|Fig. 1 Intensity plot of C/2011 L4 (March 3 2013). Numbers indicate intensity levels in greyscale (0 - black ; 255 - white)|
On last March 3rd I took some photos of C/2011 L4 (Panstars). Here I present an excercise of astrometry using one of those images to extract some meaningful data from this observation. What I do here can be applied to any other comet image or, in fact, celestial object in the sky. My aim is to show a simple example of practical applicaton of mathematics and geometry in the determination of sizes of celestial objects.
- To estimate the apparent (in minutes of arc) and real dimension of the tail (in km or mi) of C/2011 L4;
- To estimate the apparente (in minutes of arc) and real dimension of the coma (in km or mi);
For that aim, we need:
- Comet image with date;
- Scale calibration;
- Software to extract brightness levels (the so called 'isointensity' curves);
- Distance of Earth to the comet at the date;
- Comet position angle in relation to the sun at the date;
Below I coment step-by-step all procedures that I used to find the final estimates. This exercise demonstrates a practical aspect of astronomical observation, something that is fully in agreement with the objectives of this blog.
I use the image publish on last March 4 2013 reproduced below. The image was acquired on March 3rd at 22:15 UT and it is an important input for the determination of additional parameters as we will see.
|Fig. 2 Image used for the exercise. (click on the image to enlarge)|
|Fig. 3 Estimate of apparent distance between HIP 117488 and comet C/2011 L4 at the date as given by Stellarium.|
Sf = 40'/355.1 ~ 0.113'/pixel.
This is nearly 6" per pixel and corresponds to the final resolution of the image. Note that this value is the overall resolution of both the combined optics and camera setup.
The resolution above is a practical scale for the determination of the comet dimension. Using Stellarium, we find that at the observation date, the comet-Earth (observer) distance (D) was:
Since 1 AU = 149,597,870,700 meters (92,955,807.273 mi) and Sf in radians is
Sf(rad) = 0.000032763 rad/pixel
Sf(km)= Sf(rad)*D(km) = 5383.74 km/pixel (=3345.3 mi/km)
Therefore, each pixel in the image at the comet position corresponds to about 5400 km. The smallest pixel in the image, in particular the one corresponding to the "comet nucleus", is a square of ~5400 x 5400 km, much larger than the expected physical size of that nucleus.
3) Detail analysis of the cometary image
The image is in fact a matrix of intensities on an arbitrary scale (in a grey scale 8 bit image, the intensity goes from 0 to 255). If we extract a small portion of the original image (after converting it to grey scale), say, a square of 35X35 centered at HIP 117488, we get Fig. 4.
|Fig. 4 A small sample of the original image showing HIP 117488.|
|Fig. 5 Intensity surface of Fig. 3 of HIP 117488.|
|Fig. 6 3D intensity plot of the original image showing the comet and HIP 117488.|
- Apparent Tail(km) = 90.5 x Sf(km) ~ 490 000 km (= ~ 300 000 mi)
- Apparent Tail(min of arc) = 10.2'
|Fig. 7 Intensity plot of the coma.|
A minor detail
It could be argued (with reason) that the estimated tail size must be corrected for the geometrical situation shown in Fig.8. The comet tail always points towards the sun (along the Sun-comet line), while we are observing the projection of this line on the perpendicular to the Earth-comet line.
Real Tail(km) = Apparent Tail(km)/cos(alpha).
Now, again using Stellarium we have:
Des = 0.99156172 AU;
phi = 18 deg 40' (elongation angle);
as the Earth-Sun distance and elongation angle for comet Panstarrs at the date, respectively. Therefore, using the above equation we find:
alpha = 26 deg 33',
so that cos (alpha) = 0.8944249989.
The real tail length in km (mi) will be
Real Tail(km) = 90.5 x Sf(km)/cos(alpha) = 545 000 km (~340 000 mi).
Compare this with the Earth-Moon distance (384 400km). There is no accurate definition of a comet tail (that depends on the density of particles such as dust, gas etc). What we can say here, however, is that, given the "definition" of tail as determined by the smallest intensity level on Fig. 1 (30.5), the real tail extended itself for half a million kilometers in space on the date.