Shape Model of Hydra

Shape model for Hydra, a satellite of Pluto. Shape created from the resolved images of Hydra obtained by New Horizons.

The relevant information from chapter follows:

"To simultaneously fit the shape and poles of the small satellites of Pluto, we forward-modeled the New Horizons LORRI high-resolution images. In this context, “forward-modeling” refers to using a shape model to generate synthetic images with the same geometry and resolution as the real images,and using this the constrain the shape model. Forward modeling the images enables fitting all the pole and shape parameters at the same time, using as much of the information in the images as possible.Rendering a complex shape with a computer’s CPU can be very slow, so instead we performed the rendering with hardware-accelerated OpenGL, through the PyOpenGL Python interface (Fletcher and Liebscher 2005). This rendering required creating a basic bidirectional reflectance model in GLSL (OpenGL Shader Language), with parameters that could be be changed to fit the images. The GLSL shader provides a snippet of code that the GPU can execute in parallel on the image, significantly speeding up the rendering of the image. Since the color of the small satellites in the visible bands was fairly uniform (with the exception of the red region on Nix), we assumed a uniform albedo and other surface properties for the surface. Because the reflectance model was optimized for the GLSL shader code, the parameters for it were not as physically useful as for the lightcurve fits shown above. We therefore held most of the photometric parameters fixed at those in Weaver et al. (2016), with the exception of a pseudo-albedo parameter. The most complex part of the shader code was the ability to have shadows on the object. This feature was particularly important for Hydra, as the shadows visible in the highest-resolution images of Hydra provide the best constraints on its surface topography. We parameterized the shapes using the octantoid formalism of Kaasalainen and Viikinkoski (2012); this formalism has many attributes that make it mathematically useful for fitting a shape and pole from a collection of dense lightcurves. However, we did not use most of these features, and instead used octantoids because their zeroth-order shape is precisely a triaxial ellipsoid. We are thus able to fit the poles of Nix and Hydra with triaxial ellipsoids, then add complexity to their shape, then refit their poles, and so forth. By using this formalism, we were able to decouple the number of parameters
16from the number of vertices; e.g. a simple triaxial ellipsoid could be modelled with only 3 parameters,but 10,000 or more verticies. The most complex shape model was Nix, with 300 shape parameters.This procedure allowed us to create parametric shapes with an arbitrary number points, so that we could perform initial rough fitting quickly with less detailed meshes, and then gradually increase the number of triangles in the mesh as the complexity of the model required. To convert the octantoid parameters to mesh of triangles that OpenGL can interpret, we first generated a spherical mesh of uniform radius, and then used the latitude and longitude of each point on the sphere to calculate the radius at that point using the octantoid formulas. Rather than fit raw data, we generally fit stacks of images acquired at the same time, to maximize the signal-to-noise ratio and minimize cosmic ray interference. Key exceptions were the highest-resolution images of Hydra, which showed clear rotation over the course of sequence and were reduced in four separate blocks, and the high-phase panchromatic image of Nix, which was acquired in Time-Delay Integration (TDI) mode (see below). All the images were radiometrically corrected before being fit, so that the fitting code did not have to reproduce the radiometric corrections every time it ran. To reproduce the geometry of the images,we read in the World Coordinate System (WCS) and timing information from each image. We then adjusted the field-of-view and rotation of the images according to the WCS, and the rotation and distance to the object according to the time in the header and the New Horizons SPICE (Acton et al. 2018) kernels, which define the locations of the spacecraft and satellite at any given time. Since the pointing information in the headers is not precise enough, we left the positions of the objects as potentially free parameters; generally these would be fit well enough early in the fitting process, and then fixed them when using a more complex shape. The images were rendered using OpenGL onto a frame buffer, which allowed us to transfer the rendered images from GPU memory to CPU memory.We then took the perfectly-rendered images, and convolved them with the Point-Spread Function(PSF) of the images. Since the PSF of both LORRI and MVIC are larger than a single pixel, this convolution allowed us to directly model the images, without having to chance any artifacts that deconvolving the images may have introduced. The rendered and convolved images were subtracted from the real images, and the difference image squared and summed to estimate the χ2 of that image.

Finally, the per-image χ2 was normalized to a weight based on spatial resolution of the image (i.e.divided by the square of the spacecraft-target distance), and then summed to produce an overall χ2 estimate. We started with low-order shape models, optimized their χ2, and then added another order to the octantoid until the addition of orders made no difference to the solutions. Generally, it took a few hundred trials to optimize the shape parameters for each octantoid order. The unresolved images were not used for this analysis, but could provide additional constraints, especially for Styx and Kerberos.

At the time of the New Horizons flyby, Hydra was father away from the spacecraft and was viewed close to pole-on for most of the approach to Pluto, so spatial coverage of Hydra was much poorer than that of Nix. Effectively only the northern hemisphere of Hydra was observed, and most of the shape information about that hemisphere was inferred from the shadowing on its surface. The highest resolution image of Hydra was obtained as a mosaic of six pointings of LORRI, and Hydra appeared at the overlap of four of those pointings. Because of Hydra’s rapid rotation rate, it can be seen to rotate between these pointings, and the shadows do subtly change over the course of the image sequence. The dimensions of the Hydra model in Figure 4 are 52.0×36.5×29.3 km, corresponding to axial ratios of a/b = 1.4 and a/c = 1.8. Because of the viewing geometry, the a and b axes are much better constrained than the c axis. Hydra is thus roughly as prolate as Nix, and may be more flattened. There is a large concavity on the north pole of Hydra that produces the shadowing seen in the best LORRI images of Hydra. This concavity gives Hydra the appearance of being almost bilobate. In addition, there is a smaller, valley-like concavity on the larger lobe. These “lumpy” structures are reminiscent of the surface of Arrokoth, the Cold Classical Kuiper Belt Object that New Horizons visited after Pluto. The surface of Arrokoth’s larger lobe appears to be formed of many distinct units which were accreted together before the two lobes came into contact (Stern et al. 2019; McKinnon et al. 2020). Hydra’s shape may also be the result of a slow accretion process, but with larger pieces. The dichotomy of Nix’s relatively smooth shape and Hydra’s lumpy shape is one that should drive future investigations into the origin of Pluto’s small satellites."