How orbitize! Works

orbitize! turns data into orbits. More specifically, relative kinematic measure- ments of a secondary (which I will refer to as a planet throughout this section, but which could just as easily be a white dwarf, MS star, or other massive companion) and its primary are converted to posteriors over orbital parameters through Bayesian analysis.

orbitize! hinges on the two-body problem, which describes the paths of two bodies gravitationally bound to each other. The solution of the two-body problem (see Seager, 2010 Chapter 2) describes the motion of each body as a function of time, given parameters determining the position and velocity of both objects at a particular epoch. There are many basis sets (orbital bases) that can be used to describe an orbit, which can then be solved using Kepler’s equation. It is important, then, to be explicit about coordinate systems. I coded up an interactive visualization to define and help users understand our coordinate system, which you can check out here2. Blunt et al. (2020) also hosts a video of myself walking through a recording where I use and explain the coordinate system. Several still frames from that video are reproduced in Figure 4.1.

In its “standard” mode, orbitize! assumes that the user only has relative astro- metric data to fit. To obtain these measurements, an astronomer takes an image containing two point sources3 and measures the position of the planet relative to the star in angular coordinates. In theorbitize! coordinate system, relative R.A.

and decl. can be expressed as the following functions of orbital parameters (Green, 1985):

2https://github.com/sblunt/orbitize/blob/main/docs/tutorials/show-me-the-orbit.ipynb

3This is extremely simplified and not exactly accurate for high-contrast imaging, but I do not think additional detail would be beneficial here.

Figure 4.1: Stills from the show-me-the-orbit visualization of theorbitize! co- ordinate system. Positive radial velocity is described as moving away from the observer, andΩ=𝜔=0 describes an orbit with periastron pointed towards the North celestial pole.

ΔR.A. =𝜋 𝑎(1−𝑒cos E)

cos² 𝑖

2sin(𝑓 +𝜔

p+Ω) −sin² 𝑖

2sin(𝑓 +𝜔

p−Ω))

Δdecl. =𝜋 𝑎(1−𝑒cos E)

cos² 𝑖

2cos(𝑓 +𝜔

p+Ω) +sin² 𝑖

2cos(𝑓 +𝜔

p−Ω)

(4.1) where𝑎, 𝑒, 𝜔

p, Ω, and𝑖 are orbital parameters, and 𝜋 is the system parallax. f is the true anomaly, and E is the eccentric anomaly, which are related to elapsed time through Kepler’s equation and Kepler’s third law:

𝑀 =2𝜋( 𝑡 𝑃

− (𝜏−𝜏

ref)) ( 𝑃

𝑦𝑟

)²=( 𝑎 𝑎𝑢

)³( 𝑀_⊙ 𝑀_{𝑡 𝑜𝑡}

) 𝑀 =E−𝑒sin E 𝑓 =2 tan⁻¹

"√︂

1+𝑒 1−𝑒tan

𝐸 2 )

#

(4.2)

Kepler’s equation is transcendental and nonlinear, and must in general be solved numerically. orbitize! employs two Kepler solvers to convert between mean and eccentric anomaly: one that is efficient for the highest eccentricities (Mikkola, 1987), and Newton’s method in other cases, which is more efficient for the average orbit needing to be solved. See Blunt et al. (2020) for more detail.

From scrutinizing the set of equations 4.1 and 4.2, it is possible to understand how relative astrometry measurements constrain orbital parameters, and to understand a few important inherent degeneracies. First, notice that the individual component masses do not show up anywhere in this equation set. Thus, it is impossible to measure dynamical masses for either the primary or the secondary using just relative astrometry. That said, if the mass of the planet can be safely assumed to be negligible compared to the mass of the star, then the total mass derived from Keplerian analysis can be treated as a constraint on the dynamical mass of the primary. In practice, the reverse logic is often employed: an independent constraint on the mass of the primary (from e.g., spectroscopic analysis) is used as a prior on the total mass when the planet mass is small and can be ignored.

A second important degeneracy is between semimajor axis 𝑎, total mass 𝑀

tot, and parallax𝜋. If we just had relative astrometric measurements and no external knowl- edge of the system parallax, we would not be able to distinguish between a system that has larger distance and larger semimajor axis (and therefore larger total mass, assuming a fixed period) from a system that has smaller distance, smaller semimajor axis, and smaller total mass. Luckily, we live in an era where parallax measurements are excellent overall thanks to the Gaia mission (Gaia Collaboration et al., 2016), and strict priors can often be applied to parallax, breaking the degeneracy and enabling dynamical mass measurements of stars (when planet mass is negligible). However, this degeneracy is important to understand when considering the impact of potential biases in parallax or stellar mass measurements.

Figure 4.2: Posterior distributions over Ω and𝜔_𝑝 for GJ 504 b, computed via the OFTI algorithm using data from Kuzuhara et al. (2013). The 180^◦ symmetry in both posteriors, discussed in the text, is apparent.

A final degeneracy I would like to point out concerns the argument of periastron𝜔_𝑃 and the position angle of nodesΩ. Equation 4.1 is invariant to the transformation:

𝜔

′

𝑝=𝜔_𝑝+𝜋 Ω^′ = Ω−𝜋

(4.3) which creates a 180^◦ degeneracy between particular values of 𝜔_𝑝 and Ω, and a characteristic “double-peaked” structure in marginalized 1D posteriors of these parameters (see Figure 4.2 for an example). Physically, this degeneracy comes about because relative astrometry alone only constrains motion in the plane of the sky; an orbit tilted toward the observer, with the planet moving away from the observer has the same projection on the plane of the sky as an orbit tilted away from the observer, with the planet moving toward the observer. In practice, this degeneracy is handy, because if the𝜔_𝑝/Ω posteriors do not appear identical before and after 180^◦, it is generally an indication that the MCMC chains are unconverged.

Although the degeneracies between individual component masses, parallax and semimajor axis, and “the omegas” are baked into the equations describing relative

astrometric measurements, the small orbital fractions over which directly imaged planets are observed introduces additional uncertainties and degeneracies. Because direct imaging today is sensitive to only planets at large separations from their stars (≳200𝑚 𝑎 𝑠), directly imaged planets have orbital periods of in the best cases years, and in the worst cases thousands or tens of thousands of years (Bowler, 2016).

The short orbital fractions over which we can observe these objects translates to often broad orbital posteriors which are similar to the priors. A common recurring degeneracy for incomplete orbits is between eccentricity and inclination (Blunt et al., 2019; Ferrer-Chávez, Wang, and Blunt, 2021; Chapter 3 of this thesis); in the absence of complete information about the orbital acceleration over the entire orbit, the observed data can be equally well explained by a more inclined, circular orbit, and by a more face-on, eccentric orbit. Directly imaged planets often have astrometric measurements that are change approximately linearly in time, which translates to:

1) broad posteriors, and 2) strong eccentricity-inclination degeneracies.

Because the shapes of the posteriors of directly imaged planet orbits are often mul- timodal, highly covariant, and non-Gaussian, many standard posterior computation tools, such as the widely used Affine-invariant MCMC sampleremcee (Foreman- Mackey et al., 2013) struggle to converge. This motivated the development of the OFTI algorithm, and the incorporation of other specialized sampling tools, such as MCMC with parallel tempering (Vousden, Farr, and Mandel, 2016), into orbitize!.

For more details about how orbitize! sets up likelihood, defines priors, and computes posteriors, I refer readers to Blunt et al. (2020) and Blunt et al. (2017).

4.3 Jointly Fitting Radial Velocities