Surface Mapping of Earth-like Exoplanets using Single Point Light Curves

This protocol can be used to spatially resolve exoplanet features from single-point observations and is essential for evaluating the potential habitability of exoplanets. This technique can be used to reconstruct two-dimensional surface maps of Earth-like exoplanet. And this is the first technique being tested with real observations using the Earth as a proxy.

The mathematics of this technique is straightforward and can be easily adjusted for other observations. One does not need to strictly follow the coding scripts. Visual demonstration of this technique is important because one picture is worth a thousand words.

After setting up the programming environment for the attached code, enter the command to install Anaconda with Python 3.7 onto the system. After setting up the programming environment, obtain multi-wavelength light curves, view the geometry from the observations, and run the plot time series. py command to visualize the data and check their qualities.

Then enter the command to generate a geometry figure. To extract the light curve surface information, run the normalize. py command.

The output is saved in normalized light curve.csv. Enter the command to visualize the normalized light curves. A normalized light curve figure will be created.

Enter the command to decompose the normalized light curves. The resulting time series, singular values, and principal components will be saved in the appropriate output files in csv format. Use the commands to visualize the singular value decomposition result.

Figures for the singular values and principal components will be generated. Analyze the contributions and corresponding time series of the principal components to determine the one that contains surface information and compare the singular values at the diagonal of singular value matrix. An Earth-like partially cloudy exoplanet is expected to have two comparable dominant singular values.

To confirm the selection of the principal component, enter the command to obtain the power spectra of the time series of each principal component. The power spectra will be saved in periodogram.csv. Enter the command to visualize the periodograms and confirm the selection of the principal component.

A periodogram figure will be generated. The current plotting code adds dashed lines that represent the annual, semi-annual, diurnal, and half-daily cycles for reference. Select the principal component that contains the surface information and its corresponding time series.

To construct a planetary surface map, use the HEALPix random command to visualize the pixelation method. A HEALPix random figure will be created. The parameter n subside at line 17 can be changed for different resolutions.

To determine the weight of each pixel, enter the command. The output will be saved as w. npz due to its size.

Chang the n subside value at line 23 as appropriate for the other resolutions of the retrieved map. Use the plot weight. py command to visualize the weight.

A number of figures will be created in the weight folder. Merging the figures will allow illustration of how the weight of each pixel changes with time. Use the linear regression.

py command to solve the linear regression problem. The result of pixel values will be saved in the pixel value. csv file.

The value of Lambda at line 16 can be changed for different strengths of regularization as appropriate. Then run the plot map. py command to construct the retrieved maps using different regularization parameters.

Three maps will be generated. The relationship between the pixel indices and their locations on each map are described in the HEALPix document. To compute the covariance matrix of each pixel, run the covariance.

py command. The result will be saved in covariance. npz due to its size.

To visualize the covariance matrix and to map the uncertainty to the retrieved map, run the plot covariance. py command. One covariance and one uncertainty figure will be created.

Here, sample multi-wavelength observations of the Earth at 927 coordinated universal time, February 8th, 2017 are shown. Here, the time series of the two dominant principal components of the multi-wavelength light curves can be observed. The time series for the second principal component demonstrates a more regular morphology with an approximately constant daily variation and stronger diurnal cycle in its power spectrum than the first principal component.

A surface map of this proxy exoplanet consisting of the value of the second principal component at each pixel can then be constructed. Compared to the ground truth of Earth, the reconstructed map recovers all major continents after separating the surface information from the clouds. The results for the Southern Hemisphere are worse than those observed for the Northern Hemisphere due to the cloud cover over the Southern oceans.

The uncertainty of each pixel value is on the order of 10%of that in the retrieved map, suggesting a good quality of the surface mapping and a positive result. The critical requirement for applying this protocol to a future analysis is confirming that the surface information can be extracted from light curves. This technique serves as a benchmark in the surface mapping of exoplanets and may be improved with other decompensation and the regularization methods for new observations.