High-contrast medium resolution spectroscopy has been used to detect molecules such as water and carbon monoxide in the atmospheres of gas giant exoplanets. In this work, we show how it can be used to derive radial velocity (RV) measurements of directly imaged exoplanets. Improving upon the traditional cross-correlation technique, we develop a new likelihood based on joint forward modelling of the planetary signal and the starlight background (i.e., speckles). After marginalizing over the starlight model, we infer the barycentric RV of HR 8799 b and c in 2010 yielding −9.2 ± 0.5 km/s and −11.6 ± 0.5 km/s respectively. These RV measurements help to constrain the 3D orientation of the orbit of the planet by resolving the degeneracy in the longitude of ascending node. Assuming coplanar orbits for HR 8799 b and c, but not including d and e, we estimate Ω = 89 • +27 −17 and i = 20.8 • +4.5 −3.7 .

The four directly imaged planets orbiting the star HR 8799 are an ideal laboratory to probe atmospheric physics and formation models. We present more than a decade’s worth of Keck/OSIRIS observations of these planets, which represent the most detailed look at their atmospheres to date by its resolution and signal-to-noise ratio. We present the first direct detection of HR 8799 d, the second-closest known planet to the star, at moderate spectral resolution with Keck/OSIRIS (K band; R ≈ 4000). Additionally, we uniformly analyze new and archival OSIRIS data (H and K band) of HR 8799 b, c, and d. First, we show detections of water (H2O) and carbon monoxide (CO) in the three planets and discuss the ambiguous case of methane (CH4) in the atmosphere of HR 8799 b. Then, we report radial-velocity (RV) measurements for each of the three planets. The RV measurement of HR 8799 d is consistent with predictions made assuming coplanarity and orbital stability of the HR 8799 planetary system. Finally, we perform a uniform atmospheric analysis on the OSIRIS data, published photometric points, and low-resolution spectra. We do not infer any significant deviation from the stellar value of the carbon-to-oxygen ratio (C/O) of the three planets, which therefore does not yet yield definitive information about the location or method of formation. However, constraining the C/O for all the HR 8799 planets is a milestone for any multiplanet system, and particularly important for large, widely separated gas giants with uncertain formation processes.

Riemann curvature invariants are important in general relativity because they encode the geometrical properties of spacetime in a manifestly coordinate-invariant way. Fourteen such invariants are required to characterize four-dimensional spacetime in general, and Zakhary and McIntosh showed that as many as seventeen can be required in certain degenerate cases. We calculate explicit expressions for all seventeen of these Zakhary–McIntosh curvature invariants for the Kerr–Newman metric that describes spacetime around black holes of the most general kind (those with mass, charge, and spin), and confirm that they are related by eight algebraic conditions (dubbed syzygies by Zakhary and McIntosh), which serve as a useful check on our results. Plots of these invariants show richer structure than is suggested by traditional (coordinate-dependent) textbook depictions, and may repay further investigation.