Rigid Stellar Rotation and Gaussian Convolution
===============================================

The planet/stellar spectrum we observe is an integrated emission from
the top of the atmosphere over (parts of) the planet/stellar sphere. The
planet/star spin rotation yields a Doppler broadening of the
molecular/atomic line. We here demonstrate to include a rigid rotation
broadening into the simulated spectrum.

In practice, we use a specgrograph to observe the spectrum. The response
(known as instrumental profile; IP) by the spectrograph also makes the
instrumental broadening of the spectrum.

These effects are essentially a convolution of the kernel to the (raw)
spectrum.

.. code:: ipython3

    import numpy as np
    from exojax.utils.grids import wavenumber_grid

.. code:: ipython3

    nu_grid,wav,res=wavenumber_grid(23000,23100,2000,unit="AA",xsmode="premodit")


.. parsed-literal::

    xsmode =  premodit
    xsmode assumes ESLOG in wavenumber space: mode=premodit
    ======================================================================
    We changed the policy of the order of wavenumber/wavelength grids
    wavenumber grid should be in ascending order and now 
    users can specify the order of the wavelength grid by themselves.
    Your wavelength grid is in ***  descending  *** order
    This might causes the bug if you update ExoJAX. 
    Note that the older ExoJAX assumes ascending order as wavelength grid.
    ======================================================================


.. parsed-literal::

    /home/kawahara/exojax/src/exojax/utils/grids.py:145: UserWarning: Resolution may be too small. R=460768.77729497064
      warnings.warn('Resolution may be too small. R=' + str(resolution),


Let’s test using a delta function -like spectrum as an input.

.. code:: ipython3

    #1 - delta function like
    F=np.ones_like(nu_grid)
    F[1000]=0.0

The rigid rotation is characterized by a single parameter, Vsini, which
is the product of the velocity at the planet/stellar equator and sine of
the inclination of the spin axis to the observer.

.. code:: ipython3

    vsini=100.0 #km/s

We can use exojax.spec.response.rigidrot for the rigid rotation, but,
for the retrieval, we need a more efficient convolution. We try to the
latter here. We define the maximum vsini we will consider (important for
retrieval! but now just set the same value as visini) the velocity grid
for the convilution.

.. code:: ipython3

    from exojax.spec.response import ipgauss_sampling
    from exojax.spec.spin_rotation import convolve_rigid_rotation
    from exojax.utils.grids import velocity_grid
    vsini_max = 100.0
    vr_array = velocity_grid(res, vsini_max)

.. code:: ipython3

    import matplotlib.pyplot as plt
    Frot = convolve_rigid_rotation(F, vr_array, vsini, u1=0.0, u2=0.0)
    plt.plot(wav,Frot,lw=1)
    plt.show()



.. image:: Rigid_Rotation_files/Rigid_Rotation_10_0.png


You find the extreme drops at the right and left sides of the spectrum.
This is due to the convolution of the rotation kernel to the edges. So,
you need to set some margin to avoid to use the edges in the analysis.

.. code:: ipython3

    plt.plot(wav,Frot,lw=1)
    plt.xlim(23035,23065)
    plt.ylim(0.99,1.01)
    plt.show()



.. image:: Rigid_Rotation_files/Rigid_Rotation_12_0.png


We confirmed that the rotation convolution was applied to the spectrum.
The parameters of u1 and u2 are the quadratic Limb darkening
coefficients.

.. code:: ipython3

    Frot_ld = convolve_rigid_rotation(F, vr_array, vsini, u1=0.6, u2=0.4)
    plt.plot(wav,Frot,lw=1,label="u1=0,u2=0")
    plt.plot(wav,Frot_ld,lw=2,label="u1=0.6,u2=0.4")
    plt.xlim(23035,23065)
    plt.ylim(0.99,1.01)
    plt.legend()
    plt.show()



.. image:: Rigid_Rotation_files/Rigid_Rotation_14_0.png


We can also apply a Gaussian covolution and velocity shift to the
spectrum. It should be noted that beta is the standard deviation of a
Gaussian.

.. code:: ipython3

    beta1=20.0 #std of gaussian
    beta2=50.0
    RV=30.0
    Fx=ipgauss_sampling(nu_grid,nu_grid,Frot,beta1,0.0,vr_array)
    Fxx=ipgauss_sampling(nu_grid,nu_grid,Frot,beta2,RV,vr_array)

.. code:: ipython3

    plt.plot(wav[::-1],Frot,lw=1,label="Vsini=100km/s")
    plt.plot(wav[::-1],Fx,lw=1,ls="dashed",label="Vsini=100km/s $\\ast \\beta$=20km/s")
    plt.plot(wav[::-1],Fxx,lw=1,ls="dotted",label="Vsini=100km/s $\\ast \\beta$=50km/s, RV=30km/s")
    plt.legend(loc="lower left")
    plt.xlabel("wavelength $\AA$")
    plt.xlim(23030,23070)
    plt.ylim(0.99,1.005)
    plt.show()



.. image:: Rigid_Rotation_files/Rigid_Rotation_17_0.png