. The absolute value of these is degenerate with sigma.
        Units are arbitrary, only ratio of r_1/r_2, r_1/r_3, r_2/r_3 matters
        Passed to `grid.transform_grid`.
        Units: arbitrary
    r_2 : float
        Amount to scale along y-axis.
        Passed to `grid.transform_grid`.
        Units: unitless
    r_3 : float
        Amount to scale along z-axis.
        Passed to `grid.transform_grid`.
        Units: unitless
    theta : float
        Angle to rotate in xy-plane.
        Passed to `grid.transform_grid`.
        Units: radians
    amp : float
        Factor by which pressure is enhanced within the ellipsoid.
        Units: unitless
    phi0 : float
         Polar angle of nose of power law. This is CCW from the x-axis,
         after the grid rotation. See arctan2 documentation
         Units: radians
    k_r : float
        Slope of power law in radial direction.
        Units: unitless
    omega : float
        Angular freqency of the cosine term.
        Units: unitless

    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 grid.transform_grid. Units: arcsec	required
dy	float	Dec of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
dz	float	Line of sight offset of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
r_1	float	Amount to scale along x-axis. The absolute value of these is degenerate with sigma. Units are arbitrary, only ratio of r_1/r_2, r_1/r_3, r_2/r_3 matters Passed to grid.transform_grid. Units: arbitrary	required
r_2	float	Amount to scale along y-axis. Passed to grid.transform_grid. Units: arbitrary	required
r_3	float	Amount to scale along z-axis. Passed to grid.transform_grid. Units: arbitrary	required
theta	float	Angle to rotate in xy-plane. Passed to grid.transform_grid. Units: radians	required
amp	float	Factor by which pressure is enhanced within the ellipsoid. Units: unitless	required

Returns:

Name	Type	Description
new_pressure	Array	Pressure profile with ellipsoid added.

Source code in witch/structure.py

@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 `grid.transform_grid`.
        Units: arcsec
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    r_1 : float
        Amount to scale along x-axis. The absolute value of these is degenerate with sigma.
        Units are arbitrary, only ratio of r_1/r_2, r_1/r_3, r_2/r_3 matters
        Passed to `grid.transform_grid`.
        Units: arbitrary
    r_2 : float
        Amount to scale along y-axis.
        Passed to `grid.transform_grid`.
        Units: arbitrary
    r_3 : float
        Amount to scale along z-axis.
        Passed to `grid.transform_grid`.
        Units: arbitrary
    theta : float
        Angle to rotate in xy-plane.
        Passed to `grid.transform_grid`.
        Units: radians
    amp : float
        Factor by which pressure is enhanced within the ellipsoid.
        Units: unitless

    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 grid.transform_grid. Units: arcsec	required
dy	float	Dec of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
dz	float	Line of sight offset of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	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 \(2L\). Units: arcsec	required
theta	float	Angle to rotate in xy-plane. Passed to grid.transform_grid. Units: radians	required
P0	float	Amplitude of the pressure profile. Units: unitless	required
r_c	float	The critical radius of the cylindrical profile. Units: arcsec	required
beta	float	Beta value of isobeta model. Units: unitless	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

@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 `grid.transform_grid`.
        Units: arcsec
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    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 $2L$.
        Units: arcsec
    theta : float
        Angle to rotate in xy-plane.
        Passed to `grid.transform_grid`.
        Units: radians
    P0 : float
        Amplitude of the pressure profile.
        Units: unitless
    r_c : float
        The critical radius of the cylindrical profile.
        Units: arcsec
    beta : float
        Beta value of isobeta model.
        Units: unitless
    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

cylindrical_beta_2d(dx, dy, dz, L, theta, phi, P0, r_c, beta, xyz)

Same as cylindrical_beta but compute 2D profile analytically. Should be faster than 3D integration. Useful when you are not modifying the 3D grid. Also includes the LoS angle phi.

Once the grid is transformed the profile is computed as:

\[ y(R) = \frac{\sqrt{\pi} \Gamma (3\beta/2 - 1/2)}{\Gamma(3\beta /2)} \sec{\phi} P_0 r_c [1+\frac{R}{r_c}^2]^{-3\beta /2 + 1/2} \]

Note the missing factor of

\[ \frac{\sigma_T}{m_e c^2} \]

is provided by the unit conversion functionality.

Derivation from Craig Sarazin

Parameters:

Name	Type	Description	Default
dx	float	RA of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
dy	float	Dec of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
dz	float	Line of sight offset of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	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 \(2L\). Units: arcsec	required
theta	float	Angle to rotate in xy-plane. Passed to grid.transform_grid. Units: radians	required
phi	float	Angle to rotate in xz-plane Units: radians	required
P0	float	Amplitude of the pressure profile. Units: unitless	required
r_c	float	The critical radius of the cylindrical profile. Units: arcsec	required
beta	float	Beta value of isobeta model. Units: unitless	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

@jax.jit
def cylindrical_beta_2d(
    dx: float,
    dy: float,
    dz: float,
    L: float,
    theta: float,
    phi: float,
    P0: float,
    r_c: float,
    beta: float,
    xyz: tuple[jax.Array, jax.Array, jax.Array, float, float],
) -> jax.Array:
    r"""

    Same as cylindrical_beta but compute 2D profile analytically.
    Should be faster than 3D integration. Useful when you are
    not modifying the 3D grid. Also includes the LoS angle phi.

    Once the grid is transformed the profile is computed as:

    $$
    y(R) = \frac{\sqrt{\pi} \Gamma (3\beta/2 - 1/2)}{\Gamma(3\beta /2)} \sec{\phi} P_0 r_c [1+\frac{R}{r_c}^2]^{-3\beta /2 + 1/2}
    $$

    Note the missing factor of

    $$
    \frac{\sigma_T}{m_e c^2}
    $$

    is provided by the unit conversion functionality.

    Derivation from Craig Sarazin
    Parameters
    ----------
    dx : float
        RA of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    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 $2L$.
        Units: arcsec
    theta : float
        Angle to rotate in xy-plane.
        Passed to `grid.transform_grid`.
        Units: radians
    phi : float
        Angle to rotate in xz-plane
        Units: radians
    P0 : float
        Amplitude of the pressure profile.
        Units: unitless
    r_c : float
        The critical radius of the cylindrical profile.
        Units: arcsec
    beta : float
        Beta value of isobeta model.
        Units: unitless
    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, *_ = transform_grid(dx, dy, dz, 1.0, 1.0, 1.0, theta, xyz)
    rr = x[..., 0] ** 2 + y[..., 0] ** 2

    gamma_term = (
        jnp.sqrt(jnp.pi) * jax.scipy.special.gamma(3 * beta / 2 - 1 / 2)
    ) / jax.scipy.special.gamma(3 * beta / 2)

    r_term = (1 + (rr / r_c) ** 2) ** (-3 * beta / 2 + 1 / 2)

    return gamma_term * 1 / jnp.cos(phi) * (P0 * r_c) * r_term

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 grid.transform_grid. Units: arcsec	required
dy	float	Dec of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
dz	float	Line of sight offset of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	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 grid.transform_grid. Units: arbitrary	required
r_2	float	Amount to scale along y-axis. Passed to grid.transform_grid. Units: arbitrary	required
r_3	float	Amount to scale along z-axis. Passed to grid.transform_grid. Units: arbitrary	required
theta	float	Angle to rotate in xy-plane. Passed to grid.transform_grid. Units: radians	required
P0	float	Amplitude of the pressure profile Units: unitless	required
c500	float	Concentration parameter at a density contrast of 500 Units: unitless	required
m500	float	Mass at a density contrast of 500 Units: M_solar	required
gamma	float	The central slope Units: unitless	required
alpha	float	The intermediate slope Units: unitless	required
beta	float	The outer slope Units: unitless	required
z	float	Redshift of cluster Units: redshift	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

@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 `grid.transform_grid`.
        Units: arcsec
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    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 `grid.transform_grid`.
        Units: arbitrary
    r_2 : float
        Amount to scale along y-axis.
        Passed to `grid.transform_grid`.
        Units: arbitrary
    r_3 : float
        Amount to scale along z-axis.
        Passed to `grid.transform_grid`.
        Units: arbitrary
    theta : float
        Angle to rotate in xy-plane.
        Passed to `grid.transform_grid`.
        Units: radians
    P0 : float
        Amplitude of the pressure profile
        Units: unitless
    c500 : float
        Concentration parameter at a density contrast of 500
        Units: unitless
    m500 : float
        Mass at a density contrast of 500
        Units: M_solar
    gamma : float
        The central slope
        Units: unitless
    alpha : float
        The intermediate slope
        Units: unitless
    beta : float
        The outer slope
        Units: unitless
    z : float
        Redshift of cluster
        Units: redshift
    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 grid.transform_grid. Units: arcsec	required
dy	float	Dec of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
dz	float	Line of sight offset of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
r_1	float	Amount to scale along x-axis. The absolute value of these is degenerate with sigma. Units are arbitrary, only ratio of r_1/r_2, r_1/r_3, r_2/r_3 matters Passed to grid.transform_grid. Units: arbitrary	required
r_2	float	Amount to scale along y-axis. Passed to grid.transform_grid. Units: arbitrary	required
r_3	float	Amount to scale along z-axis. Passed to grid.transform_grid. Units: arbitrary	required
theta	float	Angle to rotate in xy-plane. Passed to grid.transform_grid. Units: radians	required
sigma	float	Sigma value of gaussian model. Units: arcsec	required
amp	float	Amplitude of gaussian model. Units: Jy	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

@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 `grid.transform_grid`.
        Units: arcsec
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    r_1 : float
        Amount to scale along x-axis. The absolute value of these is degenerate with sigma.
        Units are arbitrary, only ratio of r_1/r_2, r_1/r_3, r_2/r_3 matters
        Passed to `grid.transform_grid`.
        Units: arbitrary
    r_2 : float
        Amount to scale along y-axis.
        Passed to `grid.transform_grid`.
        Units: arbitrary
    r_3 : float
        Amount to scale along z-axis.
        Passed to `grid.transform_grid`.
        Units: arbitrary
    theta : float
        Angle to rotate in xy-plane.
        Passed to `grid.transform_grid`.
        Units: radians
    sigma : float
        Sigma value of gaussian model.
        Units: arcsec
    amp : float
        Amplitude of gaussian model.
        Units: Jy
    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 grid.transform_grid. Units: arcsec	required
dy	float	Dec of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
sigma	float	Sigma value of gaussian model. Units: arcsec	required
amp	float	Amplitude of gaussian model. Units: Jy	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

@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 `grid.transform_grid`.
        Units: arcsec
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    sigma : float
        Sigma value of gaussian model.
        Units: arcsec
    amp : float
        Amplitude of gaussian model.
        Units: Jy
    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 grid.transform_grid. Units: arcsec	required
dy	float	Dec of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
dz	float	Line of sight offset of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	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 grid.transform_grid. Units: arbitrary	required
r_2	float	Amount to scale along y-axis. Passed to grid.transform_grid.	required
r_3	float	Amount to scale along z-axis. Passed to grid.transform_grid.	required
theta	float	Angle to rotate in xy-plane. Passed to grid.transform_grid. Units: radians	required
P0	float	Amplitude of the pressure profile. Units: unitless	required
c500	float	Concentration parameter at a density contrast of 500. Units: unitless	required
m500	float	Mass at a density contrast of 500. Units: M_solar	required
gamma	float	The central slope. Units: unitless	required
alpha	float	The intermediate slope. Units: unitless	required
beta	float	The outer slope. Units: unitless	required
z	float	Redshift of cluster. Units: redshift	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

@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 `grid.transform_grid`.
        Units: arcsec
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    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 `grid.transform_grid`.
        Units: arbitrary
    r_2 : float
        Amount to scale along y-axis.
        Passed to `grid.transform_grid`.
    r_3 : float
        Amount to scale along z-axis.
        Passed to `grid.transform_grid`.
    theta : float
        Angle to rotate in xy-plane.
        Passed to `grid.transform_grid`.
        Units: radians
    P0 : float
        Amplitude of the pressure profile.
        Units: unitless
    c500 : float
        Concentration parameter at a density contrast of 500.
        Units: unitless
    m500 : float
        Mass at a density contrast of 500.
        Units: M_solar
    gamma : float
        The central slope.
        Units: unitless
    alpha : float
        The intermediate slope.
        Units: unitless
    beta : float
        The outer slope.
        Units: unitless
    z : float
        Redshift of cluster.
        Units: redshift
    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 grid.transform_grid. Units: arcsec	required
dy	float	Dec of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	required
dz	float	Line of sight offset of cluster center relative to grid origin. Passed to grid.transform_grid. Units: arcsec	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 grid.transform_grid. Units: arbitrary	required
r_2	float	Amount to scale along y-axis. Passed to grid.transform_grid. Units: arbitrary	required
r_3	float	Amount to scale along z-axis. Passed to grid.transform_grid. Units: arbitrary	required
theta	float	Angle to rotate in xy-plane. Passed to grid.transform_grid. Units: radians	required
beta	float	Beta value of isobeta model. Units: unitless	required
amp	float	Amplitude of isobeta model. Units: Matches unit conversion implicitly.	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

@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 `grid.transform_grid`.
        Units: arcsec
    dy : float
        Dec of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    dz : float
        Line of sight offset of cluster center relative to grid origin.
        Passed to `grid.transform_grid`.
        Units: arcsec
    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 `grid.transform_grid`.
        Units: arbitrary
    r_2 : float
        Amount to scale along y-axis.
        Passed to `grid.transform_grid`.
        Units: arbitrary
    r_3 : float
        Amount to scale along z-axis.
        Passed to `grid.transform_grid`.
        Units: arbitrary
    theta : float
        Angle to rotate in xy-plane.
        Passed to `grid.transform_grid`.
        Units: radians
    beta : float
        Beta value of isobeta model.
        Units: unitless
    amp : float
        Amplitude of isobeta model.
        Units: Matches unit conversion implicitly.
    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