Spherical harmonic maps¶

class starry.Map

The default starry map class.

This class handles light curves and phase curves of objects in emitted light.

Note

Instantiate this class by calling starry.Map() with ydeg > 0 and both rv and reflected set to False.

property N

Total number of map coefficients. Read-only

This is equal to \(N_\mathrm{y} + N_\mathrm{u} + N_\mathrm{f}\).

property Nf

Number of spherical harmonic coefficients in the filter. Read-only

This is equal to \((f_\mathrm{deg} + 1)^2\).

property Nu

Number of limb darkening coefficients, including \(u_0\). Read-only

This is equal to \(u_\mathrm{deg} + 1\).

property Ny

Number of spherical harmonic coefficients. Read-only

This is equal to \((y_\mathrm{deg} + 1)^2\).

add_spot(amp=None, intensity=None, relative=True, sigma=0.1, lat=0.0, lon=0.0)

Add the expansion of a gaussian spot to the map.

This function adds a spot whose functional form is the spherical harmonic expansion of a gaussian in the quantity \(\cos\Delta\theta\), where \(\Delta\theta\) is the angular separation between the center of the spot and another point on the surface. The spot brightness is controlled by either the parameter amp, defined as the fractional change in the total luminosity of the object due to the spot, or the parameter intensity, defined as the fractional change in the intensity at the center of the spot.

Parameters

amp (scalar or vector, optional) – The amplitude of the spot. This is equal to the fractional change in the luminosity of the map due to the spot. If the map has more than one wavelength bin, this must be a vector of length equal to the number of wavelength bins. Default is None. Either amp or intensity must be given.
intensity (scalar or vector, optional) – The intensity of the spot. This is equal to the fractional change in the intensity of the map at the center of the spot. If the map has more than one wavelength bin, this must be a vector of length equal to the number of wavelength bins. Default is None. Either amp or intensity must be given.
relative (bool, optional) – If True, computes the spot expansion assuming the fractional amp or intensity change is relative to the current map amplitude/intensity. If False, computes the spot expansion assuming the fractional change is relative to the original map amplitude/intensity (i.e., that of a featureless map). Defaults to True. Note that if True, adding two spots with the same values of amp or intensity will generally result in different intensities at their centers, since the first spot will have changed the map intensity everywhere! Defaults to True.
sigma (scalar, optional) – The standard deviation of the gaussian. Defaults to 0.1.
lat (scalar, optional) – The latitude of the spot in units of angle_unit. Defaults to 0.0.
lon (scalar, optional) – The longitude of the spot in units of angle_unit. Defaults to 0.0.

Warning

Method add_spot is deprecated as of version 1.1 and will be removed in the future. Please use method spot instead.

property amp: The overall amplitude of the map in arbitrary units. This factor multiplies the intensity and the flux and is thus proportional to the luminosity of the object. For multi-wavelength maps, this is a vector corresponding to the amplitude of each wavelength bin. For reflected light maps, this is the average spherical albedo of the body.

property angle_unit: An astropy.units quantity defining the angle unit for this map.

property deg

Total degree of the map. Read-only

This is equal to \(y_\mathrm{deg} + u_\mathrm{deg} + f_\mathrm{deg}\).

design_matrix(**kwargs)

Compute and return the light curve design matrix \(A\).

This matrix is used to compute the flux \(f\) from a vector of spherical harmonic coefficients \(y\) and the map amplitude \(a\): \(f = a A y\).

Parameters

xo (scalar or vector, optional) – x coordinate of the occultor relative to this body in units of this body’s radius.
yo (scalar or vector, optional) – y coordinate of the occultor relative to this body in units of this body’s radius.
zo (scalar or vector, optional) – z coordinate of the occultor relative to this body in units of this body’s radius.
ro (scalar, optional) – Radius of the occultor in units of this body’s radius.
theta (scalar or vector, optional) – Angular phase of the body in units of angle_unit.

draw()

Draw a map from the posterior distribution.

This method draws a random map from the posterior distribution and sets the y map vector and amp map amplitude accordingly. Users should call solve() to enable this attribute.

property fdeg: Degree of the multiplicative filter. Read-only

flux(**kwargs)

Compute and return the light curve.

Parameters

xo (scalar or vector, optional) – x coordinate of the occultor relative to this body in units of this body’s radius.
yo (scalar or vector, optional) – y coordinate of the occultor relative to this body in units of this body’s radius.
zo (scalar or vector, optional) – z coordinate of the occultor relative to this body in units of this body’s radius.
ro (scalar, optional) – Radius of the occultor in units of this body’s radius.
theta (scalar or vector, optional) – Angular phase of the body in units of angle_unit.
integrated (bool, optional) – If True, dots the flux with the amplitude. Default False, in which case this returns a 2d array (wavelength-dependent maps only).

get_latlon_grid(res=300, projection='ortho')

Return the latitude/longitude grid corresponding to the result of a call to render().

Parameters

res (int, optional) – The resolution of the map in pixels on a side. Defaults to 300.
projection (string, optional) – The map projection. Accepted values are ortho, corresponding to an orthographic projection (as seen on the sky), rect, corresponding to an equirectangular latitude-longitude projection, and moll, corresponding to a Mollweide equal-area projection. Defaults to ortho.

get_pixel_transforms(oversample=2, lam=1e-06, eps=1e-06)

Return several linear operators for pixel transformations.

Parameters

oversample (int) – Factor by which to oversample the pixelization grid. Default 2.
lam (float) – Regularization parameter for the inverse pixel transform. Default 1e-6.
eps (float) – Regularization parameter for the derivative transforms. Default 1e-6.

Returns

The tuple (lat, lon, Y2P, P2Y, Dx, Dy).

The transforms returned by this method can be used to easily convert back and forth between spherical harmonic coefficients and intensities on a discrete pixelized grid. Projections onto pixels are performed on an equal-area Mollweide grid, so these transforms are useful for applying priors on the pixel intensities, for instance.

The lat and lon arrays correspond to the latitude and longitude of each of the points used in the transform (in units of angle_unit).

The Y2P matrix is an operator that transforms from spherical harmonic coefficients y to pixels p on a Mollweide grid:

p = Y2P @ y

The P2Y matrix is the (pseudo-)inverse of that operator:

y = P2Y @ p

Finally, the Dx and Dy operators transform a pixel representation of the map p to the derivative of p with respect to longitude and latitude, respectively:

dpdlon = Dx @ p
dpdlat = Dy @ p

By combining these operators, one can differentiate the spherical harmonic expansion with respect to latitude and longitude, if desired:

dydlon = P2Y @ Dx @ Y2P @ y dydlat = P2Y @ Dy @ Y2P @ y

These derivatives could be useful for implementing total-variation-reducing regularization, for instance.

Warning

This is an experimental feature.

property inc: The inclination of the rotation axis in units of angle_unit.

intensity(lat=0, lon=0, **kwargs)

Compute and return the intensity of the map.

Parameters

lat (scalar or vector, optional) – latitude at which to evaluate the intensity in units of angle_unit.
lon (scalar or vector, optional) – longitude at which to evaluate the intensity in units of angle_unit`.
theta (scalar, optional) – For differentially rotating maps only, the angular phase at which to evaluate the intensity. Default 0.
limbdarken (bool, optional) – Apply limb darkening (only if udeg > 0)? Default True.

intensity_design_matrix(lat=0, lon=0)

Compute and return the pixelization matrix P.

This matrix is used to compute the intensity \(I\) from a vector of spherical harmonic coefficients \(y\) and the map amplitude \(a\): \(I = a P y\).

Parameters

lat (scalar or vector, optional) – latitude at which to evaluate the design matrix in units of angle_unit.
lon (scalar or vector, optional) – longitude at which to evaluate the design matrix in units of angle_unit.

Note

This method ignores any filters (such as limb darkening or velocity weighting) and illumination (for reflected light maps).

limbdark_is_physical()

Check whether the limb darkening profile (if any) is physical.

This method uses Sturm’s theorem to ensure that the limb darkening intensity is positive everywhere and decreases monotonically toward the limb.

Returns: Whether or not the limb darkening profile is physical.
Return type: bool

lnlike(*, design_matrix=None, woodbury=True, **kwargs)

Returns the log marginal likelihood of the data given a design matrix.

This method computes the marginal likelihood (marginalized over the spherical harmonic coefficients) given a light curve and its covariance (set via the set_data() method) and a Gaussian prior on the spherical harmonic coefficients (set via the set_prior() method).

Parameters

design_matrix (matrix, optional) – The flux design matrix, the quantity returned by design_matrix(). Default is None, in which case this is computed based on kwargs.
woodbury (bool, optional) – Solve the linear problem using the Woodbury identity? Default is True. The Woodbury identity is used to speed up matrix operations in the case that the number of data points is much larger than the number of spherical harmonic coefficients. In this limit, it can speed up the code by more than an order of magnitude. Keep in mind that the numerical stability of the Woodbury identity is not great, so if you’re getting strange results try disabling this. It’s also a good idea to disable this in the limit of few data points and large spherical harmonic degree.
kwargs (optional) – Keyword arguments to be passed directly to design_matrix(), if a design matrix is not provided.

Returns

The log marginal likelihood, a scalar.

load(image, extent=(- 180, 180, - 90, 90), smoothing=None, fac=1.0, eps=1e-12, force_psd=False, **kwargs)

Load an image or ndarray.

This routine performs a simple spherical harmonic transform (SHT) to compute the spherical harmonic expansion corresponding to an input image file or numpy array on a lat-lon grid. The resulting coefficients are ingested into the map.

Parameters

image – A path to an image PNG file or a two-dimensional numpy array on a latitude-longitude grid.
extent (tuple, optional) – The lat-lon values corresponding to the edges of the image in degrees, (lat0, lat1, lon0, lon1). Default is (-180, 180, -90, 90).
smoothing (float, optional) – Gaussian smoothing strength. Increase this value to suppress ringing or explicitly set to zero to disable smoothing. Default is 1/self.ydeg.
fac (float, optional) – Factor by which to oversample the image when applying the SHT. Default is 1.0. Increase this number for higher fidelity (at the expense of increased computational time).
eps (float, optional) – Regularization strength for the spherical harmonic transform. Default is 1e-12.
force_psd (bool, optional) – Force the map to be positive semi-definite? Default is False.
kwargs (optional) – Any other kwargs passed directly to minimize() (only if psd is True).

minimize(oversample=1, ntries=1, bounds=None, return_info=False)

Find the global (optionally local) minimum of the map intensity.

Parameters

oversample (int) – Factor by which to oversample the initial grid on which the brute force search is performed. Default 1.
ntries (int) – Number of times the nonlinear minimizer is called. Default 1.
return_info (bool) – Return the info from the minimization call? Default is False.
bounds (tuple) – Return map minimum in a certain latitude/longitude range, for example bounds=((0, 90), (0, 180)). Default None.

Returns

A tuple of the latitude, longitude, and the value of the intensity at the minimum. If return_info is True, also returns the detailed solver information.

property nw: Number of wavelength bins. Read-only

property obl: The obliquity of the rotation axis in units of angle_unit.

remove_prior(): Remove the prior on the map coefficients.

render(res=300, projection='ortho', theta=0.0)

Compute and return the intensity of the map on a grid.

Returns an image of shape (res, res), unless theta is a vector, in which case returns an array of shape (nframes, res, res), where nframes is the number of values of theta. However, if this is a spectral map, nframes is the number of wavelength bins and theta must be a scalar.

Note

Users can obtain the latitudes and longitudes corresponding to each point in the rendered image by calling get_latlon_grid().

Parameters

res (int, optional) – The resolution of the map in pixels on a side. Defaults to 300.
projection (string, optional) – The map projection. Accepted values are ortho, corresponding to an orthographic projection (as seen on the sky), rect, corresponding to an equirectangular latitude-longitude projection, and moll, corresponding to a Mollweide equal-area projection. Defaults to ortho.
theta (scalar or vector, optional) – The map rotation phase in units of angle_unit. If this is a vector, an animation is generated. Defaults to 0.0.

reset(**kwargs): Reset all map coefficients and attributes.

Note

Does not reset custom unit settings.

rotate(axis, theta)

Rotate the current map vector an angle theta about axis.

Parameters

axis (vector) – The axis about which to rotate the map.
theta (scalar) – The angle of (counter-clockwise) rotation.

set_data(flux, C=None, cho_C=None)

Set the data vector and covariance matrix.

This method is required by the solve() method, which analytically computes the posterior over surface maps given a dataset and a prior, provided both are described as multivariate Gaussians.

Parameters

flux (vector) – The observed light curve.
C (scalar, vector, or matrix) – The data covariance. This may be a scalar, in which case the noise is assumed to be homoscedastic, a vector, in which case the covariance is assumed to be diagonal, or a matrix specifying the full covariance of the dataset. Default is None. Either C or cho_C must be provided.
cho_C (matrix) – The lower Cholesky factorization of the data covariance matrix. Defaults to None. Either C or cho_C must be provided.

set_prior(*, mu=None, L=None, cho_L=None)

Set the prior mean and covariance of the spherical harmonic coefficients.

This method is required by the solve() method, which analytically computes the posterior over surface maps given a dataset and a prior, provided both are described as multivariate Gaussians.

Note that the prior is placed on the amplitude-weighted coefficients, i.e., the quantity x = map.amp * map.y. Because the first spherical harmonic coefficient is fixed at unity, x[0] is the amplitude of the map. The actual spherical harmonic coefficients are given by x / map.amp.

This convention allows one to linearly fit for an arbitrary map normalization at the same time as the spherical harmonic coefficients, while ensuring the starry requirement that the coefficient of the \(Y_{0,0}\) harmonic is always unity.

Parameters

mu (scalar or vector) – The prior mean on the amplitude-weighted spherical harmonic coefficients. Default is 1.0 for the first term and zero for the remaining terms. If this is a vector, it must have length equal to Ny.
L (scalar, vector, or matrix) – The prior covariance. This may be a scalar, in which case the covariance is assumed to be homoscedastic, a vector, in which case the covariance is assumed to be diagonal, or a matrix specifying the full prior covariance. Default is None. Either L or cho_L must be provided.
cho_L (matrix) – The lower Cholesky factorization of the prior covariance matrix. Defaults to None. Either L or cho_L must be provided.

show(**kwargs)

Display an image of the map, with optional animation. See the docstring of render() for more details and additional keywords accepted by this method.

Parameters

ax (optional) – A matplotlib axis instance to use. Default is to create a new figure.
cmap (string or colormap instance, optional) – The matplotlib colormap to use. Defaults to plasma.
figsize (tuple, optional) – Figure size in inches. Default is (3, 3) for orthographic maps and (7, 3.5) for rectangular maps.
projection (string, optional) – The map projection. Accepted values are ortho, corresponding to an orthographic projection (as seen on the sky), rect, corresponding to an equirectangular latitude-longitude projection, and moll, corresponding to a Mollweide equal-area projection. Defaults to ortho.
colorbar (bool, optional) – Display a colorbar? Default is False.
grid (bool, optional) – Show latitude/longitude grid lines? Defaults to True.
interval (int, optional) – Interval between frames in milliseconds (animated maps only). Defaults to 75.
file (string, optional) – The file name (including the extension) to save the figure or animation to. Defaults to None.
html5_video (bool, optional) – If rendering in a Jupyter notebook, display as an HTML5 video? Default is True. If False, displays the animation using Javascript (file size will be larger.)
dpi (int, optional) – Image resolution in dots per square inch. Defaults to the value defined in matplotlib.rcParams.
bitrate (int, optional) – Bitrate in kbps (animations only). Defaults to the value defined in matplotlib.rcParams.
norm (optional) – The color normalization passed to matplotlib.pyplot.imshow, an instance of matplotlib.colors.Normalize. Can be used to pass in minimum and maximum values. Default is None.
illuminate (bool, optional) – Illuminate the map (reflected light maps only)? Default True. If False, shows the albedo surface map.
screen (bool, optional) – Apply the illumination as a black-and-white alpha screen (reflected light maps only)? Default True. If False, a single colormap is used to plot the visible intensity.

Note

Pure limb-darkened maps do not accept a projection keyword.

Note

If calling this method on an instance of Map created within a pymc3.Model(), you may specify a point keyword with the model point at which to evaluate the map. This method also accepts a model keyword, although this is inferred automatically if called from within a pymc3.Model() context. If no point is provided, attempts to evaluate the map at model.test_point and raises a warning.

sht_matrix(inverse=False, return_grid=False, smoothing=None, oversample=2, lam=1e-06)

Return the Spherical Harmonic Transform (SHT) matrix.

This matrix dots into a vector of pixel intensities defined on a set of latitude-longitude points, resulting in a vector of spherical harmonic coefficients that best approximates the image.

This can be useful for transforming between the spherical harmonic and pixel representations of the surface map, such as when specifying priors during inference or optimization. For example, one application is when one wishes to sample in the pixel intensities p instead of the spherical harmonic coefficients y. In this case, one must compute the spherical harmonic coefficients y from the vector of pixel intensities p so that starry can compute the model for the flux:

Note

This method is a thin wrapper of the get_pixel_transforms method.

Parameters

inverse (bool, optional) – which transforms from spherical harmonic coefficients to pixels. Default is False.
return_grid (bool, optional) – shape (npix, 2) corresponding to the latitude-longitude points (in units of angle_unit) on which the SHT is evaluated. Default is False.
smoothing (float, optional) – Gaussian smoothing strength. Increase this value to suppress ringing (forward SHT only) or explicitly set to zero to disable smoothing. Default is 2/self.ydeg.
oversample (int, optional) – Factor by which to oversample the pixelization grid. Default 2.
lam (float, optional) – Regularization parameter for the inverse pixel transform. Default 1e-6.

Returns

A matrix of shape (Ny, npix) or (npix, Ny) (if inverse is True), where npix is the number of pixels on the grid. This number is determined from the degree of the map and the oversample keyword. If return_grid is True, also returns the latitude-longitude points (in units of angle_unit) on which the SHT is evaluated, a matrix of shape (npix, 2).

property solution

The posterior probability distribution for the map.

This is a tuple containing the mean and lower Cholesky factorization of the covariance of the amplitude-weighted spherical harmonic coefficient vector, obtained by solving the regularized least-squares problem via the solve() method.

Note that to obtain the actual covariance matrix from the lower Cholesky factorization \(L\), simply compute \(L L^\top\).

Note also that this is the posterior for the amplitude-weighted map vector. Under this convention, the map amplitude is equal to the first term of the vector and the spherical harmonic coefficients are equal to the vector normalized by the first term.

solve(*, design_matrix=None, **kwargs)

Solve the linear least-squares problem for the posterior over maps.

This method solves the generalized least squares problem given a light curve and its covariance (set via the set_data() method) and a Gaussian prior on the spherical harmonic coefficients (set via the set_prior() method). The map amplitude and coefficients are set to the maximum a posteriori (MAP) solution.

Parameters

design_matrix (matrix, optional) – The flux design matrix, the quantity returned by design_matrix(). Default is None, in which case this is computed based on kwargs.
kwargs (optional) – Keyword arguments to be passed directly to design_matrix(), if a design matrix is not provided.

Returns

A tuple containing the posterior mean for the amplitude-weighted spherical harmonic coefficients (a vector) and the Cholesky factorization of the posterior covariance (a lower triangular matrix).

Note

Users may call draw() to draw from the posterior after calling this method.

spot(*, contrast=1.0, radius=None, lat=0.0, lon=0.0, **kwargs)

Add the expansion of a circular spot to the map.

This function adds a spot whose functional form is a top hat in \(\Delta\theta\), the angular separation between the center of the spot and another point on the surface. The spot intensity is controlled by the parameter contrast, defined as the fractional change in the intensity at the center of the spot.

Parameters

contrast (scalar or vector, optional) – The contrast of the spot. This is equal to the fractional change in the intensity of the map at the center of the spot relative to the baseline intensity of an unspotted map. If the map has more than one wavelength bin, this must be a vector of length equal to the number of wavelength bins. Positive values of the contrast result in dark spots; negative values result in bright spots. Default is 1.0, corresponding to a spot with central intensity close to zero.
radius (scalar, optional) – The angular radius of the spot in units of angle_unit. Defaults to 20.0 degrees.
lat (scalar, optional) – The latitude of the spot in units of angle_unit. Defaults to 0.0.
lon (scalar, optional) – The longitude of the spot in units of angle_unit. Defaults to 0.0.

Note

Keep in mind that things are normalized in starry such that the disk-integrated flux (not the intensity!) of an unspotted body is unity. The default intensity of an unspotted map is 1.0 / np.pi everywhere (this ensures the integral over the unit disk is unity). So when you instantiate a map and add a spot of contrast c, you’ll see that the intensity at the center is actually (1 - c) / np.pi. This is expected behavior, since that’s a factor of 1 - c smaller than the baseline intensity.

Note

This function computes the spherical harmonic expansion of a circular spot with uniform contrast. At finite spherical harmonic degree, this will return an approximation that may be subject to ringing. Users can control the amount of ringing and the smoothness of the spot profile (see below). In general, however, at a given spherical harmonic degree ydeg, there is always minimum spot radius that can be modeled well. For ydeg = 15, for instance, that radius is about 10 degrees. Attempting to add a spot smaller than this will in general result in a large amount of ringing and a smaller contrast than desired.

There are a few additional under-the-hood keywords that control the behavior of the spot expansion. These are

Parameters

spot_pts (int, optional) – The number of points in the expansion of the (1-dimensional) spot profile. Default is 1000.
spot_eps (float, optional) – Regularization parameter in the expansion. Default is 1e-9.
spot_smoothing (float, optional) – Standard deviation of the Gaussian smoothing applied to the spot to suppress ringing (unitless). Default is 2.0 / self.ydeg.
spot_fac (float, optional) – Parameter controlling the smoothness of the spot profile. Increasing this parameter increases the steepness of the profile (which approaches a top hat as spot_fac -> inf). Decreasing it results in a smoother sigmoidal function. Default is 300. Changing this parameter is not recommended; change spot_smoothing instead.

Note

These last four parameters are cached. That means that changing their value in a call to spot will result in all future calls to spot “remembering” those settings, unless you change them back!

property u

The vector of limb darkening coefficients. Read-only

To set this vector, index the map directly using one index: map[n] = ... where n is the degree of the limb darkening coefficient. This may be an integer or an array of integers. Slice notation may also be used.

property udeg: Limb darkening degree. Read-only

property wav: The wavelength(s) at which the flux is measured in units of wav_unit.

property wav_unit: An astropy.units quantity defining the angle unit for this map.

property y

The spherical harmonic coefficient vector. Read-only

To set this vector, index the map directly using two indices: map[l, m] = ... where l is the spherical harmonic degree and m is the spherical harmonic order. These may be integers or arrays of integers. Slice notation may also be used.

property ydeg: Spherical harmonic degree of the map. Read-only