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Limb-Brightening Correction Module

Tamar Ervin Date: June 23, 2021

Limb-darkening correction functions: flatten continuum intensity images to correct for limb-darkening use fifth order polynomial and constants from Allen 1973

get_u(ll)

function to get limb darkening coefficient u (Allen 1973) based on wavelength value

Parameters

ll : float wavelength

Returns

vl : float coefficient v based on wavelength

Source code in SolAster/tools/lbc_funcs.py 
def get_u(ll):
    """
    function to get limb darkening coefficient u (Allen 1973) based on wavelength value

    Parameters
    ----------
    ll : float
        wavelength

    Returns
    -------
    vl : float
        coefficient v based on wavelength

    """

    pll = np.array([1.0, ll, ll ** 2, ll ** 3, ll ** 4, ll ** 5])
    au = -8.9829751
    bu = 0.0069093916
    cu = -1.8144591e-6
    du = 2.2540875e-10
    eu = -1.3389747e-14
    fu = 3.0453572e-19
    a = np.array([au, bu, cu, du, eu, fu])
    ul = sum(a * pll)
    return ul

get_v(ll)

function to get limb darkening coefficient v (Allen 1973) based on wavelength value

Parameters

ll : float wavelength

Returns

vl : float coefficient v based on wavelength

Source code in SolAster/tools/lbc_funcs.py 
def get_v(ll):
    """
    function to get limb darkening coefficient v (Allen 1973) based on wavelength value

    Parameters
    ----------
    ll : float
        wavelength

    Returns
    -------
    vl : float
        coefficient v based on wavelength

    """

    pll = np.array([1.0, ll, ll ** 2, ll ** 3, ll ** 4, ll ** 5])
    av = 9.2891180
    bv = -0.0062212632
    cv = 1.5788029e-6
    dv = -1.9359644e-10
    ev = 1.1444469e-14
    fv = -2.599494e-19
    a = np.array([av, bv, cv, dv, ev, fv])
    vl = sum(a * pll)
    return vl

limb_polynomial(imap)

function to calculate limb darkening correction polynomial based on IDl function: https://hesperia.gsfc.nasa.gov/ssw/gen/idl/solar/darklimb_correct.pro

Parameters

imap UNCORRECTED Sunpy map object (Intensitygram)

Returns

Lij : array_like limb-darkening polynomial array

Source code in SolAster/tools/lbc_funcs.py 
def limb_polynomial(imap):
    """
    function to calculate limb darkening correction polynomial based on IDl function:
    https://hesperia.gsfc.nasa.gov/ssw/gen/idl/solar/darklimb_correct.pro

    Parameters
    ----------
    imap
        UNCORRECTED Sunpy map object (Intensitygram)

    Returns
    -------
    Lij : array_like
        limb-darkening polynomial array

    """

    # get data
    data = imap.data

    # get header information
    header = imap.fits_header

    # wavelength
    wavelength = header['WAVELNTH']

    # coordinates
    x_center = header['CRPIX1']
    y_center = header['CRPIX2']
    radius = header['RSUN_OBS'] / header['CDELT1']

    # size
    size = imap.data.shape[0]

    # wavelength as float
    ll = 1.0 * wavelength

    # convert data to array and set nan values to zero
    array = np.array(data)
    NaNs = np.isnan(array)
    array[NaNs] = 0.0

    # get correction coefficients based on wavelength
    ul = get_u(ll)
    vl = get_v(ll)

    # make arrays of x and y coordinates
    x = np.arange(0, size, 1.0)
    y = np.arange(0, size, 1.0)

    # create grids for x, y arrays
    x_mat, y_mat = np.meshgrid(x, y)

    # make z array such that we get a circle
    z = np.sqrt((x_mat - x_center) ** 2 + (y_mat - y_center) ** 2)

    # normalize circle to radius of 1
    grid = z / radius
    out = np.where(grid > 1.0)
    grid[out] = 0.0

    # calculate polynomial
    Lij = 1.0 - ul - vl + ul * np.cos(np.arcsin(grid)) + vl * np.cos(np.arcsin(grid)) ** 2

    return Lij
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