# ISONblog

• September 3, 2013

## Measuring the Size of ISON's Nucleus: Point Spread Function

Welcome to guest poster Dr. Jian-Yang Li. Dr. Li is a research scientist at the Planetary Science Institute and a comet expert. His interests include photometry of asteroids and cometary nuclei, physical properties of cometary nuclei, and the connection between comets and asteroids. His expertise is in photometric analysis, especially from high-resolution data obtained from the ground and returned by spacecraft. He is one of the first few astronomers who applied disk-resolved photometric analysis to cometary nuclei, and found possible connections between the photometric properties and cometary activities. Dr. Li has been actively involved in Deep Impact and Dawn missions.

(If you haven't already, you might want to read Part 1: Pixelization.)

Welcome back. In my last post, I explained how we determine the size of ISON’s nucleus despite pixelization. But that’s not our only hurdle. Now let’s talk about point spread functions.

Let’s go back to our image of Rock in Landscape.

Remember that? Now let’s look at what the way that image would appear if the rock were smaller than one pixel. We’re using a single color now, because that’s how astronomical images are taken.

If one pixel = 10 centimeters, all we know now is that the rock is smaller than 10 centimeters. We need to judge the size of the rock by how much brighter that pixel is than the surrounding background.

Now that we’ve seen what pixelization means to a small rock or a cometary nucleus that is smaller than a pixel, I need to point out that, in the above example, we have been using an ideal astronomical telescope and camera.

If we use this telescope to take a picture of a star, which will be much smaller than a pixel for any traditional telescope, the star will appear in only a single pixel that is directly created by the starlight, and all other pixels will be dark.

But such a telescope can never be made in reality. In reality, all stars will form a small, round disk with long spikes, as you are familiar with in the telescopic images of the night sky. This small, round disk and spikes show off the point spread function, or how the light from a point spreads out through a telescope. The point spread function complicates the measurement of nucleus size to another level.

Let’s take our rock example again, but use an actual telescope. This is what the 8-centimeter-size rock will look like:

The rock now appears to be round, spread out to many pixels. Since the rock is smaller than the size of a pixel, we lost all information about its shape. The apparent round shape in the image is actually the shape of the point spread function, rather than the shape of the rock itself. It doesn’t matter how hard you process the image or how fancy your techniques are. The rock’s size and shape can only be measured indirectly from its brightness.

With the effect of the point spread function, we cannot simply look at the center pixel for the nucleus. We also have to consider the extended wings of the point spread function, as those pixels also contain the brightness of the nucleus.

The large area of a point spread function has two consequences. First, when we make a model of the coma, we have to look outside of the point spread function of the nucleus, some pixels away from the center pixel. This will inevitably introduce more confusion when we try to use this model to predict how bright the coma is in the center. Second, we have to predict the brightness of the coma in all the pixels affected by the point spread function of the nucleus, rather than just a single center pixel.

The presence of the point spread function also means that we have to consider its effect in making the coma model, because a point spread function essentially spreads light out and changes the distribution of brightness of any scene. If we do not consider this effect, then we will predict a much higher level of the background coma in the center region and wipe out the nucleus.

The accuracy of measuring the brightness of the nucleus strongly depends on how accurate one can predict the contribution of the coma. In most cases where we have applied this technique before, the nucleus accounted for less than a quarter of the total brightness of coma background and nucleus. So we are essentially subtracting two large numbers to get a small number. For example, assume the total brightness of coma and nucleus is 10 units, the coma is 9 units, and the nucleus is 1. If we underestimate the coma brightness by just 10% (which is fairly typical in astronomical measurements), to 8 units, then we will find a nucleus of 2 units, off by a factor of 2! The smaller the nucleus, the less accurate the measurement. That’s why it is extremely difficult to measure how accurate the coma model is.

With the significant effect of the point spread function in this process, it is key a priori knowledge that we have to apply this technique. The shape of the point spread function is determined by how a telescope is constructed and how all the optics in a telescope are laid out. Very often a slight deformation in a telescope structure, even at micrometer size, will change the point spread function significantly. Being in space and feeling no atmosphere and gravity, Hubble has an extremely stable environment and therefore point spread functions that are much better characterized than those of almost any ground-based telescope. The consistency and well-known measurements of Hubble’s point spread functions are another reason to use Hubble to measure the size of a cometary nucleus.

Comet ISON has been highly active since its discovery. From our preliminary analysis of Hubble images of this comet, its nucleus appears to be smaller than 2 km in radius. The high activity and small nucleus means that what we have in hand might be similar to our earlier example of the extremely small rock, where it is almost completely buried by the background. While we continue to refine our analysis and hope to pull out the exact size of its nucleus, let’s all hope ISON puts on a great show in the sky this holiday season.