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Density of a Kuiper Belt Object

Density is the ratio of mass to volume of a material, and it's a fundamental property of a material.  For example, ice has a density of 0.9 g cm-3, water has a density of 1.0 g cm-3, and iron has a density of 7.9 g cm-3. The Earth's interior is rich in metallic and rocky material and has an average density of about 5.5 g cm-3 while Saturn is rich in hydrogen and has an average density of only 0.7 g cm-3.  Saturn has such a low density that if you could find a lake large enough to hold Saturn, it would float in the lake. 

We were able to make the first density measurement of a Centaur (a recent escapee from the Kuiper belt on an outer planet crossing orbit). In particular, we found 5145 Pholus has a surprisingly low density of only 0.5 g cm-3.  Such a low density suggests to us that Pholus is  probably rich in icy material and contains a considerable amount of empty space in its interior (i.e. its very porous). 

Below we describe how we derived a density for Pholus using the Vatican Advanced Technology Telescope and a CCD camera. Specifically, we measured  (1) the time it takes for Pholus to spin once about its axis (its period of rotation), and (2) its three-dimensional shape. These two measurements enabled us to derive a density for Pholus. 

The football-like shape in the movie and figures below represent Pholus and are only for illustration. Because of its relatively small size and relatively large distance from the Earth, we cannot directly observe the shape of Pholus in our images. Our observed images of Pholus look like circular black dots on a white background, much like the image of the Centaur 1994 TA in the "Our Program" link at the left.  We are only able to observe the circular black dot image of Pholus and its change in brightness.

How Did We Measure the Period of Rotation of Pholus?

The movie below shows the rotation of a football-like body that represents Pholus. The figure (below the movie) shows four snap-shots during a single rotation of Pholus.  In shap-shot 'A', Pholus presents a broadside view to us. From this aspect, the largest reflecting area of the body is facing us and hence the body is brightest to us. In snap-shot 'B', Pholus has rotated one-quarter turn, and Pholus presents the "point" of its football-like shape to us. From this aspect, the smallest reflecting area of the body is facing us and hence the body is at its faintest. During one rotation of Pholus, A, B, C, D, and A face us again. The graph to the left of the snap-shots shows the variation of brightness during one rotation of Pholus.  The graph of brightness vs. time for Pholus is what we observe. The time it takes to go from A to B to C to D and back to A, is the period of rotation of Pholus.

We find it takes Pholus 10 hours to rotate once about its axis. In other words, a day on Pholus lasts 10 hours, less than half an Earth day of 24 hours.

Figure by Ron Redsteer and NAU Bilby Research Center

How Did We Measure The Three-Dimensional Shape of Pholus?

The figure below helps illustrate how we measured the shape of Pholus.  As Pholus revolves about the Sun, it presents different aspects to us. Notice the rotation axis of the body always points to the left in the figure. At point 'A', Pholus presents an equator-on aspect to us. We see the broadsides and points of the football as Pholus rotates, and so we see the maximum possible change in brightness as Pholus rotates about its axis. The difference between the maximum and minimum brightness is called the amplitude. At point 'A', we see a maximum amplitude (note the graph below point 'A'). After one-quarter of a revolution about the Sun, Pholus is at point 'B'. At this position in its orbit, we see Pholus with a pole-on aspect, and we see only one broadside of the football, not the points as it rotates about its axis. Because the same broadside faces us, we see no change in the reflecting area and no change in brightness (note the graph to the right of point 'B'). From the figure below, it is easy to see the amplitude has changed from a maximum at point 'A' to zero at point 'B'.

How do we use the observed change in amplitude as Pholus orbits the Sun to derive a shape for Pholus?  First, we create a computer model of Pholus. The model requires us to input a three-dimensional shape and an orientation for the rotation axis. For each set of inputs, the computer model outputs theoretical amplitudes.  We compare the theoretical amplitudes to the observed amplitudes. Next, we run through all possible combinations of shape and rotational axis orientation, and compare the theoretical amplitudes to the observed amplitudes. Finally, we find the theoretical amplitudes (and the associated shape and rotation axis orientation) that best matches the observed amplitudes. Presto, we have the shape of Pholus.

Figure by Ron Redsteer and NAU Bilby Research Center

How Did We Derive the Density?

KBOs may have weak internal constitutions (i.e. rubble pile type interiors) due to fracturing by past impacts. In other words, it is possible that KBOs are nearly strengthless bodies, held together primarily by self gravity. If so, then these objects will deform into football-like shapes as a result of their rotation, and we can use the physics developed by Nobel laureate Chandrasekhar to relate the measured period of rotation and shape of the object to its density. Application of Chandrasekhar’s formalism to our measured rotational period and shape of Pholus yields a density of 0.5 g cm-3.  If we could find a lake big enough, Pholus would float in it.

Scientific Paper Describing Our Pholus Density Measurement

You can find the scientific paper describing our measurements of the period of rotation, size, shape, density, and homogeneous surface color of the Centaur 5145 Pholus here (pdf file).