#7 #7

Experiment Experiment

To measure the size and height of a lunar crater

7.1 Instruments needed • Telescope (925 CGE Pro) • CCD (NexImage planetary imager) • Computer & imaging software

7.2 Overview The surface of the moon is littered with various size of craters. With a telescope, one can observe these craters and with simple techniques, their sizes and depths (heights) can be measured. In this experiment, we shall learn how to do this.

7.3 Determining the size of a lunar feature To determine the size of any lunar feature, we, in principle, have to translate the image of the feature from a CCD image to the real size. This can simply be done by measuring the size of the image of an object located at a known distance from simple geometrical consideration. It however, may not as simple as it sounds, as the lunar surface is curved and the position of the moon with respect to the earth is not fixed.

(a)

Calculating the size of a crater

The size of a lunar crater can be computed from its CCD image, by comparing how much sky (angular measure) can be captured by one pixel of the CCD with a fixed combination of optics (telescope and the CCD), which is known as plate scale that we learned in the previous experiment. As we know, the plate scale p can be written as, wpixel p = 206265 ⇥ , (7.1) f measured in arc seconds per pixel, wpixel is the width of one pixel (assuming square pixel as in our setup), and f is the focal length of the optical combination, which in our case is the focal length of the telescope. The plate scale, p now denotes the angular portion of sky which is captured by one pixel of the CCD. If now the earth-moon distance is D, the linear distance l covered by one pixel of the CCD on moon’s surface is, l = D ⇥ tan p.

(7.2)

So, if the diameter of the image of a crater is say dpixel in terms of the number of pixels covered by the diameter on the image, then the actual diameter d of the crater is, d = l ⇥ dpixel . 20

(7.3)

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7.3 24 pixels

36 pixels to earth

moon

Figure 7.1: Shortening of a feature due to curvature of the surface. The above relation would be right if the surface of the moon had been flat, which is not true. So, we have to multiply the above relation by a correction factor tcf , known as the tilt correction factor which takes care of the curvature of the moon (see Fig.7.1). The tilt correction factor can be found by, tcf =

1 , cos(lat) ⇥ cos(long)

(7.4)

where the latitude and longitude of the crater on the surface of the moon, known as selenographic co-ordinates, which can be found out from a map of the crater. So, the actual diameter of the crater would be, d = l ⇥ dpixel ⇥ tcf .

(b)

(7.5)

Calculating the depth of a crater

The depth of a lunar crater can be determined by measuring the length of its shadow. However things become complicated as the length of the shadow changes according to the position of the sun. In order to understand this we have to consider the diagram shown in Fig.7.2. In the figure the crate is located at C. The sub-earth point which defines the selenographic co-ordinate is marked as L. The pole is denoted by P. The terminator defines the boundary of a moon-day i.e. if we stand at the terminator, we would see the sun at the horizon and if we stand at the point B, we will see the sun directly overhead. The point B is known as the sub-solar point. BO is the latitude of the subsolar point and LO is its longitude (in selenographic co-ordinates). PY is the perpendicular from the pole through the crater on the lunar equator, so CY is the selenographic latitude of the crater and LY (= ) is the longitude. The angle ↵ is elevation of the sun as looked from the bottom of the crater (see Fig.7.3, denoted as ✓). Now we can apply the laws of spherical trigonometry to determine the angle ✓ as, sin ↵ = x = sin CY sin BO + cos CY cos BO sin(LO + ) ↵ = sin

1

(7.6)

x

Now, as before we have to measure the length of the shadow of the crater which in pixel terms is dpixel , so that the actual length of the shadow is given by, L = l ⇥ dpixel ⇥ tcf , 21

(7.7)

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7.4 P

T

↵

90

C

↵ B

Dark side O Y

L

Equator Illuminted side

Terminator

Figure 7.2: Geometry of the shadow of a crater. and the crater depth H is now given by, H = L ⇥ tan ↵.

(7.8)

7.4 Procedure • Choose a moonlit night when the moon is in its crescent stage. Note that the caters are best viewed when sunlight falls on the moon at an angle not perpendicularly as in a full-moon night. Prepare your telescope and the CCD for capturing lunar image. • Select a portion of the image where the craters are distinctly visible. Also mark in your notebook (with the help of a rough sketch), the portion of the moon, you are imaging. • For every image that you take, note down the exact time, which is required for finding the sun’s position with reference to the moon and the exact earth-moon distance. At a given date and at a certain time, you can find out the lunar parameters from the site http://www.lunar-occultations.com/rlo/ephemeris.htm • Now from the captured image, find out to which portion of the moon, your image corresponds. You can find out this by comparing your image to the those found in lunar atlas in the site http://www.lunasociety.org/atlas/index.shtml (see Fig.7.4). In the lunar image of this site, you can click on each segment of the image and get a magnified view of the segments along with the crater names. The crater name can then be searched in the site

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7.4

Figure 7.3: The depth and the shadow of a crater.

Figure 7.4: Lunar atlas as will be seen in the site http://www.lunasociety.org/atlas/index.shtml

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7.4

http://the-moon.wikispaces.com to get the exact parameter of a crater, which can then be compared to the values, you have calculated.

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