Observing the Planet Mars -- By: Jeffrey D. Beish (Version 03-Mar-2016)
Chapter 10 -- MEASURING SOLAR SYSTEM OBJECTS
The astronomical micrometer is a most useful tool for the telescopic observer in recording the positions and sizes of a planet’s atmospheric and/or surface features. While there are many different micrometer designs used in astronomy, only three types will be discussed here.
To find the linear size of celestial objects with a telescope one must use a measuring device such as a micrometer. Micrometers are usually made with adjustable webs, needlepoints, or an eyepiece reticle with graduated lines ruled in the glass. The webs, needlepoints, or reticle lines are positioned at the focal plane, in focus with the image, and magnified. This article will feature the Bi-Filar Micrometer (See Figure 10-1).
Figure 10-1. TOP: A Bi-Filar Micrometer produced by Ron Darbinian. It was sold with either a 12mm or 27mm eyepiece attached to the black box that contains the webs and web holding mechanism. The silver handle to right is the micrometer thimble and spindle to adjust the movable web. A protractor shown is shown behind the black box and is free to rotate 360 degrees to measure the position angle of the double stars (PDF). BOTTOM: Darbinian micrometer thimble and spindle for reading separation in millimeters.
After the image is positioned at the focal plane of the eyepiece, between the webs the telescope drives are adjusted so the object, such a Jupiter, is centered in the movable (M) and centerline (T) webs, and the fixed (F) and centerline (T) webs. The separation is read from the micrometer thimble and spindle and noted in either in millimeters (mm) or in fractions of an inch (See Figure 10-1).
Let’s use a micrometer to measure Mars from pole to pole. After the image is positioned between the webs or points at the focal plane the telescope drives are adjusted so the image just touches the north and south limbs of the planet. The separation is read from the micrometer thimble and spindle dial and noted. You can find the separation by subtracting the micrometer "zero," that is, the dial reading where the webs or points are centered on each other. To determine the micrometer zero one positions the movable web exactly over the fixed web, then reading the micrometer. The Darbainian B-Filar Micrometer this author uses has a micrometer zero at 10.6772. I found this by superimposing the movable web on the fixed web and adjusted the micrometer thimble on either side and back to center several times and then recording the center position. The average of a dozen attempts worked out well.
In these examples the "direct-indirect" method
will be used [Peek, 1981]. This method eliminates the need to use
the micrometer "zero" and reduces any mechanical error by half. To
measure the disk of Mars with Direct-Indirect method: the disk is
measured directly (Di) between the fixed (F) and movable (M)
webs, then Mars is positioned across the fixed web to the other side (See
Figure 10-2). The movable web is adjusted across the fixed web and the disk is
then measured indirectly (In) with the opposite screw adjustments.
Finally, the thickness of the webs (Wt) is subtracted. The results are
averaged and the separation of the webs is determined by: separation =
(Di - In) / 2 – web thickness [Beish et al, 1986].
Figure 10-2. Appearance of Mars in micrometer eyepiece using ALPO method. For measuring the disk of Mars on the DIRECT side, as shown in the left image, using the telescope drive slow motion controls Mars is situated between the movable web (M) and the fixed web (F) and the micrometer separation is recorded. In the right image Mars is driven down and the movable web (M) is adjusted up and across the fixed web (F) in the INDIRECT side and the micrometer separation is recorded. The web (T) is used for centering. Measuring the apparent disk (D) of Mars will be used in both methods described below.
For example, on January 27, 2010 Mars is observed in perfect seeing using a Ron Darbinian Bi-filar micrometer with web thicknesses of 0.012 mm (Wt = 0.012 mm) in the focuser of a 16-inch (406.4mm) f/6.9 telescope with a focal length of 2806.7 mm. Since many of the objects subtend very small angles, usually in the seconds of arc (arcsec), we must increase the effective focal length (EFL) of our telescope to allow the image to be large enough to be separated by several increments. A large image also results in a higher apparent resolution in the micrometer readings. The Barlow lens is a good way to accomplish this. If a 2.65x Barlow is used with the micrometer in the above telescope then the EFL will be:
EFL = 2806.7mm x 2.65 = 7,438 millimeters
Measuring the apparent disk diameter (D) we find Direct (Di) = 11.1980 mm and Indirect (In) = 10.1564 mm, and web thickness of 0.012 mm, therefore the separation of the webs is determined by:
D = (Di - In) / 2 - Wt = (11.1980 – 10.1564) / 2 - 0.012 = 0.5088 mm
WHAT DOES THE SEPARATION MEAN?
To reduce our measurements into some useful terms we must first establish what our readings mean. Each increment on the micrometer dial head is a linear measure of the image at the focal plane that represents the apparent angle of the object being measured. The linear separation between two points in the magnified image is directly proportional to the apparent angle of the object and the effective focal length of the telescope.
example, if the telescope has an effective focal length (EFL) of 7,438
millimeters and the reading on the dial of the micrometer indicates a
separation of 0.5088 millimeters, the measurement represents an angle of 14.11
arc-seconds. This is accomplished by means of the mathematical formula;
(206,265 x 0.5088 mm) / 7,438 mm = 104,947.6 / 7,438 = 14.1 arcsec
The ratio between the focal length and the linear measurement is the micrometer "screw constant" (in arc-seconds per inch or millimeter). This is found by using the formula: SC = 206,265 / EFL. The screw constant or image scale for a 16-inch (406.4mm) f/6.9 aperture telescope with an EFL of 7,438 millimeters is: 206,265 / 7,438 mm or 27.73 arc-seconds per millimeter.
Another method to determine the screw constant is to record the time it takes an object, such as a star, to cross the field of the micrometer from one point to another point at a prescribed separation with the telescope’s drive off. The time in seconds that the object takes to cross the separation can be converted to an angle in arc-seconds and is the screw constant and can be found by the formula:
constant = (dT x
where dT is the average time in seconds derived from several timings, d is the declination of the star
Figure 10-3. With telescope drive off cause a star at or very near the celestial equator will drift along the centerline web of the micrometer between the fixed and movable webs. Record time it takes the drifting star to pass from the fixed to moveable webs that is set to a predetermined separation.
For example, a test star at 0° d takes an average of 9.22 seconds to cross 5 millimeters in the focal plane, then the screw constant is: SC = (9.22 x 15.041068 x Cos 0°) / 5 = 27.73 arcsec/mm. Of course, to be accurate one must repeat this process several times to find an average time. Then multiplying the screw constant by the micrometer measurement in the above, Mars pole to pole is 0.5088 mm or 27.73 arcsec/mm, the apparent diameter (AppDia) of the image is:
AppDia = 0.5088 x 27.73 = 14.1 arcsec
If one should calculate, or look up in the Ephemeris, the apparent diameter of Mars for that date they would find the polar diameter to be 14.11 seconds of arc. Of course, this usually doesn’t happen in real life because of the difficulties involved with using a micrometer at the telescope in Earth’s turbulent atmosphere. Even under perfect "seeing" the observer is compelled to use a deep red filter in conjunction with the micrometer to reduce atmospheric and irradiation effects. Irradiation of bright objects, especially planets in the eyepiece, is evidently a physiological effect originating in the eye itself and occurs between adjoining areas of unequal brightness.
Also, the image seldom remains exactly positioned between the webs of the micrometer and must constantly reposition the telescope slow motion controls. To reduce the observational systematic errors one must make many measurements during the observing period and average the readings.
PRACTICAL MEASUREMENTS WITH MICROMETER
If we were to measure the apparent size of the Great Red Spot (GRS) on Jupiter and desire to know how large it is with respect to the size of the Earth, then we find the relative angular size of the GRS to the disk of Jupiter and convert that to kilometers. The diameter of Jupiter is 142,984 kilometers. So, if we measure the apparent size of Jupiter to be 1.8432 mm or 44.56 arcsec by micrometer and measure the GRS to be 0.17559 mm or 4.244 arcsec, then the size of the GRS in kilometers is; (4.244 / 44.56) x 142,984 = 13,618 km. Apparently the GRS is about the same size of our 12,756 km-diameter planet, Earth.
Another example, if we used a 16-inch telescope to find the smallest feature on Mars we must calculate the resolution of the telescope. If we use the theoretical Dawes limit of resolution: 115.8 / aperture in millimeters (4.56 arcsec / aperture in inches) the resolution of a 16” telescope is: 115.8 / 406.4mm = 0.285 arcsec. If we observe Mars at it’s maximum apparent angle of 25.1 arcsec and want to know what is the smallest feature we can see on the planet then multiply the diameter of planet (6,792 kilometers) by the ratio of the telescope resolution to the apparent angle of Mars at that time:
smallest feature = 6792 * (0.285 / 25.1) = 77 km
The Dawes limit may not be the best measure to use for this. That is a subject for the future debate.
POLAR CAP WIDTHS WITH MICROMETER
Figure 10-4 illustrates the method to measure the latitude or width of the Mars polar cap using a Bi-Filar micrometer. First, the polar cap breadth, or East to West span across the sphere of the planet, is measured. Then the apparent disk of Mars is measured from north to south to be used in a standard spherical geometric equation:
Latitude = cos-1 (C/D),
where C is the breadth of the cap and D is the apparent diameter
of the disk.
Figure 10-4. Mars as seen and measured in a Bi-filar micrometer. Show are three webs, ‘T’ for centerline, ‘F’ is the fixed web, and ‘M’ for movable web. LEFT: the apparent polar diameter of Mars with a thimble and spindle dial reading of 11.7157 mm. Micrometer zero at 10.6772.
Using the 16-inch telescope mentioned above we measure the apparent polar diameter of Mars a reading may be; AppDia = 11.1860 – 10.6772 = 0.5088 mm. Now, measure the extent of the east and west edges of the polar cap we may find that the AppWidth = 10.8302 – 10.6772 = 0.153 mm. Applying the differences or distances between extents using the equation above, we find the Latitude = cos -1 (0.153 / 0.5088) = 72.5 degrees.
To quickly find the width of the polar cap in degrees, where C =
0.153 mm and D = 0.5088 mm from above measurements use this equation: W = 2 sin-1
(C/D), the W = 2 sin-1 0.153 / 0.5088 = 35.0°.
PRACTICAL MEASUREMENTS WITH A FILM CAMERA
In deep sky (wide field) photography we will want to calculate the area of the sky captured on a 25.2 x 37.7mm film frame. The width of field (W) is found with: W = (57.3 x S) / F in degrees or 206265 S/F in seconds of arc, where S = film size and F = focal length of the telescope or camera lens. For example, using a 6” f/4 Newtonian at prime focus the focal length will be 609.6mm (24 inches), so (57.3 x 25.2) / 609.6 = 2.4º by (57.3 x 37.7) / 609.6 = 3.5º.
The apparent sizes for the Moon or Sun is around 32 minutes of arc and additional mathematics are required to find the actual linear size they would be reproduced on the film. The linear diameter of a 32 arcmin or 1920 arcsec object on film we use: S = (W x F) / 206265, where 206265 is the number of second of arc in a full circle (360º). In this example we find: (1920 x 609.6) / 206265 = 5.7mm. To reveal more Lunar or Solar details we must increase the effective focal length (efl) of the 6” f/4 telescope, so some projection is required. Using a 2.947x Barlow to increase the efl to 1797mm, a 2.947x Barlow (46.482 mm) could be used and would give us a: (1920 x 1797) / 206265 = 16.7mm. The effective focal ration is 4 x 2.947 = f/11.8.
Figure 10-6. ~32 arcmin Moon at f/11.8.
Figure 10-7. Lunar crater at f/100.
Increasing the Fr to f/100, results in 15,240mm efl to magnify the image of a lunar crater many times. A 6-inch f/4 is not really the best choice to shot the Moon with, but this one with a near perfect mirror figure and 25% central obstruction was used to produce a fairly shape image of the Moon. If we used this telescope to photograph Mars, say in 1988 at 23 arcsec, the image would be too small to effectively evaluate the planet Mars with. (23 x 22860) / 206265 = 2.5mm.
Figure 10-8. Now, this is more like it! Compare the 1988 Mars with a 6” f’4 (Fr = f/150) with the 2003 Mars with a 16” f/6.9 (Fr = f/20). The tiny, distorted image on the left demonstrated how difficult it is to reproduce a crummy Mars image from film.
REDUCED SYSTEMATIC ERRORS WITH CCD CAMERA
Using CCD images to record the positions and sizes of a planet’s surface features can be easier that using a micrometer at the telescope. Very short exposure times make measuring planetary features a pleasure. No more waiting for those moments of steady seeing to fit a bright planet between two fine wires of the micrometer. A typical observational while using a micrometer at the telescope might require you to wait several minutes to get one set of measurements and past practices required at least eight or ten measurement sets should be made during any observational period. So, this could take up quite a long time for an observer and may seem more like work than having fun [Beish at al, 1986].
The typical CCD camera allows exposures of Mars in tenths of a second as opposed to 2 to 5 seconds for the film method and is ready to take the next image right away. So, one doesn’t even have to wind the film, wait for the telescope to stop vibrating, then wait for that moment of good seeing to take an image. With the CCD, you just shoot away and take as many images as desired. Surely some of the images will be exposed in the moments with steady seeing.
As stated above, the CCD camera chip records images into thousands of cells, called "pixels," that can be stored on the hard disk of your computer. This image array can be used to analyze every pixel of the planet’s image, including the background sky. If the image was taken in steady sky with a very short exposure you can count the pixels at each limb of Mars’ image -- including the edges of the polar cap.
PRACTICAL MEASUREMENTS WITH CCD CAMERA
Systematic errors are reduced significantly when using Web cam CCD images to measure features on planets. A typical set of Web cam images of a planet taken in fair to good seeing conditions easily replaces the time and effort used at the telescope using a micrometer. Also, the reduction of the images can be done in the comfort of your home instead of peering at a planet in the micrometer eyepiece.
Figure 10-9. Toucan Pro
In the examples below a web-camera was attached to a 16” f/6.9 telescope (EFL 2806.7mm) and software that represents the 659 x 494 pixel CCD chip as 640 x 480 pixels on a PC screen, and a projection magnification of: 206,265 / 2806.7 mm = 73.49 mm/arcsec. Considering that this CCD chip pixel size is 5.6 mm then the arcsec/pixel = F / 206.265 mm = 2806.7 / (206.265 * 5.6) = 2.43 arcsec / pix.
Using a typical image processing program the image is rotated on the screen so the disk is seen pole to pole relative to the image frame. If one desires to find the apparent diameter and linear image size of Mars on the CCD chip then the cursor is placed at each limb of Mars’ image and the pixel positions of each are recorded.
POLAR CAP MEASUREMENTS FROM A TYPICAL CCD IMAGE
Using CCD images and image processing software to measure Mars’ polar caps is also useful. In figure 10-12 a CCD image of Mars taken with a deep red filter to cut through the atmospheres of both Earth and Mars. Using a typical image processing program the image is rotated on the screen so the disk is seen pole to pole relative to the image frame. The cursor is placed at the north and south edge or limb of the image and the pixel positions of each are recorded. Next, read the cursor/pixel position of the east and west edges of the polar cap.
Figure 10-12. Typical Mars image as it would appear on 02-27-2010 (De = 12.4 ° ). Cursor placed at extremes of north-south of image (264) and (35) yielding D = 264 – 35 = 229 pixels. Then at extremes of the left (75) and right (135) of the polar cap to yield C = 135 - 75 = 60 pixels.
Now, one has only to count the number of pixels between the extremes of the image and apply this to the desired equations for determining the latitude of polar cap boundary. So, one has only to record the two pixel locations to determine the distance between the features on Mars. Taking the difference of the positions, apply these values to the proper conversion equations to determine the latitude. Either method 1 or method 2 can be used here without serious systematic errors.
With the extent of the north limb (pixel 264) and south limb (pixel 35) of the disk and extent of the east (pixel 135) and west (pixel 75) positions we can take the difference: north-south difference is 264 - 35 = 229 and east-west difference is 135 - 75 = 60. Applying the differences or distances between extents to standard equation:
Latitude = cos -1 (C / D) = cos -1 (60 / 229) = 74.8°
To compute width (W) of the polar cap this equation is used: W = 2 sin-1 (C/D) , where C = 60 and D = 229 from above:
W = 2 sin-1 (60 / 229) = 2 * sin-1 (0.262009) = 30.4°
Figure 10-13. Image of Mars on 08-28-2003 by
Jeff Beish. Cursor placed at extremes of north south of image (163) and then at extremes of north south of a feature (321) and the left (307) and right (337) of the polar cap to yield D = 321 – 163 = 158 and C = 337 – 307 = 30.
Latitude = cos-1 (C / D) = cos-1 (30 / 158) = 79.1 degrees
Width (W) = 2 sin-1 (C
/ D) = 2 sin-1 30 / 158 = 2 * sin-1 (0.1898734) = 21.9°
FIND LATITUDE / LONGITUDE OF A SPOT ON CCD IMAGE
We can find the latitude (q) and longitude (f) of a particular feature on Mars’ globe using simple trigonometry. Since an image displayed on a PC screen is a two dimensional representation of the planet we must first convert linear dimensions to spherical dimensions. If the measurements are in the vertical scale of this image we add the Declination of Earth (De) to the vertical angle V to find the latitude of the feature. If the measurements are in the horizontal scale of this image we add the Central Meridian (CM) to the horizontal angle H to find the longitude of a feature.
As illustrated in figure 10-14, a ratio is derived by dividing the distance (S) from the center of an image to a feature P(x,y) on the image and dividing that by the radius (R) of the image, then taking the inverse sine of that ratio to establish angles (q) and (f), where q = sin-1 Py / R + De and f = sin-1 Px / R + CM.
Figure 10-14. Plot representing the disk of Mars showing the linear coordinates (S and R) used to calculate angles ( q ) and ( f ), where q = sin-1 Py / R + De and f = sin-1 Px / R + CM.
The orientation of Mars’ image on the PC screen is dependent on the type of telescope and optical projection used to take the image and the axis of rotation of the camera in the focuser. To make it simple for this author the image is displayed with south pole at the top and the morning limb to the right, as if the image was taken with a simply inverting telescope and a Barlow projection.
One caveat must be mentioned when using images for precise measurements; many of the images are not normalized or corrected for the north-south orientation. A simple method that is used by Dr. Don Parker is to include a star trail in at least one frame to indicate the direction of west and west as illustrated in the figure below (see Figure 10-15). With the preceding drift (P) and following drift (F) direction on the frame one can then determine the north-south orientation of the image by reckoning that the points north or south are 90 from the east-west line in the trail. Since the Position Angle of Axis for that date was ~345º then the original image was adjusted left by 15º to normalize the image north and south.
Figure 10-15. Image of Mars taken Don Parker on 08-19-2003 De = -18.7, ° ) Pos. Angle of Axis = ~345 ° ). Direction of the east-west orientation, or preceding drift (P) and following drift (F) as indicted by the star trail in the lower left corner of the frame.
An example is illustrated in figure 10-16 where the south pole is at the top and the morning limb is at the right within the frame.
Figure 10-16. Image of Mars taken Dave Tyler on 11-02-2007 at 0340 UT (CM = 28.2º, De = 7.1º). Cursor is placed at top and bottom of illuminated disk (Tp and Bp) to find the pixel positions of the south and north poles. In this example the pixels at Tp = 112 and Bp = 380. Because the terminator is hiding the left edge of the disk the cursor is placed at right edge of the illuminated disk (Rp) and reads 446. The cursor is then placed over a suspected dust cloud within Nilokeras (Xp, Yp) and is 372, 295.
Read the top pixel position (Tp) and move the cursor to the bottom and read the (Bp) pixel, where the radius is the difference between Bpix and Tpix divided by 2. The radius of the horizontal disk is the same as the height radius (south to north), so the radius is:
R = (Bp – Tp) / 2 = (380 – 112 / 2) =134 pixels.
In this example the Sunlit morning limb is to the right edge (Rp) of the image and it reads = 446. From the above image the De = 7.1º and the CM = 28.2º.
Next we find the
vertical center (Vc) =
(Bp – R) = (295 – 134) = 238
Sy = Yp – Vc = 295 - 238 = 57
Latitude (q) = sin-1 Py/R + De = sin-1 (57 / 134) = 25.2º + 7.1º = 32.3ºN.
horizontal center (Hc)
= (Rp – R) = (446 – 134) = 312
Sx = Xp - Hc = 372 – 312 = 60
Longitude (f) = sin-1 Px/R + CM = sin-1 (60 / 134) = 26.6º + 28.2º = 54.8ºW.
NOTE: The classical location
for Nilokeras is 28º N and 55º W.
An interesting image of Mars taken by Don Parker on December 01, 2007 shows the beginning formation of the “W-clouds” in Tharsis-Candor-Ophir. Three of the white clouds are over the Tharsis volcanoes, Arsia Mons (9ºS, 120ºW), Pavonis Mons (0º, 113ºW) and Ascraeus Mons (11ºN, 103ºW). An orographic cloud appears over Olympus Mons (18ºN, 133ºW), but is not as white as the other three clouds.
Figure 10-17. A blue light (UV-filter) image of Mars taken Don Parker on 12-01-2007 at 0457 UT (CM = 143º, De = 5.6º). Cursor is placed at top and bottom of illuminated disk (Tp and Bp) to find the pixel positions of the south and north poles. In this example the pixels at Tp = 130 and Bp = 360. Because the terminator is hiding the left edge of the disk the cursor is placed at right edge of the illuminated disk (Rp) and reads 440.
In this example we find R = 115, Vc = 245 and Hc = 325. Therefore, orographic cloud #1, Xp, Yp = 278, 212, Vs = -33 and Hs = -47 so the results will be: Latitude (q) = 5.6º + sin-1 (-33/ 115) = 5.6º + (-16.7º) = -11.1º and Longitude (f) = 143º + sin-1 (-47 / 115) = 143º + (-24.1º) = 118.9º. This cloud is very close to Arsia Mons (9º S, 120º W).
#2: Latitude = -0.9º and longitude = 110.4º. Very close to Pavonis Mons (0º, 113ºW).
Orographic cloud #3: Latitude = 12.7º and longitude = 97.6º. Close to Ascraeus Mons (11ºN, 103ºW).
Orographic cloud #4: Latitude = 18.7º and longitude = 130º. Very close to Olympus Mons (18ºN, 133ºW).
For those interested in finding Olympus Mons on CCD images Damian Peach captured the image below and the huge "donut" shaped feature was measured and using my short computer routine (CCDSPOT.exe) the following latitude and longitude of the volcano was gleamed:
Figure 10-18. Image of Mars taken by Damian Peach on 12-06-2007 (CM = 104.3º, De = 4.8º) with Olympus
shown as a “donut” circular feature at pixels X = 293, and Y = 266. Cursor is placed at top and bottom of illuminated disk (Tp and Bp) to find the pixel positions of the south and north poles. In this example the pixels at Tp = 40 and Bp = 400. Because the terminator is hiding the left edge of the disk the cursor is placed at right edge of the illuminated disk (Rp) and reads 390. Mons
In this example we find R = 180, Vc = 220 and Hc = 210. Therefore, Olympus Mons, Xp, Yp = 293, 266, Vs = 46 and Hs = 83, so the results will be: Latitude = 4.8º + sin -1 (46/ 180) = 4.8º + 14.8º = 19.6º and longitude = 104.3º + sin-1 (83 / 180) = 104.3º + 27.5º = 131.8º. The caldera of Olympus Mons is shown in NASA’s Atlas of Mars at 18.4ºN, 133.1ºW. An error of 1.2ºN (~4 pix) and 1.3ºW (~4 pix) with the calculations in the above.
The polar cap can also be measured using CCDSPOT. Illustrated below is an image of Mars by Parker on January 23, 2010 at 0442UT.
Figure 10-19. Image of Mars taken by Don Parker on 01-23-2010 at 0442UT (CM = 168º, De = 15.4º) with north polar cap exposed with a south edge at 69.5 º latitude. The feature at pixels X = 190, and Y = 294 along the central meridian line at the edge of the cap. Cursor is placed at top and bottom of illuminated disk (Tp and Bp) to find the pixel positions of the south and north poles. In this example the pixels at Tp = 18 and Bp = 323. Because the terminator is hiding the left edge of the disk the cursor is placed at right edge of the illuminated disk (Rp) and reads 343.
HOW LONG ARE THOSE SHADOWS ON MARS
One case in point is the volcano Olympus Mons, one of the tallest topographic features on the planet that stands at a height of 22 kilometers relative to its surrounding area; we can then determine the length of the shadow cast by this structure and may be able to image with our telescopes and camera. Remember that Olympus Mons has a slope angle of only is 2.5 degrees from the edge of its caldera is to about 50 miles out and then from there the grade increases to only 5 degrees and continues 162 miles to the very edge or scarp of the volcano. Further shadowing would be seen to cover the slope from the 5-degree slope plane out to miles to the edge of the volcano.
A good example of this can be seen in an image below of Mars by Coral Gables, Florida amateur imager, Don Parker, on September 04, 2005 at 0859UT using a 16-inch telescope and ST9XE CCD camera. In the 0859UT image, Olympus Mons is positioned near the northeastern terminator and is casting an apparent shadow towards the phase terminator.
Figure 10-20. Image of Mars taken by Don Parker on 09-04-2005 (CM = 142.3º, De = -24.6º) with Olympus
shown as a triangular feature near the northeastern limb. Mons
Olympus Mons is located at 133.4°W and 18.6°N and is 22-Km high relative to the surrounding plains. The central meridian (CM) at 0859UT would be at 142.3° and the Declination of Earth (Ds) would be -24.6 . On that date the Phase Angle (i) was 41.4°, so the Solar angle would be CM + i = 142.3° + 41.4° = 183.7° (noonday Sun). We can use the mathematical methods of the Spherical Law of Cosines to find the shadow angle f :
Cos(f) = cos(90° -
or the "haversine" formula:
a = sin2(D lat/2) + cos (lat
Then apply f to find the shadow length (S) using the equation: h Tan f, where h = height of the object and f = angle of the shadow cast by the volcano derived from the latitude (d 1) and longitude (a1) of the solar angle and the latitude (d2)and longitude (a2) of the Olympus Mons (See image below).
Figure 10-21. Diagram showing mathematical expression for a shadow cast my feature by the Sun on Mars
We find for the Spherical Law of Cosines: a1 = 183.7°, d1 = -24.6°, a2 = 133.4°, d2 = 18.6°
Cos(f) = cos(90° - (-24.6°)) cos(90° -
18.6°) + sin(90° - (-24.6°)) sin(90° - 18.6°) cos(183.7°
= cos(114.2°) * cos(71.4°) + sin(114.2°) * sin(71.4°) * cos(54.4°)
= -0.416281 * 0.318959 + 0.909236 * 0.947768 * 0.638768
f = 65.34°
or the "haversine" formula where: D lat = -24.6° - 18.6° = -43.2°, D long = 183.7° - 133.4° = 50.3°, so then:
= sin2(-43.2/2) + cos
(-24.6) cos (18.6) sin2(
= 0.135516 + 0.909236 * 0.947768 * 0.180616
f = 2 * atan2(v0.291161, v(1 - 0.291161)) = 65.34°
Hence: the shadow length (S) using the equation: h Tan f or 22 tan 65.34° = 47.9 Km
The 22-km high volcano would produce a shadow of 47.9-Km, a few degrees away from the terminator, in the Parker image. Since the average radius of the huge volcano 300 kilometers (186 miles) then much of the terminator side of Olympus Mons and the shadow would be in limb darkening and dusty atmosphere. Using my short computer routine (Shadows) one may determine lengths of shadows on Mars images.
There are several other large volcanoes on Mars that possibility could cast a shadow dark and long enough to be resolved by ground-based telescopes. Arsia Mons (9-km relief), Ascraeus Mons (15-km relief), Pavonis Mons (11-km relief) and Elysium Mons (13.9-km relief) are such examples. If they are far enough away from the limbs and contrast is not reduced by limb darkening, the ever-present dusty haze, and clouds, then one may very well see a shadow.
Since the atmosphere of this planet is quite thin the Martian sky appears quite clear; however, the extension of the atmospheric mass near the Mars' limb, as well as fine dust particles, CO2 and/or H2O hazes and ice crystals near the bright limb or terminator greatly reduces contrast of surface features. Confusion may exist between a shadow and dark surface materials within the feature. If an observer is lucky enough to catch Mars see a particular topographic feature on Mars when its atmosphere is clear then a shadow cast by the feature may be seen or imaged from earth.
Measure CCD Images with WinJUPOS
First, from the menu bar select “Recording” then “Image Measurement…” and “Open Image (F7)” to open an image file. Fill in the UT date and UT time and then hit “Ephemerides (F8)” to compute the ephemeris.
Second, select the tab “Opt.” and make sure the “Normal Image” is checked in the “Image orientation” panel.
Next, select the “Adj” tab. Note: For a simple inverted image, that is south at the top and preceding limb to the left, hit the “Backspace” key to invert the frame so that the “N” is to the bottom and “P” is to the left. Now, the image and fame are orientated to match the north-south and sunlit east or west limbs. To center the frame and adjust the frame size to fit the general outline of the planet use the following instructions from his help file:
[R] rotate image 1º clockwise [L]
[Arrow keys] removes frame
[PageUp] increases frame
[PageDown] decreases frame
[Backspace Key] rotate frame 180°
Finally , use the “Pos” tab and position the cursor over the feature to be measured and Left click to mark the feature, record and/or file away the positions.
Figure 10-22. After image is loaded (1) the frame may be too small and with wrong orientation. Use [Backspace Key] to invert frame (2). Adjust frame to fit image (3) and use the “Pos” tab (4) to measure a particular feature. Image: December 06, 2007 at 0458UT by Damian Peach. The latitude and longitude of the cloud was found to be: 49.2º N, 74.7º W.
An example of measuring features on Mars, such as the shadow from Olympus Mons, etc., can be digested after reading this.
In the discussions above simple geometry is used to convert linear measurements of a two dimensional presentation of the disk image of Mars to angles to be used to find the latitude or width of the polar cap. First, the geometry of a circle will be discussed and two simple equations will then be used to establish an angle from the equator to the edge of Mars’ polar cap.
Figure 10-23. Geometry of a circle. Let S = length of arc subtended by q, l = chord subtended by arc S, R = radius of the circle, D = diameter, h = rise, q = central angle in radians and f angle of latitude from equator to edge of the rise.
l = 2 SQR (R2 – d2) = 2R sin q /2 = 2d tan q /2
d =1/2 SQR (4 R2 – l2)
h = R – d
q = 2 sin-1 l /D
f = cos-1 l / D
Beish, J.D., D.C. Parker, and C.F. Capen, "Calculating Martian Polar Cap Latitudes," J.A.L.P.O., Vol. 31, No. 7-8, April 1986.
B.M., The Planet Jupiter: The Observer’s
Handbook, Rev. ed,