TTV and RV fit

TTV and RV fit#

In this example we will model the planetary system around TOI-1130 (Huang et al. 2020) , using the Transit Timing Variations (TTVs) from [Borsato et al. 2024] (https://ui.adsabs.harvard.edu/abs/2024A%26A…689A…52B/abstract) and the Radial Velocity (RV)measurements from Korth et al. 2023

Note

Both PyORBIT and TRADES measure and predict the times of inferior cinjunction \(T_c\) , which may differ from the central times of transit for eccentric orbits. For short, we will call the former transit times

Preparing the datasets#

The dataset must be prepared following the instructions in Times of inferior conjunction (transit time) for the TTV files, and How to properly format a dataset for the RV files.

Keep in mind the following facts when preparing the dataset files:

  • The code will assume that the transit time measurements included in a file belong to the same planet, so you cannot mix \(T_c\) measurements of different planets within the same file.

  • You cannot include in the same file two \(T_c\) measurements referring to the same transits, e.g. from two different instruments.

Each dataset will have its corresponding jitter parameter. To use the same jitter parameter across different datasets, use the common_jitter model.

 1inputs:
 2  TCdata_b:
 3    file: ../dataset_pyorbit/borsato24_t0_planetb_MOD.dat
 4    kind: transit_time
 5    models: ttv_b
 6  TCdata_c:
 7    file: ../dataset_pyorbit/borsato24_t0_planetc_MOD.dat
 8    kind: transit_time
 9    models: ttv_c
10  HARPS_RVdata:
11    file: ../dataset_pyorbit/korth23_harps_rvs.dat
12    kind: RV
13    models:
14      - radial_velocities
15  PFS_RVdata:
16    file: ../dataset_pyorbit/korth23_pfs_rvs.dat
17    kind: RV
18    models:
19      - radial_velocities

The planet model#

When modelling the radial velocities or the transit of non-interacting planets, the relative geometry of the orbits (how the orbital planes are positioned with respect to each other) is not relevant, and that allows us to fix several orbital parameters without affecting the final outcome. For example, it is customary to fix the longitude of the ascending note \(\Omega\) (either to \(0^{\circ}\) or \(180^{\circ}\)) and set the eccentricity to zero (circular orbit). For the modelling of radial velocity, we don’t even need to know the orbital inclination or the mass of the planets.

Dynamical modelling works differently. First of all, we need to include the true masses of the planets (and therefore also the mass of the star) to properly compute how they interact with each other. We must also know how the orbits are oriented with each other, using one of the planet as reference. These are the orbital parameter we have to include for each planet:

Name

Symbol

Parameter

Unit

P

\(P\)

Orbital period of the planet

days

mean_long

\(L_0\)

Mean longitude of the orbit at \(T_{\rm ref}\)

degrees

e

\(e\)

eccentricity of the orbit

adimensional

omega

\(\omega_p\)

argument of periastron of the planet

degrees

M_Me

\(M_{\rm p}\)

planet mass in Earth masses

\(M_\oplus\)

i

\(i\)

orbital inclination with respect to the plane of the sky

degrees

Omega

\(\Omega\)

longitude of the ascending note

degrees

To force the use of the mass and the orbital inclination, we have to activate the corresponding flags in the planet model:

    planets:
      b:
        use_mass: True
        use_inclination: True

In transit fitting, the maximum value allowed for the inclination is set to \(90^{\circ}\), as two inclinations equidistant from \(90^{\circ}\) would produce the same signal. For the same reason, the impact parameter \(b\) is forced to be positive.

In dynamical modelling, mutual inclinations play a role: two planets \(i_b=87^{\circ}\) and \(i_c=93^{\circ}\) would produce the same transit signal of two planets with \(i_b=i_c=87^{\circ}\), but different dynamical signature.

After fixing the quadrant of the reference planet by forcing its inclination below \(90^{\circ}\), we must allow for the other planets to expler both quadrants. If we are using the impact parameter \(b\), we have to allow for negative values for all the planets except the first one.

For similar reasons, we fix the longitude of the ascending node the first planet \(\Omega\) to \(180^{\circ}\), and leave it free for the other planets.

In the example below, planet \(b\) acts as the reference planet. For planet \(c\), we constrained \(\Omega\) to avoid retrograde orbits.

20  common:
21    planets:
22      b:
23        orbit: dynamical
24        parametrization: Eastman2013
25        use_mass: True
26        use_inclination: True
27        boundaries:
28          P: [4.00, 4.10]
29          M_Me: [10., 30.0]
30          i: [85., 90.]
31          e: [0.00, 0.20]
32        priors:
33          i: ['Gaussian', 87.5, 0.1]
34          P: ['Gaussian', 4.074554, 0.01]
35        spaces:
36          P: Linear
37          M_Me: Linear
38        fixed:
39          Omega: 180.000
40      c:
41        orbit: dynamical
42        parametrization: Eastman2013
43        use_mass: True
44        use_inclination: True
45        boundaries:
46          P: [8.30, 8.40]
47          M_Me: [300.0, 350.0]
48          i: [85., 95.]
49          e: [0.00, 0.20]
50          Omega: [90.0, 270.]
51        priors:
52          #i: ['Gaussian', 87.6, 0.1]
53          P: ['Gaussian', 8.3501898, 0.01]
54        spaces:
55          P: Linear
56          M_Me: Linear