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.

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 3.04x Barlow is used with the micrometer in the above telescope then the EFL will be:

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

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.

For example, if the telescope has an effective focal length (EFL) of 8,532 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 Reading) /Effective Focal Length.  The number of arc-seconds in one radian is equal to 205,265. In our example, we substitute the values above to give:

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 8,532 millimeters is: 206,265 / 8,532 mm or 24.18 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:

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.

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.

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 (6,787 kilometers) by the ratio of the it’s resolution to the apparent angle of Mars at that time:

The Dawes limit may not be the best measure to use for this. That is a subject for the future debate.

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:

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°.
 
 

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.

Increasing the Fr to f/100, results in 15,240mm efl to magnify the image of a lunar crater many times.  A 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 ascetic, 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.

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.

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.

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.

With a prime focal length of 2806.7mm the 15.1 seconds of arc image of Mars on the CCD chip was a bit too small, so an eyepiece projection will be used to increase the effective focal length (EFL).  We can accomplish this by increasing the EFL using a positive projection, such as the ToUcam Pro-II with adapter and sleeve that holds a 9 mm eyepiece. To find the number of pixels an object will occupy on the chip:

                                                       Where P = diameter of the object in pixels
                                                                    F = effective focal length as calculated (mm)
                                                                     i = apparent diameter of the object (arc-seconds)
                                                                m m =  pixel size of the camera (microns)

For a positive projection we find the magnification:  M = B / F - 1.   So, for the figure below the distance (B) = 43.66mm, so 43.66 / 9 - 1 = 3.85x (efr = 3.85 x f/6.9 = f/26.6 and Efl = 10,814mm). The image scale is: 206265 /10,814= 19.1 arcsec/ mm and P = Fi / 206.265 m m = (10,814 * 15.1) / (206.265 * 5.6) = 142;  a nice fit.

Adding ½-inch to the projection distance results in:  57.15 / 9 - 1 = 5.35x (efr = 5.35 x f/6.9 = f/36.9 and efl = 15,016mm).  Image scale: 206265 / 15,016 = 13.74 arcsec/mm and P = = Fi / 206.265 m m = (15,016 * 15.1) / (206.265 * 5.6) = 197 pixels on the PC screen.

Figure 10-10.  Calculating a positive eyepiece projection magnification using distance (B) and eyepiece focal length (fl), to find magnification: M = B / f -1.  An eyepiece with a FL of 9mm and a projection distance of 43.66mm and 56.36mm respectively;  M = 43.66 / 9 - 1 = 3.85x, RIGHT: 57.15 / 9 - 1 = 5.35x.
 
 

Figure 10-11.  Mars images on a 640 x 480 pixel screen with three simulated images of Mars using 16-inch f/6.9 Newtonian with CCD at f/6.91 or projection at f/26.6 and at f/36.9.
 

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:

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.

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.

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.

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.

Finally, the 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 showing 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.

Orographic cloud #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 Mons 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.

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 size of the frame to fit the general outline of the planet use the following instructions from his help file:

[R]  rotate image 1º clockwise   [L]  1º counterclockwise
[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 10-19. 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.

Note: Using WinJUPOS one can find highly accurate positions of features on Mars. A tutorial can be found here.

Also, a neat Webcam capture program is found here with the WxCapture discussion group here.

Appendix:  Geometry

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 10-20.  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.

                                 Equations:   S = R q = 1/2 D q = D cos ^-1 d / R
                                                      l = 2 SQR ( R ² – d ² )  = 2 R sin q /2  = 2 d tan q /2
                                                      d =1/2 SQR (4 R ² – l ² )
                                                      h = R – d
                                                      q = 2 sin ^-1 l /D
                                                      f = cos ^-1 l / D
 
 

REFERENCES

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.

Peek, B.M., The Planet Jupiter: The Observer’s Handbook, Rev. ed, London; Faber and Faber Limited, 1981.
 
 

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