(Mono-)Transiting planet¶
The way in which the Keplerian function is parameterized in kima is most useful for RV detection, but can in some cases become a bottleneck.
One such case is when a planet is observed to transit only one or two times, the
so-called mono-transits or duo-transits. Here, the orbital period is uncertain
but the time of transit is known relatively well, and so the standard
parameterization in \(P, K, e, M_0, \omega\) — which is used for the
known_object as well — is not ideal.
Note
The (old) work-around of using the Gaussian_from_Tc prior for M0 will not help in cases like this.
To allow for this corner case, a transiting_planet mode has been added.
This is much alike the known_object, except that the Keplerian is
parameterized directly in terms of \(P, K, e, \omega, T_c\). In this way,
independent priors can be assigned to these parameters.
Let's see an example.
Simulating a dataset¶
First, we get some standard imports out of the way
We'll create a function to generate a simple RV dataset containing a planet. We define the time of periastron and then calculate the time of conjunction from it. To make things interesting, we remove a few data points so that the orbit is slightly harder to constrain.
def create_data():
np.random.seed(43) #(1)!
# random times and uncertainties
t = np.sort(np.random.uniform(0, 100, 57))
err = np.random.uniform(0.1, 0.3, t.size)
# define orbital parameters
P, K, e, w = 30, 2, 0.6, 0.1
Tp = 15
M0_epoch = t[0] + np.ptp(t) / 2 #(2)!
M0 = 2 * np.pi * (M0_epoch - Tp) / P
f = np.pi / 2 - w
E = 2 * np.arctan(np.tan(f / 2) * np.sqrt((1 - e) / (1 + e)))
# calculate the time of conjunction
Tc = Tp + P / (2 * np.pi) * (E - e * np.sin(E))
# calculate Keplerian signal and add white noise
v = kima.keplerian(t, P, K, e, w, M0, M0_epoch)
v = v + np.random.normal(loc=0.0, scale=0.05, size=t.size)
# remove some points to make things interesting
t = t[v < 1]
err = err[v < 1]
v = v[v < 1]
return (t, v, err), P, K, Tp, e, w, M0, M0_epoch, Tc
- so you can reproduce the same dataset
- use the mid time of the observations as the epoch
Let's plot the data together with the true Keplerian model
def plot_data_and_truth():
tt = np.linspace(t[0], t[-1], 1000)
vv = kima.keplerian(tt, P, K, e, w, M0, M0_epoch)
fig, ax = plt.subplots()
ax.errorbar(t, v, err, fmt='o', label='data')
ax.plot(tt, vv, color='g', label='true Keplerian')
ax.legend()
ax.set(xlabel='Time [days]', ylabel='RV [m/s]')
return fig
plot_data_and_truth()
A simple fit¶
Let's now fit these data, assuming only that we know there is one Keplerian signal
def simple_model():
data = RVData(t, v, err) #(1)!
model = RVmodel(fix=True, npmax=1, data=data) #(2)!
model.Jprior = distributions.Fixed(0.0) #(3)!
return model
model1 = simple_model()
- read data from the simulated arrays
- create the model with one Keplerian
- since we didn't simulate any jitter, let's just assume it's zero
Run the model for a few thousand steps
# Seeding random number generators. First seed = 1777118690.
# Generating 8 particles from the prior...done.
# Sampling...
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# Took 63.433 seconds
and now load the results
Loading files took 0.09 seconds
log(Z) = 13.49
Information = 17.27 nats
BMD = 6.75
Effective sample size = 1224.4
Plotting a few posterior samples together with the data already shows that we didn't recover the true Keplerian signal very well
def plot_samples_and_truth(res):
fig = res.plot_random_samples(legend=False, isolate_transiting_planet=False)
tt = np.linspace(t[0], t[-1], 1000)
vv = kima.keplerian(tt, P, K, e, w, M0, M0_epoch)
fig.axes[0].plot(tt, vv, color='g')
return fig
plot_samples_and_truth(res1)
and indeed the posteriors for the orbital parameters show that, while the period was recovered, the semi-amplitude and eccentricity are a bit off from the true values
def plot_corner(res, true_values, **kwargs):
fig, axs = res.corner_planet_parameters(true_values=true_values, **kwargs)
fig, axs = fig[0], axs[0] #(1)!
return fig
corner_planet_parametersreturns a list of Figures and Axes- adjust the axis limits slightly
Assuming \(T_c\) is known¶
In our convoluted example, we do have information about the time of conjunction \(T_c\). If we assume we don't know the orbital period, then we are basically in a similar situation to a mono-transit or a duo-transit.
Let's fit the data again with a model that uses one transiting planet, allowing us to assign a prior to \(T_c\) directly.
def new_model():
data = RVData(t, v, err)
model = RVmodel(fix=True, npmax=0, data=data) #(1)!
model.Jprior = distributions.Fixed(0.0)
model.set_transiting_planet(1)
model.TR_Pprior = [distributions.LogUniform(1.0, data.get_timespan())] #(2)!
model.TR_Kprior = [distributions.Uniform(0.0, 5 * data.get_RV_span())]
model.TR_eprior = [distributions.Uniform(0, 1)]
model.TR_wprior = [distributions.UniformAngle()]
model.TR_Tcprior = [distributions.Gaussian(Tc, 0.001)] #(3)!
return model
model2 = new_model()
- now there's only one transiting planet, so
npmax=0 - broad priors for the period and other parameters
- assume the time of transit is known precisely
Let's run the model again
# Seeding random number generators. First seed = 1777118856.
# Generating 8 particles from the prior...done.
# Sampling...
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# Took 77.217 seconds
Loading files took 0.09 seconds
log(Z) = 13.16
Information = 18.42 nats
BMD = 6.39
Effective sample size = 880.9
and now the true Keplerian signal is much better recovered
The posteriors for the orbital parameters (of the transiting planet) show that just by setting a prior on \(T_c\), we are now able to recover the true values of the semi-amplitude and eccentricity
def plot_comparison():
fig, axs = plt.subplots(1, 3, constrained_layout=True, figsize=(10, 4))
axs[0].axvline(P, color='k', lw=2, label='true value')
axs[1].axvline(K, color='k', lw=2, label='true value')
axs[2].axvline(e, color='k', lw=2, label='true value')
kw = dict(density=True, histtype='step')
for r in [res1, res2]:
if hasattr(r, 'TR') and r.TR:
axs[0].hist(r.TRpars[:, 0], **kw, label='transiting planet')
axs[1].hist(r.TRpars[:, 1], **kw, label='transiting planet')
axs[2].hist(r.TRpars[:, 3], **kw, label='transiting planet')
else:
axs[0].hist(r.posteriors.P.ravel(), **kw, label='normal Keplerian')
axs[1].hist(r.posteriors.K.ravel(), **kw, label='normal Keplerian')
axs[2].hist(r.posteriors.e.ravel(), **kw, label='normal Keplerian')
axs[0].set(xlabel='P [days]', ylabel='posterior')
axs[1].set(xlabel='K [m/s]', ylabel='posterior')
axs[2].set(xlabel='e', ylabel='posterior')
fig.legend(*axs[0].get_legend_handles_labels(),
ncols=3, loc='right', bbox_to_anchor=[1, 1.04])
return fig
plot_comparison()
