Quasi-periodic with cosine kernel

Quasi-periodic with cosine kernel#

This kernel has been introduced by Perger et al. 2021. The kernel has been implemented in PyORBIT without relying on any other package.

If we define \(\tau = t_i-t_j\) :

(10)#\[G(\tau ) = \exp{\frac{-2 \tau^2}{P_\mathrm{dec}^2}} * \left [ H_\mathrm{amp}^2 \exp{-\frac { \sin^2{( \pi \tau / P_\mathrm{rot} )}}{2 O_\mathrm{amp} ^2}} + C_\mathrm{amp}^2 \cos \frac{ 4\pi \tau}{P_\mathrm{rot}} \right ]\]

where \(P_\mathrm{rot}\) is equivalent to the rotation period of the star, \(O_\mathrm{amp}\) is the coherence scale, and \(P_\mathrm{dec} \) is usually associated with the decay time scale of the active regions. Within PyORBIT, the amplitude of the quasi-periodic part of the kernel \(h_1\) and the amplitude of the cosine part \(h_2\) have been labeled as Hamp and Camp respectively.

Important

As for the quasi-periodic kernel, mind the possible presence of a factor 2 in the denominator of the aperiodic variation (i.e., \(2 \lambda\) rather than \(\lambda\))

Model definition and requirements#

model name: tinygp_quasiperiodic_cosine

  • required common object: activity

  • this model relies on tinygp

  • alias: tinygp_quasiperiodiccosine

model name: gp_quasiperiodic_cosine

  • required common object: activity

  • direct implementation relying only on scipy and numpy

Model parameters#

The following parameters will be inherited from the common model (column Common?: common) or a different value will be assigned for each dataset (column Common?: dataset)

Name

Parameter

Common?

Definition

Notes

Prot

Rotational period of the star \(\theta\)

common

activity

Pdec

Decay time scale of active regions \(\lambda\)

common

activity

Oamp

Coherence scale \(w\)

common

activity

Hamp

Amplitude of the kernel

dataset

activity

Camp

Amplitude of the cosine part of the kernel

dataset

activity

Keywords#

Model-wide keywords, with the default value in boldface.

hyperparameters_condition

  • accepted values: True | False

  • activate the conditions \( \lambda ^ 2 > (3/4 \pi) \theta ^2 w ^ 2 \) (adapted from Rajpaul 2017 and Rajpaul et al. 2021 to take into account the factor 2 in the denominator of the aperiodic variation) to ensure that the QP function has at least one non-trivial turning point.

rotation_decay_condition

  • accepted values: True | False

  • if activated, it ensures that the decay time scale of the activity regions \(\lambda\) is at least twice the rotational period of the star \(\theta\)

use_stellar_rotation_period

  • accepted values: True | False

  • if activated, the parameter Prot from the activity common model will be replaced by the parameter rotation_period from the star_parameters common model. In this way, a unique parameter can be used by different models, e.g., stellar activity and Rossiter-McLaughlin modeling. It can also be useful if you want to use independent GP hyperparameters over several observational seasons while using a single parameter for the rotational period of the star.

use_stellar_activity_decay

  • accepted values: True | False

  • if activated, the parameter Pdec from the activity common model will be replaced by the parameter activity_decay from the star_parameters common model.