Measure the Polar Caps of
Mars (
Rev. 13-Feb-2018 )

Discussed in this article will
be the method of using a Ron Darbinian Bi-Filar micrometer (See
Figure **M3-1** ) at the telescope and reduction of this
data to a useable form. Also, using digital images and image
processing software to analyze these images and to refine the
polar cap measurements that reduces systematic errors is
presented. The separation is read from
the micrometer thimble and spindle and noted in either in
millimeters.

Figure M3-1.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. BOTTOM: Darbinian micrometer thimble and spindle for reading separation in millimeters.

In our first example, Mars is observed January 27, 2010 in
perfect seeing using a 16-inch (406.4mm) f/6.9 telescope with
an effective focal length of 8,532 millimeters (3.04x Barlow
used). We find the screw constant or image scale then becomes;
206,265 / 8,532 mm = 24.18 arcsec/mm. We might have measured
the north polar cap when the planet was 12.3 seconds of arc in
apparent diameter and a Sub-Earth Point or Declination of the
Earth (**De**) of 14.9°.

The reason for the measurements
is to determine the apparent latitude (**L**) of the
east-west edges of the polar cap using a standard spherical
geometric equation:

( **1**
)
L = cos^{-1} (C/D),

Where **C** is the
measured width of the polar cap measured and **D** is the measured diameter of the disk of
Mars.

USING DIRECT-INDIRECT METHOD

The method used in this article
for measuring the polar caps with a bi-filar micrometer is the
Direct-Indirect Method [**Peek**, 1981].
This method is desirable because it reduces possible mechanical
errors in the micrometer and eliminates the need to compute the
micrometer zero. In this method, half of the measurements are
made on one side of the fixed web and half on the other
side.

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 **M3-3** ). 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* [

Figure M3-2. 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.

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

Second, the micrometer is
rotated 90 degrees to the right, as the arrow indicates in
Figure **M3-3** , and the webs are then returned to their
initial positions -- that is with web (**M)** back across
web (**F)**. The polar cap breadth or East to West
span across the sphere of the planet is measured directly and
indirectly.

We find the polar cap width (Figure M3-3. Appearance of Mars in micrometer eyepiece using ALPO method. In the left image, the apparent polar cap (C) is situated between the movable web (M) and the fixed web (F) in DIRECT side. Right image Mars is moved right by the telescope drive and the apparent polar cap (C) is situated between the fixed web (F) and the movable web (M) in the INDIRECT side. The movable web (M) is adjusted right and across the fixed web (F). The web (T) is used for centering.

C = (Di - In) / 2 - Wt = (10.8422 - 10.5122) / 2 - 0.012 = 0.33 / 2 - 0.012 = 0.153 mm

Next, usung equation (**1**)
the apparent latitude (**L**) is: L
= cos^{-1} (0.153 / 0.5088) =
72.5°

To compute width
(**W**) of the polar cap this equation is
used: **W = 2 sin**** ^{-1} (C/D)**
, where C = 0.153 mm and D = 0.5088 mm from
above:

(

**Systematic error** due to
“astronomical seeing”: If the image of Mars would expand
(blur) by one-micrometer web, or 0.012 mm, we would find the
cap width and disk diameter vary from 0.153 mm to
0.165 mm and the disk from 0.5088 mm to 0.5208 mm
respectively, then: L =
cos^{-1} (0.165 / 0.5208) =
71.5°, with an error of -1°.

ALTERNATE METHOD TO MEASURE POLAR CAP

The alternate method often uses
drawings to determine the polar cap latitudes of Mars
[**Dollfus**, 1973] and may be subject to
additional systematic errors due to personal errors or hand-eye
coordination problems.

First, find the planetocentric
angle (β) of the polar cap by measuring
the north-south thickness of the polar cap as shown in figure
M3-4. The apparent disk (0.5088 mm) is measured as in figure
M3-2 and the radius is determined by dividing this diameter by
2. Then take the absolute value the Declination of Earth
(**De**) to find the planetocentric latitude of the Earth
(**|∅|**) and apply these to the
following geometric equation:

[2] β
= cos^{-1} (1 – d / r) -
(**|∅|**)

whereβis the planetocentric angle of the polar cap,Deis the Declination of Earth,dis the thickness of the cap, andris the radius of the apparent disk (See FigureM3-2).

Figure M3-4. 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.

Using the apparent disk diameter (**D**)
as we found in Figure M3-2 (0.5088 mm) and then measuring the
north to south thickness of the cap, we find:

d = (10.73294 - 10.62146) / 2 - 0.012 = 0. 04374 mm

With a
β = cos^{-1}
(1 – d / r) - |16.6°| =
cos^{-1}(1 – (0.04374 /
0.2544)) - 16.6°

= cos^{-1} (0.82806) = 34.1° -
16.6° = 17.5°

**Note:** The latitude
of the polar cap at the central meridian = 90 - β, or 90° - 17.5° = 72.5°

**Systematic error** due to
“astronomical seeing”: If the image of Mars would expand
(blur) by one-micrometer web, or 0.012 mm, we would find the
cap width and disk diameter vary from 0.04374 mm to 0.05574 mm
and the disk radius from 0.5088 /2 mm to 0.5208 / 2 = 0.2604
mm, respectively, then:

β = cos^{-1}
(1 - (0.05574 / 0.2604)) - |16.6°| = 38.2° -
16.6° = 21.6°,

tthen the latitude would be: 90° - 21.6° = 68.4°. The result is an error of 4.1° (72.5° - 68.4°).

EFFECT OF PHASE DEFECT ON POLAR CAP MEASUREMEMNTS

There is one other problem --
the Martian phase. The Red Planet can have a significant phase
defect, or terminator, near quadrature when Earth-based
observers will see less than a fully illuminated disk. However,
if the angle of the phase defect (**terminator**) is in the opposite hemisphere from the
**De**, i.e., the **De** is positive and the **PA of
Defect** is negative, then the terminator cusp will be clear
of the edge of the polar cap. In this case there will be no
problems with our calculations. **Note:** Shortly
before or after opposition, the phase becomes undetectable as
it approaches 99%. Phase corrections for polar cap measurements
are unnecessary for phases of 99% or more.

Now for a practical example we
find the Red Planet near quadrature and usually a phase
terminator will be present. For example, on November 01,
2009 there will be a 39.4° phase angle (**i**)
present, so Earth-based observers will only 88.7% of the
illuminated disk of Mars.

From the example in the above FiguresFigure M3-5. Simulated view of Mars on 11-01-2009 with a phase angle (i) at 39.4°, Positional Angle of Axis (P) = 355.5° and the Positional Angle of the defect (Q) = 284.8°. Figure M3-6. Illustrates the incidence angle of the Sun, ("Is"). Maximum Phase (Qm) = 289.3°, measured counterclockwise from North Pole of Mars to the greatest defect of illumination.

( **3** )
Correction for the north polar cap: AC = C / k = 0.153 /
0.887 = 0.1725

( **4**
) Corrected
latitude B = cos ** ^{-1}** (AC / D) = cos

Where k = 0.5 (1+cos i) and "i" is the angle of defect, or, "phase angle."

A difference in computed for the north polar cap latitude from 72.5° to 70.2° is -2.3° and that may be considered a significant difference.

Another problem leading to
systematic errors in polar cap measurement is the fact that the
cap may be clear of the phase altogether. Or it may be tilted
sufficiently towards the earth so the cap is less embedded in
the shadow, again making equation (**3**) less accurate. The
phase defect or "terminator" can cover some or all of
the polar cap at times; Figure **M3-6** illustrates the
effects of a positive or negative effect of the Martian axis
position angle and the position angle of the defect. Formula
(**4**) applies only when the incidence angle of the Sun,
("**Is**") is 0°. This angle is derived as
follows:

[4**a**]
Is = 270° - (Q - P), (before
opposition)

[4**b**] Is = 90° - (Q - P), (after opposition)

[4

where "Q" is the Position Angle of the Defect, "P" is the Position Angle of the Axis, and remembering to normalize the results to under 360°.

Figure M3-6. Illustrates the incidence angle of the Sun, ("Is").

**PHASE ANGLE EQUATIONS** :
Phase (**k’**) is the ratio of the polar diameter to
the illuminated portion of the equatorial diameter. The
equation for phase is:

[5b] or if Is <0° then k’ = {(1 + cos i)(1 + cos Is)}/ 4 [5c] North Polar Cap: After Opposition and if Is >0° then k’ = {(1 + cos i)(1 + cos Is)}/ 4

[5d] or if Is <0° then k’ = (1 + cos i) / (1 + cos Is) [5e] South Polar Cap: Before opposition and if Is >0° then k’ = {(1 + cos i)(1 + cos Is)}/ 4

[5f] or if Is <0° then k’ = (1 + cos i) / (1 + cos Is)

[5h] or if Is <0° then k’ = {(1 + cos i)(1 + cos Is)}/ 4

Note: Shortly before or after opposition, the phase becomes undetectable as it approaches 99%. Phase corrections for polar cap measurements are unnecessary for phases of 99% or more.

In Figure **M3-5** (11-01-2009) the phase
(**i**) = 39.4°, Q = 284.8° and P = 355.5°, so we calculate:
Is = 270° - (284.8° – 355.5°) = 340.7deg;. In
the example we find that Mars has not reached opposition and it
has a phase that is tilted with an angle Is = 340.7°, so we
will use formula [ **5b** ] that results with:

k’ = (1 + cos 39.4°) / (1 + cos 340.7°) = 1.7727336 / 1.943801 = 0.912

Now that the part of the phase
defect which affects the cap has been corrected (**k’**), we
can rewrite equation (**4**) above to yield the correct cap
width:

[ **6**
]
AC = C/k’

and, finally, use equation
(**3**) to yield the latitude (**B**) of the edge of the
Polar Cap:

B = cos^{-1}
(AC / D), where AC = 0.153 / 0.912 =
0.1678

The Areocentric
corrected latitude is; B = cos^{-1}
(0. 1678/ 0.5088) = 70.7°.

The difference between the corrected latitude using only the phase and when considering all the angles is 72.5° - 70.7° or 1.8°. This as a large systematic error.

POLAR CAP MEASUREMENTS WITH CCD CAMERA

While we found a less than two degree (1.8°) systematic error in latitude using photographs to measure the polar caps of Mars these systematic errors are reduced significantly when using CCD images. A typical set of CCD images of Mars 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 Mars at the micrometer eyepiece.

Figure M3-8. Typical Don Parker CCD image of Mars as shown on PC screen using image processing program (south at top). Cursor placed at extremes of north - south and east - west of image. Resulting pixel locations are 10 to 178, and 76 to 120. Differences or distances are: 178 - 10 = 168 and 120 - 76 = 44.

In Figure **M3-7** the CCD
image of Mars was 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. Using the pixel positions of the image on the PC
screen is the same as using the fixed and movable webs of the
micrometer, as all we need for the calculations is simple
ratio.

The apparent disk of Mars is
measured from north (pixel-**178**) to south
(pixel-**10**) that yields a difference of 178 – 10 =
**168** pixels. The extent of the east (pixel-**76**) to
west (pixel-**120**) edge of the polar cap yields a
difference of 120 - 76 = **44** pixels. 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. Applying the differences
or distances between extents using the equation above, we
find:

L =
cos^{-1} (44/ 168) =
cos^{-1} (0.2619048) =
74.8°

Figure M3-9. Another Don Parker CCD image of Mars taken at closest approach on 27 August 2003 as shown on PC screen using image processing program. Cursor placed at extremes of north - south and east - west of image.

In FigureIn figure M3-8 the
apparent disk of Mars is measured from north (pixel-**285**)
to south (pixel-**38**) that yields a difference of 285 - 38
= **247** pixels. The east (pixel-**134**) to west
(pixel-**187**) extents of the edge of the polar cap that
yields a difference of 187 - 134 = **53** pixels:

Latitude =
cos** ^{-1}** (53/ 247) =
cos

Since the south polar cap is
offset from the pole by approximately 7° in the general
direction of 30° - 60° Areographic longitude we usually use the
width (**W**) of the cap to study this region and can be
found using the equation:

W =
2*sin** ^{-1}** (53 / 247)
= 2*12.39 = 24.8°

TECHNIQUE

Using a typical image processing
programs an image is opened to the screen and usually the image
gamma and/or brightness and contrast, and image scale is
increased 2x to 3x to show the pixel patterns at the edges of
Mars’ limbs and polar cap. If the image is not sharp then
some image sharpening may be necessary until the pixels can be
distinguished (see Figure **M3-9**
).

To find the very edge of the feature, especially in a fuzzy image, one clicks the proper icon to read the “Red, Green, Blue” density levels of the dark background and then drag it onto the image at a point where the levels appear to indicate an exposed pixel. The real edge can be assumed to be somewhere between the background and the apparent exposed pixels. (Note: pixel X-Y is displayed in bottom toolbar in Paint Shop and “dropper” icon is equivalent to Adobe Photoshop “Info palette window” using the “eyedropper tool”).

In this example the cursor
‘X-Y’ pixel is set near edge in background to (**74, 80**)
and the RGB levels may read ’43.” Moving the cursor right
into the brighter edge, the RGB levels begin to increase to
read ‘63” at ‘X-Y’ pixel (**80, 80**). Then, move the
cursor further right until the RGB levels become stead at
‘83’. Move back until it begins to decrease and record
RGB levels ‘63” at ‘X-Y’ pixel (**86, 80**). For a
movement of +/- 6 pixels we find the RGB levels average:
(83 + 43) /2 or 63. That is the assumed real edge
of the feature. In Figure M3-9, the image us of a
magnified section of the evening side of the north polar
cap.

Figure M3-10. A 3X magnified section of the evening side of the north polar cap of Mars as shown on PC screen using image processing program. Cursor move and at least tree X-Y positions and RGB levels recorded.

Reference

Dollfus, Audouin, (1973), "New optical measurements of planetary diameters IV. Size of the north polar cap of Mars," Icarus 18, 142.

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Caps