Emission Spectroscopy with Equilibirum Chemistry ================================================ Last update: August 24th (2025) Hajime Kawahara for v2.1 In this getting started guide, we will use ExoJAX to simulate a high-resolution emission spectrum from an atmosphere with CO molecular absorption and hydrogen molecule CIA continuum absorption as the opacity sources. We assume the thermochemical equilibrium. We will then add appropriate noise to the simulated spectrum to create a mock spectrum and perform spectral retrieval using NumPyro’s HMC NUTS. First, we recommend 64-bit if you do not think about numerical errors. Use jax.config to set 64-bit. (But note that 32-bit is sufficient in most cases. Consider to use 32-bit (faster, less device memory) for your real use case.) .. code:: ipython3 from jax import config config.update("jax_enable_x64", True) 1. Loading a molecular database using mdb ----------------------------------------- ExoJAX has an API for molecular databases, called ``mdb`` (or ``adb`` for atomic datbases). Prior to loading the database, define the wavenumber range first. .. code:: ipython3 from exojax.utils.grids import wavenumber_grid nu_grid, wav, resolution = wavenumber_grid( 22920.0, 23000.0, 3500, unit="AA", xsmode="premodit" ) print("Resolution=", resolution) .. parsed-literal:: xsmode = premodit xsmode assumes ESLOG in wavenumber space: xsmode=premodit Your wavelength grid is in *** descending *** order The wavenumber grid is in ascending order by definition. Please be careful when you use the wavelength grid. Resolution= 1004211.9840291934 .. parsed-literal:: /home/kawahara/exojax/src/exojax/utils/grids.py:85: UserWarning: Both input wavelength and output wavenumber are in ascending order. warnings.warn( Then, let’s load the molecular database. We here use Carbon monoxide in Exomol. ``CO/12C-16O/Li2015`` means ``Carbon monoxide/ isotopes = 12C + 16O / database name``. You can check the database name in the ExoMol website (https://www.exomol.com/). .. code:: ipython3 from exojax.database.exomol.api import MdbExomol mdb = MdbExomol(".database/CO/12C-16O/Li2015", nurange=nu_grid) .. parsed-literal:: /home/kawahara/exojax/src/exojax/utils/molname.py:197: FutureWarning: e2s will be replaced to exact_molname_exomol_to_simple_molname. warnings.warn( /home/kawahara/exojax/src/exojax/utils/molname.py:91: FutureWarning: exojax.utils.molname.exact_molname_exomol_to_simple_molname will be replaced to radis.api.exomolapi.exact_molname_exomol_to_simple_molname. warnings.warn( /home/kawahara/exojax/src/exojax/utils/molname.py:91: FutureWarning: exojax.utils.molname.exact_molname_exomol_to_simple_molname will be replaced to radis.api.exomolapi.exact_molname_exomol_to_simple_molname. warnings.warn( .. parsed-literal:: HITRAN exact name= (12C)(16O) radis engine = vaex Molecule: CO Isotopologue: 12C-16O ExoMol database: None Local folder: .database/CO/12C-16O/Li2015 Transition files: => File 12C-16O__Li2015.trans Broadener: H2 Broadening code level: a0 .. parsed-literal:: /home/kawahara/anaconda3/envs/myenv39/lib/python3.9/site-packages/radis-0.16-py3.9.egg/radis/api/exomolapi.py:687: AccuracyWarning: The default broadening parameter (alpha = 0.07 cm^-1 and n = 0.5) are used for J'' > 80 up to J'' = 152 warnings.warn( 2. Computation of the Cross Section using opa --------------------------------------------- ExoJAX has various opacity calculator classes, so-called ``opa``. Here, we use a memory-saved opa, ``OpaPremodit``. We assume the robust tempreature range we will use is 500-1500K. .. code:: ipython3 from exojax.opacity import OpaPremodit opa = OpaPremodit(mdb, nu_grid, auto_trange=[500.0, 1500.0], dit_grid_resolution=1.0) .. parsed-literal:: /home/kawahara/exojax/src/exojax/opacity/premodit/core.py:28: UserWarning: dit_grid_resolution is not None. Ignoring broadening_parameter_resolution. warnings.warn( .. parsed-literal:: OpaPremodit: params automatically set. default elower grid trange (degt) file version: 2 Robust range: 485.7803992045456 - 1514.171191195336 K OpaPremodit: Tref_broadening is set to 866.0254037844389 K max value of ngamma_ref_grid : 9.450919102366303 min value of ngamma_ref_grid : 7.881095721823979 ngamma_ref_grid grid : [7.88109541 9.4509201 ] max value of n_Texp_grid : 0.658 min value of n_Texp_grid : 0.5 n_Texp_grid grid : [0.49999997 0.65800005] .. parsed-literal:: uniqidx: 0it [00:00, ?it/s] .. parsed-literal:: Premodit: Twt= 1108.7151960064205 K Tref= 570.4914318566549 K Making LSD:|####################| 100% Then let’s compute cross section for two different temperature 500 and 1500 K for P=1.0 bar. opa.xsvector can do that! .. code:: ipython3 P = 1.0 # bar T_1 = 500.0 # K xsv_1 = opa.xsvector(T_1, P) # cm2 T_2 = 1500.0 # K xsv_2 = opa.xsvector(T_2, P) # cm2 Plot them. It can be seen that different lines are stronger at different temperatures. .. code:: ipython3 import matplotlib.pyplot as plt plt.plot(nu_grid, xsv_1, label=str(T_1) + "K") # cm2 plt.plot(nu_grid, xsv_2, alpha=0.5, label=str(T_2) + "K") # cm2 plt.yscale("log") plt.legend() plt.xlabel("wavenumber (cm-1)") plt.ylabel("cross section (cm2)") plt.show() .. image:: equilibrium_chemistry_files/equilibrium_chemistry_15_0.png 3. Atmospheric Radiative Transfer --------------------------------- ExoJAX can solve the radiative transfer and derive the emission spectrum. To do so, ExoJAX has ``art`` class. ``ArtEmisPure`` means Atomospheric Radiative Transfer for Emission with Pure absorption. So, ``ArtEmisPure`` does not include scattering. We set the number of the atmospheric layer to 200 (nlayer) and the pressure at bottom and top atmosphere to 100 and 1.e-5 bar. Since v1.5, one can choose the rtsolver (radiative transfer solver) from the flux-based 2 stream solver (``fbase2st``) and the intensity-based n-stream sovler (``ibased``). Use ``rtsolver`` option. In the latter case, the number of the stream (``nstream``) can be specified. Note that the default rtsolver for the pure absorption (i.e. no scattering nor reflection) has been ``ibased`` since v1.5. In our experience, ``ibased`` is faster and more accurate than ``fbased``. .. code:: ipython3 from exojax.rt import ArtEmisPure art = ArtEmisPure( nu_grid=nu_grid, pressure_btm=1.0e1, pressure_top=1.0e-5, nlayer=100, rtsolver="ibased", nstream=8, ) .. parsed-literal:: rtsolver: ibased Intensity-based n-stream solver, isothermal layer (e.g. NEMESIS, pRT like) Let’s assume the power law temperature model, within 500 - 1500 K. :math:`T = T_0 P^\alpha` where :math:`T_0=1200` K and :math:`\alpha=0.1`. .. code:: ipython3 art.change_temperature_range(500.0, 1500.0) Tarr = art.powerlaw_temperature(1200.0, 0.1) Sets chemistry presets .. code:: ipython3 from exogibbs.presets.ykb4 import prepare_ykb4_setup # chemical setup chem = prepare_ykb4_setup() idx_co = chem.species.index("C1O1") print("idx for CO=",idx_co, "JANAF name", chem.species[idx_co]) # check index of CO idx_h2 = chem.species.index("H2") print("idx for H2=",idx_h2, "JANAF name", chem.species[idx_h2]) # check index of H2 print("element:", chem.elements) .. parsed-literal:: idx for CO= 26 JANAF name C1O1 idx for H2= 1 JANAF name H2 element: ('C', 'H', 'He', 'K', 'N', 'Na', 'O', 'P', 'S', 'Ti', 'V', 'e-') Sets solar abundance (AAG21) as the elemental vector. Do not forget e-! .. code:: ipython3 from exojax.utils.zsol import nsol import jax.numpy as jnp solar_abundance = nsol() nsol_vector = jnp.array([solar_abundance[el] for el in chem.elements[:-1]]) # no solar abundance for e- element_vector = jnp.append(nsol_vector, 0) print("element_vector:", element_vector) .. parsed-literal:: Database for solar abundance = AAG21 Asplund, M., Amarsi, A. M., & Grevesse, N. 2021, arXiv:2105.01661 element_vector: [2.66271344e-04 9.23260873e-01 7.57398483e-02 1.08473694e-07 6.24200958e-05 1.53223166e-06 4.52193620e-04 2.37314585e-07 1.21709487e-05 8.61637180e-08 7.33372179e-09 0.00000000e+00] The mass mixing ratio of CO (MMR) should be computed based on the thermochemical equilibirum. .. code:: ipython3 from exogibbs.api.equilibrium import equilibrium_profile, EquilibriumOptions from exojax.atm.atmconvert import vmr_to_mmr from exojax.database.molinfo.mass import isotope_molmass # Thermodynamic conditions Pref = 1.0 # bar, reference pressure opts = EquilibriumOptions(epsilon_crit=1e-11, max_iter=1000) res = equilibrium_profile( chem, Tarr, art.pressure, element_vector, Pref=Pref, options=opts, ) nk_result = res.x vmr_co = nk_result[:, idx_co] vmr_h2 = nk_result[:, idx_h2] mean_molecular_weight = 2.33 ## assume constant (not accurate) molmass = isotope_molmass("12C-16O") mmr_profile = vmr_to_mmr(vmr_co, molmass, mean_molecular_weight) mmr_profile_h2 = vmr_to_mmr(vmr_h2, isotope_molmass("1H2"), mean_molecular_weight) import matplotlib.pyplot as plt fig = plt.figure() ax = fig.add_subplot(111) ax.plot(mmr_profile, art.pressure, label="CO") ax.plot(mmr_profile_h2, art.pressure, ls="--", label="H2") ax.invert_yaxis() ax.legend() ax.set_xscale("log") ax.set_yscale("log") ax.set_xlabel("mmr") ax.set_ylabel("Pressure (bar)") plt.show() .. parsed-literal:: HITRAN exact name= (12C)(16O) HITRAN exact name= H2 .. parsed-literal:: /home/kawahara/exojax/src/exojax/utils/molname.py:91: FutureWarning: exojax.utils.molname.exact_molname_exomol_to_simple_molname will be replaced to radis.api.exomolapi.exact_molname_exomol_to_simple_molname. warnings.warn( /home/kawahara/exojax/src/exojax/utils/molname.py:91: FutureWarning: exojax.utils.molname.exact_molname_exomol_to_simple_molname will be replaced to radis.api.exomolapi.exact_molname_exomol_to_simple_molname. warnings.warn( .. image:: equilibrium_chemistry_files/equilibrium_chemistry_26_2.png Surface gravity is also important quantity of the atmospheric model, which is a function of planetary radius and mass. Here we assume 1 RJ and 10 MJ. .. code:: ipython3 from exojax.utils.astrofunc import gravity_jupiter gravity = gravity_jupiter(1.0, 10.0) In addition to the CO cross section, we would consider `collisional induced absorption `__ (CIA) as a continuum opacity. ``cdb`` class can be used. .. code:: ipython3 from exojax.database.contdb import CdbCIA from exojax.opacity import OpaCIA cdb = CdbCIA(".database/H2-H2_2011.cia", nurange=nu_grid) opacia = OpaCIA(cdb, nu_grid=nu_grid) .. parsed-literal:: H2-H2 Before running the radiative transfer, we need cross sections for layers, called ``xsmatrix`` for CO and ``logacia_matrix`` for CIA (strictly speaking, the latter is not cross section but coefficient because CIA intensity is proportional density square). See `here `__ for the details. .. code:: ipython3 xsmatrix = opa.xsmatrix(Tarr, art.pressure) logacia_matrix = opacia.logacia_matrix(Tarr) Convert them to opacity .. code:: ipython3 dtau_CO = art.opacity_profile_xs(xsmatrix, mmr_profile, mdb.molmass, gravity) #vmrH2 = 0.855 # VMR of H2 dtaucia = art.opacity_profile_cia(logacia_matrix, Tarr, vmr_h2, vmr_h2, mean_molecular_weight, gravity) Add two opacities. .. code:: ipython3 dtau = dtau_CO + dtaucia Then, run the radiative transfer. As you can see, the emission spectrum has been generated. This spectrum shows a region near 4360 cm-1, or around 22940 AA, where CO features become increasingly dense. This region is referred to as the band head. If you’re interested in why the band head occurs, please refer to `Quatum states of Carbon Monoxide and Fortrat Diagram `__. .. code:: ipython3 F = art.run(dtau, Tarr) fig = plt.figure(figsize=(15, 4)) plt.plot(nu_grid, F) plt.xlabel("wavenumber (cm-1)") plt.ylabel("flux (erg/s/cm2/cm-1)") plt.show() .. image:: equilibrium_chemistry_files/equilibrium_chemistry_38_0.png You can check the contribution function too! You should check if the dominant contribution is within the layer. If not, you need to change ``pressure_top`` and ``pressure_btm`` in ``ArtEmisPure`` .. code:: ipython3 from exojax.plot.atmplot import plotcf .. code:: ipython3 cf = plotcf(nu_grid, dtau, Tarr, art.pressure, art.dParr) .. image:: equilibrium_chemistry_files/equilibrium_chemistry_41_0.png 4. Spectral Operators: rotational broadening, instrumental profile, Doppler velocity shift and so on, any operation on spectra. ------------------------------------------------------------------------------------------------------------------------------- The above spectrum is called “raw spectrum” in ExoJAX. The effects applied to the raw spectrum is handled in ExoJAX by the spectral operator (``sop``). First, we apply the spin rotational broadening of a planet. .. code:: ipython3 from exojax.postproc.specop import SopRotation sop_rot = SopRotation(nu_grid, vsini_max=100.0) vsini = 10.0 u1 = 0.0 u2 = 0.0 Frot = sop_rot.rigid_rotation(F, vsini, u1, u2) .. code:: ipython3 fig = plt.figure(figsize=(15, 4)) plt.plot(nu_grid, F, label="raw spectrum") plt.plot(nu_grid, Frot, label="rotated") plt.xlabel("wavenumber (cm-1)") plt.ylabel("flux (erg/s/cm2/cm-1)") plt.legend() plt.show() .. image:: equilibrium_chemistry_files/equilibrium_chemistry_45_0.png Then, the instrumental profile with relative radial velocity shift is applied. Also, we need to match the computed spectrum to the data grid. This process is called ``sampling`` (but just interpolation though). Below, let’s perform a simulation that includes noise for use in later analysis. .. code:: ipython3 from exojax.postproc.specop import SopInstProfile from exojax.utils.instfunc import resolution_to_gaussian_std sop_inst = SopInstProfile(nu_grid, vrmax=1000.0) RV = 40.0 # km/s resolution_inst =70000.0 beta_inst = resolution_to_gaussian_std(resolution_inst) Finst = sop_inst.ipgauss(Frot, beta_inst) nu_obs = nu_grid[::5][:-50] from numpy.random import normal noise = 500.0 Fobs = sop_inst.sampling(Finst, RV, nu_obs) + normal(0.0, noise, len(nu_obs)) .. code:: ipython3 fig = plt.figure(figsize=(12, 6)) ax = fig.add_subplot(211) plt.plot(nu_grid, Frot, label="rotated") plt.plot(nu_grid, Finst, label="rotated+IP") plt.ylabel("flux (erg/s/cm2/cm-1)") plt.legend() ax = fig.add_subplot(212) plt.errorbar(nu_obs, Fobs, noise, fmt=".", label="rotated + RV + IP (sampling)", color="gray",alpha=0.5) plt.xlabel("wavenumber (cm-1)") plt.legend() plt.show() .. image:: equilibrium_chemistry_files/equilibrium_chemistry_48_0.png 5. Retrieval of an Emission Spectrum ------------------------------------ Next, let’s perform a “retrieval” on the simulated spectrum created above. Retrieval involves estimating the parameters of an atmospheric model in the form of a posterior distribution based on the spectrum. To do this, we first need a model. Here, we have compiled the forward modeling steps so far and defined the model as follows. The spectral model has six parameters. .. code:: ipython3 from jax import jit soleve_thermochemical_equilibirum = jit(lambda T, P, b_element_vector: equilibrium_profile(chem, T, P, b_element_vector, Pref=Pref, options=opts)) .. code:: ipython3 def fspec(T0, alpha, g, RV, vsini, b_element_vector_in): #molecule Tarr = art.powerlaw_temperature(T0, alpha) xsmatrix = opa.xsmatrix(Tarr, art.pressure) # MMR profile from equilibrium chemistry res = soleve_thermochemical_equilibirum(Tarr, art.pressure, b_element_vector_in) nk_result = res.x vmr_co = nk_result[:, idx_co] mmr_arr = vmr_to_mmr(vmr_co, molmass, mean_molecular_weight) vmr_h2 = nk_result[:, idx_h2] #opacity dtau = art.opacity_profile_xs(xsmatrix, mmr_arr, molmass, g) #continuum logacia_matrix = opacia.logacia_matrix(Tarr) dtaucH2H2 = art.opacity_profile_cia(logacia_matrix, Tarr, vmr_h2, vmr_h2, mean_molecular_weight, g) #total tau dtau = dtau + dtaucH2H2 F = art.run(dtau, Tarr) Frot = sop_rot.rigid_rotation(F, vsini, u1, u2) Finst = sop_inst.ipgauss(Frot, beta_inst) mu = sop_inst.sampling(Finst, RV, nu_obs) return mu Let’s verify that spectra are being generated from ``fspec`` with various parameter sets. .. code:: ipython3 fig = plt.figure(figsize=(12, 3)) plt.plot(nu_obs, fspec(1200.0, 0.09, gravity_jupiter(1.0, 1.0), 40.0, 10.0, element_vector),label="model") plt.plot(nu_obs, fspec(1100.0, 0.12, gravity_jupiter(1.0, 10.0), 20.0, 5.0, element_vector),label="model") .. parsed-literal:: [] .. image:: equilibrium_chemistry_files/equilibrium_chemistry_54_1.png NumPyro is a probabilistic programming language (PPL), which requires the definition of a probabilistic model. In the probabilistic model ``model_prob`` defined below, the prior distributions of each parameter are specified. The previously defined spectral model is used within this probabilistic model as a function that provides the mean :math:`\mu`. The spectrum is assumed to be generated according to a Gaussian distribution with this mean and a standard deviation :math:`\sigma`. i.e. :math:`f(\nu_i) \sim \mathcal{N}(\mu(\nu_i; {\bf p}), \sigma^2 I)`, where :math:`{\bf p}` is the spectral model parameter set, which are the arguments of ``fspec``. .. code:: ipython3 from numpyro.infer import MCMC, NUTS import numpyro.distributions as dist import numpyro from jax import random .. code:: ipython3 from exogibbs.api.chemistry import element_indices_by_name, update_element_vector # Compute indices once (outside jit/NumPyro tracing) _idx_CO = element_indices_by_name(chem, ['C', 'O']) _idx_C, _idx_O = map(int, list(_idx_CO)) .. code:: ipython3 def model_prob(spectrum): # atmospheric/spectral model parameters priors logg = numpyro.sample("logg", dist.Uniform(4.0, 5.0)) RV = numpyro.sample("RV", dist.Uniform(35.0, 45.0)) T0 = numpyro.sample("T0", dist.Uniform(1000.0, 1500.0)) alpha = numpyro.sample("alpha", dist.Uniform(0.05, 0.2)) vsini = numpyro.sample("vsini", dist.Uniform(5.0, 15.0)) logZ = numpyro.sample("logZ", dist.Uniform(-1.0, 1.0)) # logC [solar] scale = 10**logZ # Build element vector in a JAX-safe way (scale C/O; set e- to 0) element_vector_in = update_element_vector( element_vector, scale_indices=jnp.array([_idx_C,_idx_O]), scales=jnp.array([scale,scale]), ) mu = fspec(T0, alpha, 10**logg, RV, vsini, element_vector_in) # noise model parameters priors sigmain = numpyro.sample("sigmain", dist.Exponential(1.0e-3)) numpyro.sample("spectrum", dist.Normal(mu, sigmain), obs=spectrum) Note that we did not account for the effects of limb darkening. However, in actual analyses, one possible approach might be to use an uninformative prior, such as the one proposed by Kipping. .. code:: python from exojax.postproc.limb_darkening import ld_kipping q1 = numpyro.sample('q1', dist.Uniform(0.0,1.0)) q2 = numpyro.sample('q2', dist.Uniform(0.0,1.0)) u1,u2 = ld_kipping(q1,q2) Now, let’s define NUTS and start sampling. .. code:: ipython3 rng_key = random.PRNGKey(0) rng_key, rng_key_ = random.split(rng_key) num_warmup, num_samples = 500, 1000 #kernel = NUTS(model_prob, forward_mode_differentiation=True) kernel = NUTS(model_prob, forward_mode_differentiation=False) Since this process will take several hours, feel free to go for a long lunch break! .. code:: ipython3 mcmc = MCMC(kernel, num_warmup=num_warmup, num_samples=num_samples) mcmc.run(rng_key_, spectrum=Fobs) mcmc.print_summary() .. parsed-literal:: sample: 100%|██████████| 1500/1500 [15:38:27<00:00, 37.54s/it, 255 steps of size 8.95e-03. acc. prob=0.94] .. parsed-literal:: mean std median 5.0% 95.0% n_eff r_hat RV 40.06 0.08 40.06 39.95 40.20 676.46 1.00 T0 1207.13 14.56 1206.39 1183.92 1230.80 395.24 1.00 alpha 0.11 0.01 0.11 0.09 0.14 419.41 1.00 logZ -0.04 0.06 -0.04 -0.14 0.06 399.38 1.00 logg 4.32 0.12 4.31 4.12 4.52 401.06 1.00 sigmain 498.59 15.17 497.60 474.93 525.32 613.82 1.00 vsini 9.65 0.16 9.65 9.37 9.90 623.04 1.00 Number of divergences: 0 .. parsed-literal:: After returning from your long lunch, if you’re lucky and the sampling is complete, let’s write a predictive model for the spectrum. .. code:: ipython3 from numpyro.diagnostics import hpdi from numpyro.infer import Predictive import jax.numpy as jnp .. code:: ipython3 # SAMPLING posterior_sample = mcmc.get_samples() pred = Predictive(model_prob, posterior_sample, return_sites=['spectrum']) predictions = pred(rng_key_, spectrum=None) median_mu1 = jnp.median(predictions['spectrum'], axis=0) hpdi_mu1 = hpdi(predictions['spectrum'], 0.9) .. code:: ipython3 fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(15, 4.5)) ax.plot(nu_obs, median_mu1, color='C1') ax.fill_between(nu_obs, hpdi_mu1[0], hpdi_mu1[1], alpha=0.3, interpolate=True, color='C1', label='90% area') ax.errorbar(nu_obs, Fobs, noise, fmt=".", label="mock spectrum", color="black",alpha=0.5) plt.xlabel('wavenumber (cm-1)', fontsize=16) plt.legend(fontsize=14) plt.tick_params(labelsize=14) plt.show() .. image:: equilibrium_chemistry_files/equilibrium_chemistry_67_0.png .. code:: ipython3 #save the result import arviz idata = arviz.from_numpyro(mcmc, posterior_predictive=predictions, coords = {"wavenumber": nu_obs,},dims = {"spectrum": ["wavenumber"],}) arviz.to_netcdf(idata, "posterior_logZ.nc") .. parsed-literal:: 'posterior_logZ.nc' You can see that the predictions are working very well! Let’s also display a corner plot. Here, we’ve used ArviZ for visualization. .. code:: ipython3 import arviz pararr = ['T0', 'alpha', 'logg', 'logZ', 'vsini', 'RV'] arviz.plot_pair(arviz.from_numpyro(mcmc), kind='kde', divergences=False, marginals=True) plt.show() .. image:: equilibrium_chemistry_files/equilibrium_chemistry_70_0.png We see the strong degeneracy between metalicity and gravity!!!