Planets

Parameters definition

Each planet is defined by the following parameters

Name	Parameter	Unit
P	Orbital period of the planet	days
K	Radial velocity semiamplitude	m/s
Tc	Central time of transit	days
mean_long	Mean longitude of the orbit, assuming \(\Omega=0\)	degrees
e	eccentricity of the orbit	adimensional
omega	argument of periastron of the planet \(\omega_p\)	degrees
e_coso	\( e \cos{\omega_p}\) in Ford 2006 parametrization	adimensional
e_sino	\( e \sin{\omega_p}\) in Ford 2006 parametrization	adimensional
sre_coso	\( \sqrt{e} \cos{\omega_p}\) in Eastman et al. 2013 parametrization	adimensional
sre_sino	\( \sqrt{e} \sin{\omega_p}\) in Eastman et al. 2013 parametrization	adimensional
M_Me	planet mass in Earth masses	\(M_\oplus\)
R_Rs	planet radius in stellar radii	\(R_\star\)
a_Rs	semi-major axis of th eorbit in stellar radii	\(R_\star\)
b	impact parameter	\(R_\star\)
i	orbital inclination with respect to the plane of the sky	degrees
Omega	longitude of the ascending note \(\Omega\)	degrees
lambda	Spin-orbit alignment angle \(\lambda\)	degrees
delta_occ	Depth of the occulation	Normalized flux
phase_amp	Amplitude of the phase curve, if no occultation was present	Normalized flux
phase_off	Offset of the peak of the phase curve (light travel time effect not included)	degrees

Warning

PyORBIT uses the argument of pericenter of the planet \(\omega_p\), while other packages may use the argument of pericenter of the star \(\omega _\star\). Some papers report the latter without specifying the \(\star\) pedix.

Keywords

The default keyword is highlighted in boldface.

orbit

• accepted values: circular | keplerian | dynamical

• define if the planet is moving on a circular orbit (\(e=0\), \(\omega=90°\)), a standard Keplerian, or on an orbit computed through N-body integration

parametrization

• accepted values: Standard | Standard_Tcent | Ford2006 | Ford2006_Tcent | Eastman2013 | Eastman2013_Tcent

• define the parametrization for eccentricity and argument of periastron: (\(e\), \(\omega_p\)) for Standard, (\(e \sin{\omega_p}\), \(e \cos{\omega_p}\)) for Ford2006 (Ford 2006), (\(\sqrt{e} \sin{\omega_p}\), \(\sqrt{e} \cos{\omega_p}\)) for Eastman2013 (Eastman et al. 2013).

• Appending _Tcent to the parametrization label will replace the mean longitude mean_long with the central time of transit Tc.

use_inclination

• accepted values: True | False

• if True, the inclination of the planet i replaces the impact parameter b.

use_semimajor_axis

• accepted values: True | False

• if True, the scaled semimajor axis of the planet a_Rs replaces the stellar density (defined in the star section).

use_time_inferior_conjunction

• accepted values: True | False / overridden by parametrization

• alternative way to replace the mean longitude mean_long with the central time of transit Tc when set to True. It is overridden by parametrization if its value ends with _Tcent

use_mass_for_planets

• accepted values: True | False

• if False, the mass of the planet replaces the radial velocity semiamplitude. It should be activated with those methods that allow the determination of the true mass of a planet, e.g., TTVs

Examples

The most basic example is provided by a non-transiting planet in a circular orbit. For purely coding reasons, the dictionary of a planet (in this case, b) cannot be empty, so we provide just one of the keywords:

common:
  planets:
    b:
      orbit: circular

The code will automatically pick the parametrization to be employed (here, the mean longitude as the planet is not transiting), the way the parameter space is explored (here, Base-2 Logarithm for period and RV semi-amplitude), the boundaries for the parameters (in sampler space):

----- common model:  b
mean_long     id:   0  s:Linear      b:[      0.0000,     360.0000]   p:Uniform   []
P             id:   1  s:Log_Base2   b:[     -1.3219,      16.6096]   p:Uniform   []
K             id:   2  s:Log_Base2   b:[     -9.9658,      10.9658]   p:Uniform   []
omega         derived (no id, space, bound)                           p:None   []
e             derived (no id, space, bound)                           p:None   []

The default behavior of PyORBIT is to set boundaries and parametrizations according to hard-coded when not explicitly provided in the configuration files. While in most cases the default values are fine (e.g., angles, offsets, jitter parameters), sometimes it may be useful to restrict the range explored by the parameters:

common:
  planets:
    b:
      orbit: circular
      boundaries:
        P: [0.50, 500.0]
        K: [0.01, 300.0]

Warning

Boundaries and priors must be expressed in the physical/natural space, even if the parameters space will be explored in logarithmic space. If the parameters are explored in logarithmic space, the boundaries must be strictly positive.

The logarithmic parametrization allows an efficient exploration of the parameter space across several orders of magnitude. When the parameter space is restricted by tight boundaries or a prior is provided, as in the case of transiting planets (i.e., well-known period and RV semi-amplitude within an expected range), then the space exploration can be switched to Linear (if it is not already in that space):

common:
  planets:
    b:
      orbit: circular
      use_time_inferior_conjunction: True
      boundaries:
        P: [2.20, 2.25]
        Tc: [2456194.00, 2456194.1.0]
        K: [190.0, 220.0]
      priors:
        P: ['Gaussian', 2.218574944, 0.000000030]
        Tc: ['Gaussian', 2456194.067619, 0.000034]
      spaces:
        P: Linear
        K: Linear

Finally, keep in mind that spaces, boundaries, and priors may change depending on the datasets you are fitting. In the case of transit photometry, for example, you may want to specify boundaries for the scaled planetary radius, while dropping the boundaries for the RV semi-amplitude.

common:
  planets:
    b:
      orbit: keplerian
      parametrization: Eastman2013
      use_time_inferior_conjunction: True
      use_inclination: False    # can be omitted when default value is used
      use_semimajor_axis: False # can be omitted when default value is used
      boundaries:
        P: [2.20, 2.25]
        Tc: [2456194.00, 2456194.1.0]
        R_Rs: [0.00, 1.00]
        e: [0.00, 0.90]
      priors:
        P: ['Gaussian', 2.218574944, 0.000000030]
        Tc: ['Gaussian', 2456194.067619, 0.000034]
      spaces:
        P: Linear
        K: Linear

In this case, we also required to use a keplerian orbit with the Eastman et al. 2013 parametrization.