For many years the ALPO Mars Section has engaged in micrometer measurements of Mars to plot the retreat and reformation of the planet's polar caps. The small apparent size of Mars renders micrometer measurement very difficult and laborious task amateur astronomers are not likely to enjoy. You might compare this type of observing to having a tooth filled without the benefit of Novocain. However, it is an important method for studying the behavior of Mars' polar caps and their effect on the planet's weather.  Adding to this difficulty is the variable "astronomical seeing" conditions that causes the apparent disk of Mars to blur or expand. Clouds and hazes at the Martian polar regions also hamper measurements.

During the 1960's and 1970's experiments in measuring Martian polar caps from photographs were carried out by the ALPO Mars Section. A two degree (2°) systematic error in latitude was found by the late Charles F. ("Chick") Capen using this method even on photographs taken with a variety of telescopes and apertures. Measurements made by camera were compared with those made with bi-filar micrometer using the same telescope and plate scale. Photographs taken during less than ideal "astronomical seeing" resulted in increased errors.

No empirical method has been found to correct for these systematic errors. So, we adopted the bi-filar micrometer, and in some cases the reticule micrometer, as our main polar cap measuring device.

In recent years we have attempted to employ an alternate method of measuring the Martian polar caps used by other observing organizations. We have found this method to be outmoded and to produce high rates of systematic errors. We also feel that measuring the Martian polar cap from drawings produces errors in excess of those by the photographs or micrometer.  Also, we use a deep red filter in the optical system while measuring the Martian polar caps to penetrate through the planet's atmosphere. A red filter also reduces the effects of poor seeing and irradiation of the bright planet in the micrometer eyepiece.

The current ALPO Mars Recorders continued to use an old and proven Indirect-Direct Method of measuring the polar caps with a bi-filar micrometer [Peek, 1981]. This method is desirable because it reduces possible mechanical errors in the bi-filar micrometer and eliminates the need to compute the micrometer zero.

In both methods discussed below the disk of Mars is measured using the Direct-Indirect method discussed next (See Figure 1).

To measure with Indirect-Direct method an object: first indirectly (In) measured between the fixed and movable webs, then, the object is positioned across the fixed web to the other side. The movable web (M) is adjusted across the fixed web (F) and the object is then measured directly (Di) with the opposite screw adjustments [Beish et al, 1986]. The results are averaged and the separation of the webs are determined by:

Figure 1. Appearance of Mars in micrometer eyepiece using ALPO method. In the left image, Mars is situated between web (M) and web (F) in Direct side. Right image Mars is moved down by the telescope drive and situated between web (F) and web (M) in the Indirect side. The movable web (M) is adjusted down and across the fixed web (F). The web (T) is used for centering. Measuring the apparent disk (D) of Mars will be used in both methods described below.
 

We will begin by discussing the alternate method of measuring the polar latitudes of Mars. The alternate method often uses uses drawings to determine the polar cap latitudes of Mars and may be subject to additional systematic errors due to personal errors or hand-eye coordination problems.

First, the north-south thickness of the polar cap is measured. Then the apparent disk is measured and the radius is determined by dividing this diameter by 2 and used along with the Declination of Earth (De) in the following geometric equation [Minami, 1993], [Minami, 1993]:

Figure 3. Appearance of Mars in micrometer eyepiece using ALTERNATE method. In the left image, the apparent polar cap (C) is situated between web (M) and web (F) in Direct side. Right image Mars is moved down by the telescope drive and the apparent polar cap (C) is situated between web (F) and web (M) in the Indirect side. The movable web (M) is adjusted down and across the fixed web (F). The web (T) is used for centering.
 

For the ALTERNATE method: cap thickness (d) = 0.0273 mm, radius (r) of Mars’s disk = 0.64 / 2 or 0.32 mm

This calculation results in the colatitude of the cap, so to obtain the latitude simply subtract the colatitude from 90 degrees:

However, even using a high quality bi-filar micrometer a slight misalignment or position error due to blurring of the image in the webs while measuring the thickness of the cap can produce large errors in colatitude. Another problem arises when the De approaches zero or becomes less than the desired tilt angle of the polar cap of interest. Also, the phase defect or terminator can cover much of the backside of the polar cap near some oppositions, therefore, causing a large observational and calculation error!

ALTERNATE method: cap thickness (d) = 0.0393 mm, radius (r) = 0.332 mm

Second, 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 2. Appearance of Mars in micrometer eyepiece using ALPO method. In the left image, the apparent polar cap (C) is situated between web (M) and web (F) in Direct side. Right image Mars is moved left by the telescope drive and the apparent polar cap (C) is situated between web (F) and web (M) in the Indirect side. The movable web (M) is adjusted left and across the fixed web (F). The web (T) is used for centering.
 

This results in straightforward latitude of the edge of each side of the polar cap, or the colatitude can be found by the equation: arcsin (1 -C/D). [Beish, et al, 1986]. This method and calculation has proven quite good and produces less than 0.5 degree systematic errors (see Figure 1 and Figure 2).

For example, let's consider a typical telescope and micrometer used by ALPO's Mars Recorders during the last apparition. We might have measured the north polar cap of Mars when the planet was 10 seconds of arc apparent diameter and a declination of the Earth (De) of 10 degrees. Our observing period would be under perfect seeing conditions and then under less than perfect conditions where the planet would expand by one-micrometer web or 0.012 mm. Using this author's micrometer screw constant of 9.526 arcsec/mm the expansion of 0.012mm would correspond to slightly more than 1/8th of an arcsec during less than perfect seeing. Not a very large difference in seeing or image blurring.

Now we will use the standard ALPO micrometer method, measuring Mars' polar cap width (east-west) resulting in a cap width of 0.153mm and the apparent disk diameter as 0.640mm and for the alternate method a cap thickness (north-south) of 0.0273mm: where the cap width (C) = 0.153 mm and disk (D) = 0.640 mm

Now, let's add the one 0.012mm web image expansion or blur due to less than perfect seeing conditions and we have a cap width of 0.165mm, and a apparent disk of 0.652mm, or a cap thickness of .0393mm and a disk radius of .332mm:

Measuring the polar cap in less than perfect seeing conditions that would expand and contract Mars’s disk by 0.012mm or one web thickness, a systematic error can be calculated. In the ALTERNATE method, the latitude of the cap during perfect seeing would be 76.2 degrees. During less than perfect latitude of 71.8 results. In the ALPO method, the latitude of the cap during perfect seeing would be 76.2 degrees. However, it is measured 75.3 degrees during less than perfect seeing.

A difference in the measurements from the ALTRNATE method results in an error of: 76.2 - 71.8 = -4.4 degrees. The ALPO method results in an error of: 76.2 - 75.3 = 0.9 degrees. A significant difference.

Observers using micrometers are encouraged to use both methods and determine for themselves that is best for their equipment and conditions.

Beish, J.D., Parker, D.C., and Capen, C.F., "Calculating Martian Polar Cap Latitudes", J.A.L.P.O., Vol. 31, Nos. 7-8, April 1986.

Minami, M., "The Recession of the North Polar Cap," Communications in Mars Observations, OAA Mars Section, No. 138, 25 October 1993.

Minami, M., "Tendency of the NPC Recession in 1992/93," Communications in Mars Observations, OAA Mars Section, No. 139, 25 November 1993.

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