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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.

Arguments:

dx: RA of cluster center relative to grid origin

dy: Dec of cluster center relative to grid origin

dz: Line of sight offset of cluster center relative to grid origin

theta: Angle to rotate in xy-plane

P0: Amplitude of the pressure profile

c500: Concentration parameter at a density contrast of 500

m500: Mass at a density contrast of 500

gamma: The central slope

alpha: The intermediate slope

beta: The outer slope

z: Redshift of cluster

xyz: Coordinte grid to calculate model on

Returns:

model: The gnfw model
Source code in witch/structure.py 
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@jax.jit
def 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.

    Arguments:

        dx: RA of cluster center relative to grid origin

        dy: Dec of cluster center relative to grid origin

        dz: Line of sight offset of cluster center relative to grid origin

        theta: Angle to rotate in xy-plane

        P0: Amplitude of the pressure profile

        c500: Concentration parameter at a density contrast of 500

        m500: Mass at a density contrast of 500

        gamma: The central slope

        alpha: The intermediate slope

        beta: The outer slope

        z: Redshift of cluster

        xyz: Coordinte grid to calculate model on

    Returns:

        model: The gnfw model
    """

    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.

Arguments:

pressure: The pressure profile

xyz: Coordinate grids, see make_grid for details

dx: RA of ellipsoid center relative to grid origin

dy: Dec of ellipsoid center relative to grid origin

dz: Line of sight offset of ellipsoid center relative to grid origin

r_1: Amount to scale ellipsoid along x-axis

r_2: Amount to scale ellipsoid along y-axis

r_3: Amount to scale ellipsoid along z-axis

theta: Angle to rotate ellipsoid in xy-plane

amp: Factor by which pressure is enhanced at peak of exponential

xk: Power of exponential in RA direction

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

yk: Power of exponential in Dec direction

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

zk: Power of exponential along the line of sight

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

Returns:

new_pressure: Pressure profile with ellipsoid added
Source code in witch/structure.py 
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@jax.jit
def 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.

    Arguments:

        pressure: The pressure profile

        xyz: Coordinate grids, see make_grid for details

        dx: RA of ellipsoid center relative to grid origin

        dy: Dec of ellipsoid center relative to grid origin

        dz: Line of sight offset of ellipsoid center relative to grid origin

        r_1: Amount to scale ellipsoid along x-axis

        r_2: Amount to scale ellipsoid along y-axis

        r_3: Amount to scale ellipsoid along z-axis

        theta: Angle to rotate ellipsoid in xy-plane

        amp: Factor by which pressure is enhanced at peak of exponential

        xk: Power of exponential in RA direction

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

        yk: Power of exponential in Dec direction

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

        zk: Power of exponential along the line of sight

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

    Returns:

        new_pressure: 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.

Arguments:

pressure: The pressure profile

xyz: Coordinate grids, see make_grid for details

dx: RA of ellipsoid center relative to grid origin

dy: Dec of ellipsoid center relative to grid origin

dz: Line of sight offset of ellipsoid center relative to grid origin

r_1: Amount to scale ellipsoid along x-axis

r_2: Amount to scale ellipsoid along y-axis

r_3: Amount to scale ellipsoid along z-axis

theta: Angle to rotate ellipsoid in xy-plane

amp: Factor by which pressure is enhanced at peak of power law

phi0: Polar angle of nose of power law

k_r: Slope of power law in radial dirction

k_phi: Slope of power law in polar direction

Returns:

new_pressure: Pressure profile with ellipsoid added
Source code in witch/structure.py 
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@jax.jit
def 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.

    Arguments:

        pressure: The pressure profile

        xyz: Coordinate grids, see make_grid for details

        dx: RA of ellipsoid center relative to grid origin

        dy: Dec of ellipsoid center relative to grid origin

        dz: Line of sight offset of ellipsoid center relative to grid origin

        r_1: Amount to scale ellipsoid along x-axis

        r_2: Amount to scale ellipsoid along y-axis

        r_3: Amount to scale ellipsoid along z-axis

        theta: Angle to rotate ellipsoid in xy-plane

        amp: Factor by which pressure is enhanced at peak of power law

        phi0: Polar angle of nose of power law

        k_r: Slope of power law in radial dirction

        k_phi: Slope of power law in polar direction

    Returns:

        new_pressure: 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 cos dependant structure to 3d pressure profile.

Arguments:

pressure: The pressure profile

xyz: Coordinate grids, see make_grid for details

dx: RA of ellipsoid center relative to grid origin

dy: Dec of ellipsoid center relative to grid origin

dz: Line of sight offset of ellipsoid center relative to grid origin

r_1: Amount to scale ellipsoid along x-axis

r_2: Amount to scale ellipsoid along y-axis

r_3: Amount to scale ellipsoid along z-axis

theta: Angle to rotate ellipsoid in xy-plane

amp: Factor by which pressure is enhanced at peak of power law

phi0: Polar angle of nose of power law

k_r: Slope of power law in radial dirction

omega: Angular freqency if cos term

Returns:

new_pressure: Pressure profile with ellipsoid added
Source code in witch/structure.py 
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@jax.jit
def 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 cos dependant structure to 3d pressure profile.

    Arguments:

        pressure: The pressure profile

        xyz: Coordinate grids, see make_grid for details

        dx: RA of ellipsoid center relative to grid origin

        dy: Dec of ellipsoid center relative to grid origin

        dz: Line of sight offset of ellipsoid center relative to grid origin

        r_1: Amount to scale ellipsoid along x-axis

        r_2: Amount to scale ellipsoid along y-axis

        r_3: Amount to scale ellipsoid along z-axis

        theta: Angle to rotate ellipsoid in xy-plane

        amp: Factor by which pressure is enhanced at peak of power law

        phi0: Polar angle of nose of power law

        k_r: Slope of power law in radial dirction

        omega: Angular freqency if cos term

    Returns:

        new_pressure: 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 * (1 - jnp.float_power(1 + r, -1.0 * k_r)) * jnp.cos(omega * phi)
    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.

Arguments:

pressure: The pressure profile

xyz: Coordinate grids, see make_grid for details

dx: RA of ellipsoid center relative to grid origin

dy: Dec of ellipsoid center relative to grid origin

dz: Line of sight offset of ellipsoid center relative to grid origin

r_1: Amount to scale ellipsoid along x-axis

r_2: Amount to scale ellipsoid along y-axis

r_3: Amount to scale ellipsoid along z-axis

theta: Angle to rotate ellipsoid in xy-plane

amp: Factor by which pressure is enhanced at peak of exponential

Returns:

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

    Arguments:

        pressure: The pressure profile

        xyz: Coordinate grids, see make_grid for details

        dx: RA of ellipsoid center relative to grid origin

        dy: Dec of ellipsoid center relative to grid origin

        dz: Line of sight offset of ellipsoid center relative to grid origin

        r_1: Amount to scale ellipsoid along x-axis

        r_2: Amount to scale ellipsoid along y-axis

        r_3: Amount to scale ellipsoid along z-axis

        theta: Angle to rotate ellipsoid in xy-plane

        amp: Factor by which pressure is enhanced at peak of exponential

    Returns:

        new_pressure: 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

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.

Arguments:

dx: RA of cluster center relative to grid origin

dy: Dec of cluster center relative to grid origin

dz: Line of sight offset of cluster center relative to grid origin

r_1: X-axis scaling. Units are arbitrary, only radio of r_1/r_2, r_1/r_3, r_2/r_3 matters

r_2: Y-axis scaling.

r_3: Z-axis scaling.

theta: Angle to rotate in xy-plane

P0: Amplitude of the pressure profile

c500: Concentration parameter at a density contrast of 500

m500: Mass at a density contrast of 500

gamma: The central slope

alpha: The intermediate slope

beta: The outer slope

z: Redshift of cluster

xyz: Coordinte grid to calculate model on

Returns:

model: The gnfw model
Source code in witch/structure.py 
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@jax.jit
def 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.

    Arguments:

        dx: RA of cluster center relative to grid origin

        dy: Dec of cluster center relative to grid origin

        dz: Line of sight offset of cluster center relative to grid origin

        r_1: X-axis scaling. Units are arbitrary, only radio of r_1/r_2, r_1/r_3, r_2/r_3 matters

        r_2: Y-axis scaling.

        r_3: Z-axis scaling.

        theta: Angle to rotate in xy-plane

        P0: Amplitude of the pressure profile

        c500: Concentration parameter at a density contrast of 500

        m500: Mass at a density contrast of 500

        gamma: The central slope

        alpha: The intermediate slope

        beta: The outer slope

        z: Redshift of cluster

        xyz: Coordinte grid to calculate model on

    Returns:

        model: The gnfw model
    """
    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 = jnp.sqrt(r_1**2 + r_2**2 + r_3**2)

    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.

Arguments:

dx: RA of gaussian center relative to grid origin

dy: Dec of gaussian center relative to grid origin

dz: Line of sight offset of gaussian center relative to grid origin

r_1: Amount to scale along x-axis

r_2: Amount to scale along y-axis

r_3: Amount to scale along z-axis

theta: Angle to rotate in xy-plane

sigma: Sigma of the gaussian

amp: Amplitude of the gaussian

xyz: Coordinte grid to calculate model on

Returns:

model: The gaussian
Source code in witch/structure.py 
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@jax.jit
def 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.

    Arguments:

        dx: RA of gaussian center relative to grid origin

        dy: Dec of gaussian center relative to grid origin

        dz: Line of sight offset of gaussian center relative to grid origin

        r_1: Amount to scale along x-axis

        r_2: Amount to scale along y-axis

        r_3: Amount to scale along z-axis

        theta: Angle to rotate in xy-plane

        sigma: Sigma of the gaussian

        amp: Amplitude of the gaussian

        xyz: Coordinte grid to calculate model on

    Returns:

        model: The gaussian
    """
    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 3d. 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.

Arguments:

dx: RA of gaussian center relative to grid origin

dy: Dec of gaussian center relative to grid origin

sigma: The effective, beam-convolved half-width of the point source.

amp: Amplitude of the gaussian

xyz: Coordinte grid to calculate model on

Returns:

model: The gaussian
Source code in witch/structure.py 
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@jax.jit
def gaussian(dx, dy, sigma, amp, xyz):
    """
    Standard gaussian profile in 3d.
    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.

    Arguments:

        dx: RA of gaussian center relative to grid origin

        dy: Dec of gaussian center relative to grid origin

        sigma: The effective, beam-convolved half-width of the point source.

        amp: Amplitude of the gaussian

        xyz: Coordinte grid to calculate model on

    Returns:

        model: The gaussian
    """
    x, y, z = 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.

Arguments:

dx: RA of cluster center relative to grid origin

dy: Dec of cluster center relative to grid origin

dz: Line of sight offset of cluster center relative to grid origin

r_1: Amount to scale along x-axis

r_2: Amount to scale along y-axis

r_3: Amount to scale along z-axis

theta: Angle to rotate in xy-plane

P0: Amplitude of the pressure profile

c500: Concentration parameter at a density contrast of 500

m500: Mass at a density contrast of 500

gamma: The central slope

alpha: The intermediate slope

beta: The outer slope

z: Redshift of cluster

xyz: Coordinte grid to calculate model on

Returns:

model: The gnfw model
Source code in witch/structure.py 
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@jax.jit
def 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.

    Arguments:

        dx: RA of cluster center relative to grid origin

        dy: Dec of cluster center relative to grid origin

        dz: Line of sight offset of cluster center relative to grid origin

        r_1: Amount to scale along x-axis

        r_2: Amount to scale along y-axis

        r_3: Amount to scale along z-axis

        theta: Angle to rotate in xy-plane

        P0: Amplitude of the pressure profile

        c500: Concentration parameter at a density contrast of 500

        m500: Mass at a density contrast of 500

        gamma: The central slope

        alpha: The intermediate slope

        beta: The outer slope

        z: Redshift of cluster

        xyz: Coordinte grid to calculate model on

    Returns:

        model: The gnfw model
    """
    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.

Arguments:

dx: RA of cluster center relative to grid origin

dy: Dec of cluster center relative to grid origin

dz: Line of sight offset of cluster center relative to grid origin

r_1: Amount to scale along x-axis

r_2: Amount to scale along y-axis

r_3: Amount to scale along z-axis

theta: Angle to rotate in xy-plane

beta: Beta value of isobeta model

amp: Amplitude of isobeta model

xyz: Coordinte grid to calculate model on

Returns:

model: The isobeta model
Source code in witch/structure.py 
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@jax.jit
def 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.

    Arguments:

        dx: RA of cluster center relative to grid origin

        dy: Dec of cluster center relative to grid origin

        dz: Line of sight offset of cluster center relative to grid origin

        r_1: Amount to scale along x-axis

        r_2: Amount to scale along y-axis

        r_3: Amount to scale along z-axis

        theta: Angle to rotate in xy-plane

        beta: Beta value of isobeta model

        amp: Amplitude of isobeta model

        xyz: Coordinte grid to calculate model on

    Returns:

        model: The isobeta model
    """
    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