How To Calculate The Size Of A Crater On The Moon

The bore and heights of lunar craters

Abstract

In this experiment you lot will use images of the Moon (taken with the School's telescope) to identify bear upon craters on the lunar surface. You will then continue to measure the diameters of the craters and their heights. Finally, yous will compare the results you have obtained with published values.

Introduction

Before the advent of spaceflight our knowledge of the Moon was limited to what could be gleaned from meticulous observations of its surface using gound-based drawings and photographs. Eventually, space missions, such as Apollo, revolutionized our understanding of the lunar topography and geology (really selenology).

Here you will use images taken with our accuse-couple device (CCD) detector attached to our 10" telescope to make measurements of the diameters and heights of craters.

Section 1: The data

The data consist of a series of 8 images that were taken on thirteen March 2003. The images were taken along the terminator of the Moon, which is the line that separates the day and dark hemispheres. You will run into that substantial amounts of the images are a uniform white, and that some testify streaks of white in the vertical management. This is where the CCD images are "saturated"; the Moon is very bright and some of the pixels in the photographic camera have get full of charge, which may have and so leaked into surrounded pixels. You lot should ignore regions of the images that are saturated, every bit they practice not incorporate useful data.

Image 1: 2003 March xiii 19:25 UT	Image 2: 2003 March 13 19:27 UT	Image 3: 2003 March 13 19:28 UT	Prototype iv: 2003 March 13 nineteen:30 UT
Image v: 2003 March 13 19:31 UT	Epitome vi: 2003 March thirteen 19:32 UT	Image seven: 2003 March thirteen 19:32 UT	Image 8: 2003 March 13 19:33 UT

The images in the above table are links to jpeg format copies of the original images and y'all should download these to your user space. On the Macs in the lab y'all will find an application called ImageJ (in the Applications folder) which is suitable for viewing these images.

The lunar surface comprises dark, flat regions known as maria (singular mare, latin for bounding main), brighter regions which are highlands, and of course the ubiquitous touch on craters. The craters are especially conspicuous near the terminator, where their shadows are longest. Many craters incorporate interior planes that are as flat as the maria, and mountainous walls that define their edges. The largest crater on the virtually side of the Moon is Clavius (bore 235km), and the curvature of the lunar surface is such that if you lot were continuing in the heart of this crater yous would not be able to see its walls (which are 6.6 km high).

Use the online lunar atlas to identify the master features on each prototype. Print out a copy of each of the images and annotate them with the names of the features. Hint: as a starting signal compare this atlas image with images number vii and 8.

Department two: Measuring the diameters of craters

Identify five craters that you wish to measure. These should show a measurable shadow (so await nearly the terminator) and should announced in this listing of craters. Make certain you write downward the latitude and longitude of the craters. The breadth and longitude are in the selenographic coordinate system, where the naught-point is in the middle of the disc facing Globe.

Question: Why did you select your detail craters?

You now need to mensurate the diameter of the crater in pixels. In ImageJ the ten and y co-ordinates of the pixel under the cursor are shown in the ImageJ window. Due to projection effects (foreshortening) the crater will appear every bit an ellipse; always measure the longest centrality. Estimate your error on this value. Nosotros know that the pixel scale of the image is 0.73 arcseconds per pixel (remember that an arcsecond is ane/sixty of an arcminute which is in turn one/threescore of a degree). You can now summate the apparent angular diameter of the crater. In order to decide the physical diameter of the crater we need to know the lunar distance, which (since its orbit about the Earth is eccentric) varies with fourth dimension. Utilise this lunar ephemeris program and the UT fourth dimension of the epitome that you are interested in to find the lunar altitude at the time of the observation (note down all the information given, you'll need it later). Now convert the lunar distance (given in Globe radii) to metres, and hence calculate the bore of your crater in kilometres and its associated error. Compare this with the value in the database, and comment on the accuracy of your measurements. Hint: you may find the pocket-sized bending approximation useful.

Section three: The heights of craters

The principle behind measuring the heights of lunar craters is quite simple:

For a given crater the Sun is at an elevation α. The quantity measured is the shadow length s. It is and then picayune to piece of work out the superlative h of the crater, measured from the crater flooring to the meridian of the crater rim.

There is a subtly here though. The shadow length observed volition just be the same as the measured length s if the crater is viewed directly from higher up. Otherwise the shadow is foreshortened, and the projected length we mensurate is shorter than the true length. We need to compensate for this factor, so divide the apparent length of the shadow by the cosine of the crater'southward selenographic longitude to get the truthful length.

Question: Why doesn't the foreshortening modify with latitude along the terminator?

We now need to summate α, which is the Sun's elevation viewed from the crater at the time the image was taken. This requires some spherical trigonometry:

The to a higher place figure shows the lunar disc, with the crater marked at C. The sub-earth point which defines the zero-point of selenographic coordinates is marked East, while the pole is denoted P. The terminator, which divides the illumated part of the disc from the night part is also shown. If you were continuing on the lunar surface at S at the time of the observation the Sunday would be directly overhead, and this is called the sub-solar bespeak. Patently the angular altitude from Southward, through the crater C, to the terminator at T, is xc degrees. (If you lot were standing on the terminator then the Sun would be on the horizon). Nosotros are interested in the bending of Sun above the horizon at C, which we accept called α.

The bending CY is the selenographic latitude of the crater, while bending EY is its longitude. Similarly, angle SX is the selenographic breadth of the sub-solar signal, and EX is its longitude (this quantities are given by the ephemeris plan). We can at present apply the cosine equation for a spherical triangle to get:

cos(CS)=cos(PC)cos(PS)+sin(PC)sin(PS)cos(CPS)

hither CPS is the vertex angle, and is the deviation in longitudes between the sub-solar signal and the crater (be careful about signs here). Solve this equation for CS, and hence α (remember since your crater is close to the terminator you expect this angle to be small). Yous are now in a position to calculate your crater acme, and its associated error.

Compare your crater heights with those in the literature - try this page for depths of craters (in anxiety, and so convert to metres). Annotate on the accuracy of your measurements.

Question: How could you improve the accuracy of the experiment?

Comments and dead links should be sent to acreman@astro.ex.ac.united kingdom of great britain and northern ireland
Last revised: $Date: 2012/09/26 fourteen:07:08 $
Version: $Revision: i.viii $