This note discusses the procedure for drafting the position of a terminator on a blank for planetary sketching either using a common Microsoft Windows tool (MS-Paint) and a Microsoft Office tool (MS Picture Manager) and by hand. The sketch blanks approximate the outline of the body and terminator that will be seen in a direct view eyepiece of a telescope tracking on an equatorial mount and where a reticule is used to indicate the direction to the celestial north pole (NCP). The terminator is pre-drafted based on a target observing date and time from ephemeris data.
The hand-drawing method updpates a procedure after Sidgwick, J.B. 1971. Observational Astronomy for Amateurs. Dover. pp. 121-124. Sidgwick's method approximates the terminator using a circle created with a drafting compass and is limited to 85%-100% illuminated fractions.
An 85%-100% illuminated fraction is the general range of the illuminated fractions of Mars that is visible to Earth based observers. Because of their extreme distance, superior gas giants (Jupiter and Saturn) generally have nearly 100% illuminated fractions at all optically visible Earth orbital positions. Inferior planets Venus and Mercury and the Moon have illuminated fractions that range from 0%-100%.
The variation on Sidgwick's hand-drawing method used here yields sketch blanks in the illuminated fraction range of 0%-100% for use with inferior planets Mercury and Venus. It can also be applied to preparing sketch blanks for whole, half and quarter Moon image drawings. This method also has limited application to the Sun and Saturn. Although these solar system bodies do not display a terminator per se, part of the technique discussed here includes looking up and marking the apparent position of the north pole of the body. For the Sun and Saturn that apparent position can move more than 10° in a year.
With proficiency, it is estimated a blank can be prepared in less than 10 minutes. A few simple multiplication and division calculations are most difficult math used to draw the terminator. A simple sine is calculated for the equator.
Online application alternatives
The easiest method to generate blanks with a terminator is to use one of two online applications. The NASA/JPL Solar System Live Planetica will show a realistic view of a planet as seen from Earth, magnified several hundred times, for a configurable date and time. Set the NASA/JPL Planetica to view a planet from the Earth with an apparent field of view that takes up one-half the width of the screen. The resulting image can be printed and the outline of the planet and shadow traced. The United States Naval Observatory Astronomical Applications Department has an online application that displays the current view of the Moon with terminator and latitude lines. Choose the option for "Data Services" and then "What The Moon Looks Like Today". The lunar view is based on a geocentric position. There may be minor differences in the appearance of the Moon from your topocentric position. Jim Mosher's Lunar Terminator Visualization Tool
(LTVT) is a good selenographic planetarium program that can be used for preparing lunar terminator blanks by tracing. Another is Chevalley's Virtual Moon Atlas (VMA). The USNO lunar application only projects an image for the current date and time. LTVT and VMA can be particularized for a specific date and time.
A net search for software that prepares blanks was done but nothing was found. If you are aware of any shareware or freeware for this function, please let the above author know.
Procedure overview
Ephemeris data is gathered that is necessary to construct the drawing. The planetary disk is drawn on a celestial NSEW grid. A tic mark is added to the circle which indicates the direction of the planetary north pole. An east-west trending line indicating the apparent direction of the bright limb is added. The size of the dark limb is marked on the bright limb line. A north-south trending line perpendicular to the east-west trending bright limb line is drawn. Using a lookup table, the radius of a circle that intersects the greatest size of the dark limb and the two points where the planetary disk intersects the north-south line to the bright limb. The arc of the circle that crosses the planetary disk is marked - approximating the terminator. The blank is finalized by removing construction lines and single or double reversing the image using a copier, as needed, depending on the number of reflections in the telescope used.
Nomenclature for ephemeris use
To read an ephemeris you need a basic understanding of this terminology that astronomers use to describe the many permutations of the three-way orientation that the Earth can take with respect to a planet and the Sun.
Figure 4 shows the relationship between the Sun's illumination of a planet and the view that we see of it from Earth. The view above the planet is shown in the top half of Figure 4; the view of the planet from Earth in the bottom half of Figure 4. Key to the relationship between the two views is the phase angle or in the nomenclature used by the NASA/JPL Horizons Ephemeris Generator the Sun-Target-Observer angle, abbreviated "S-T-O". The S-T-O angle is arc b to d in Figure 4. The phase angle S-T-O includes a right-angle and arc b-c in Figure 4. Arc a-b is also a right-angle. Therefore, arc a-c equals arc b-d in Figure 4, which is arc a-c also is equal to the phase angle S-T-O.
The phase angle modeled in the top half of Figure 4 (arc a-c) is proportional to the x axis dimension of the illuminated bright-limb fraction and the dark limb fraction of the planet as seen from the Earth. Arc a-c in the top half of the drawing is proportional to the length of the line e-f in the bottom half of Figure 4. Line e-f in the bottom half of Figure 4 represents the percent of the dark limb of the planet as seen by Earth (when divided by the planet's apparent diameter D). The length of line e-f, which is usually measured in arcsecs, is called the defect of illumination in professional astronomical jargon. In Figure 4, it is denoted by the variable q. Conversely, arc c-d in the top half is proportional to line
f-g in the bottom half and represents the illuminated or bright-limb fraction of the planet's disk as seen from Earth. The illuminated fraction (the length of line f-g) is denoted with the variable i in Figure 4.
The direction of the Sun in Figure 4 is indicated by two parameters. The first defines the direction of the Sun in x-y plane of the ecliptic and is indicated by the phase angle (S-T-O). Since the phase angle and illuminated fraction vary proportionally, the phase angle is also indicated by the illuminated fraction of the planet, or i, in the bottom half of Figure 4.
The relative direction to the Sun in the x-z axis of the ecliptic is indicated in the bottom half of Figure 4 by the position angle of the Sun on the disk of the planet as seen from the Earth. That position angle is arc NCP-e-g in the bottom half of Figure 4 and is also labeled angle pa_bl for "position angle of the bright limb". In this case Figure 4 is not a particularly informative schematic. It shows a planet with no inclination above or below ecliptic plane, so the Sun shines directly on the planet in the x-z plane of the ecliptic and has the uninteresting position angle of 270°. A planet below the ecliptic would have a position angle of the bright limb that is greater than 270°; a planet above the ecliptic would have a position angle of the bright limb that is less than 270°.
Another important component to representing the orientation of the planet is the relative tilt of its axis of rotation relative to the Earth. This value is indicated relative to the North Celestial Pole - that common frame of reference that you establish in your eyepiece by turning off the tracking mechanism on your equatorial mount and watching the direction of field drift. The relative position of a planet's north pole (labeled "npp" in Figure 4). In professional astronomer jargon, the relative position of a planet's north pole as seen in the eyepiece is indicated by the north planet position angle. The position angle of the north planet pole is labeled "pa_npp" in Figure 4. In Figure 4, the NPP could be located at any number of positions - it could even be obscured on the side of the planet facing away from the Earth - and it still would have the same position angle. You need a second characteristic - the distance between the planet's disk center as seen from the Earth to the planet's North Planetary Pole in order to locate it. This distance is known as the north pole angular distance, is usually given in arcsecs and is labeled h in the bottom half of Figure 4. Where the value is positive, the NPP is on the side of the planet facing Earth; if negative, the NPP is on the side facing away from the Earth. You can also estimate this distance and the NPP's position angle using an eyepiece reticule.
For the purpose of drawing a planetary blank, the significance of the mark indicating the position angle of the north planetary pole is twofold. First, the pole probably is not located at the tick mark made on the circumference of the blank; it is probably located elsewhere along the position angle line. Second, the position angle of the NPP gives an indication of the axis of rotation of the planet at the eyepiece.
As noted above, in Figure 4 arc b-d (the phase angle) equals arc a-c and arc a-c and line e-f (the dark fraction) are proportional, as are line f-g (the illuminated fraction). The illuminated fraction (the phase) and the phase angle can be related mathematically with a series of equations that are detailed in the Math Appendix. That you understand how these equations work is not necessary to draw a blank planetary sketch template with a terminator on it. However, to read an ephemeris you need a basic understanding of this terminology that astronomers' use to describe the relative orientation of the Earth to a planet and to `the Sun.
Normal ranges by which these apparent orientation characteristics of the planets change over the year, as seen from Earth, are summarized as follows:
Table 1 - Approximately normal annual ranges of characteristics describing the apparent positions of the major planets and the Moon
Linear radius on the drawing of the circle that approximates the terminator
r'
-
-
-
Computed: r'=y*r
Optional: Phase angle
delta
24. Sun-Target-Obsrvr angl [S-T-O]
Phase Angle
Phase
-
Optional: Apparent brightness
V
9. Vis mag. & Surf Brt [APmag]
Mag.
-
-
Optional: Observer's sub-latitude (Tilt - used to draw equator or Saturn rings)
beta
14. Obs sub-lng & obs sub-lat [Ob-lat]
Sub-Earth Pt. Lat.
Libration in latitude
-
Notes: All position angles are in the equatorial coordinate system. Some ephemeris programs use counterclockwise (CWW) rotation, others may report a negative clockwise rotation.
For the NASA/JPL Horizon application, the number and name of the menu selection is listed, followed by the corresponding field label on outputted reports.
(Calculate | Physical ephemeris | Body | Illumination or Rotation). Two calculations need to be run for each body. One to generate the "Illumination" report and a second to generate the "Rotation" report.
Linear radius on the drawing of the circle that approximates the terminator
r'
-
27.5mm
Computed: r'=y*r
Optional: Phase angle
delta
24. Sun-Target-Obsrvr angl [S-T-O]
26.5
-
Optional: Apparent brightness
V
9. Vis mag. & Surf Brt [APmag]
-3.8
-
Optional: Observer's sub-latitude (Tilt - used to draw equator or Saturn rings)
beta
14. Obs sub-lng & obs sub-lat [Ob-lat]
-
n/a
-
Materials for MS-Windows drawing
MS-Paint
Optionally, MS-Picture Manager, an MS-Office component, or any other drawing product that will allow you to rotate an image by a flexible number of degrees
Draw the outline of the body and bisect it with two perpendicular lines.
In MS-Paint, first draw a square of sufficient size by holding down the SHIFT key and dragging the rectangle draw tool. If the SHIFT key is depressed while dragging, MS-Paint forces the creation of square.
Put your cursor on the upper left-hand corner of the square. Select the ellipse tool. Hold the SHIFT key down. Drag the ellipse tool towards the other corner of the square. This will force the creation of a circle inscribed inside the square.
To draw perfectly horizontal and vertical lines in MS-Paint, hold the SHIFT key down while dragging the line tool.
The end result is Figure 5. It represents the outline of the body with the line of the position angle of the bright limb. A line perpendicular to the position angle of the bright line marks the cusps where the bright limb connects with the outline of the planetary body.
Mark the defect of illumination on the subscribing square and add the terminator with the ellipse tool.
Draw a linearly proportioned marker with the paint tool at points a and b at each end of the rectangle as shown in Figure 6. The length of the line represents the proportional linear amount of the defect-of-illumination. SeeTable 2, above.
Position your cursor over the edge of the left marker at point a. Dragging the ellipse tool, move the cursor to the edge of the second marker at point b. This will draw the terminator.
This is as far as you can go with Ms-Paint alone. You have constructed the planet body outline and the dark and bright limbs separated by a terminator.
Optionally, print the drawing and using a protractor, mark the position of the North Celestial Pole - which is the inverse of the position angle of the bright limb. Hand-rotate the drawing on a copier glass plate to make the final blank.
Rotate the image using MS-Office Picture Manager and draw North Celestial Pole
The position angle of the bright limb is 106° in the working example. Close the image in MS-Paint and reopen it using MS-Picture Manager.
Using the "Picture - Rotate" menu option in MS-Office Picture Manager, rotate the figure -6° to properly orient the position angle of the bright limb to 106°.
In this rotation, the North Celestial Pole is now at the top of the drawing.
Close the rotated image and reopen in MS-Paint. Recall that in MS-Paint, holding the SHIFT key down while using the line drawing tool forces the creation of true vertical and horizontal lines. Use this feature to create two more lines that bisect
planetary body. Label the North Celestial Pole.
This result is Figure 10, which shows the terminator properly oriented with respect to the North Celestial Pole.
Close and reopen the drawing with MS-Picture Manager. Again, using the rotation feature, rotate the drawing so the North Planetary Pole (NPP) is at the top of the drawing. For the working example, position angle of the NPP is 3.4° Rotate the drawing as close to -3.4° as possible. MS-Picture Manager will accept decimal negative rotations if typed in the applicable dialogue box.
Close and reopen the drawing with MS-Paint. Draw a single true vertical line and mark it as the NPP
Optionally, you can mark the position of the North Planetary Pole on the position angle line. In the working example, the NPP is 5.1 arcsecs along the NPP position angle line. The NPP distance is negative, so the pole is obscured behind the planet. Determine the proportional linear value of 5.1 arcsecs and mark that distance along the NPP position angle line. Mark it with a minus sign, if the pole is behind the planet, or a plus sign if the pole is on the Earth-side of the disk. In the working example, the planetary happens to be sitting right on the edge of body's outline.
Close and reopen the drawing using MS-Picture Manager. Now rotate the drawing back so the North Celestial Pole is at the top of the drawing,
You have now completed construction of the drawing as shown in Figure 12. The blank shows the terminator and the planet's rotational axis as it will appear in a reticule aligned to the North Celestial Pole.
In general, Sidgwick's construction procedure is followed. The procedure illustrated here differs from Sidgwick's method in an important respect. Sidgwick drafts directly to a double reversed view. Here, we construct the image in direct view on the assumption that a copier can be used to reverse the drawing the required number of times to make the final blank.
Begin by drawing the body's outline with a compass and adding the bright limb position angle line.
Draw the body's outline and bisect it with two perpendicular lines. This defines the North Celestial Pole (NCP). Label the diagram with the NCP.
With your protractor, mark the position angle of the bright limb - 110.2° - draw the bright limb line. Extend it beyond the circle on the side away from the limb. Draw a second line perpendicular to the bright limb line. This marks the cusps where the terminator meets the outline of the planet's body - at points c and c'.
Compute the linear size of the defect of illumination and mark it as point i along the bright limb line d - O - p. (The small letter designations used in this illustration match those used by Sidgwick (1971) at page 123.) In Figure 17, the linear diameter of Mercury is 218 pixels. The defect of illumination is 1 arcsec out of an angular diameter on August 18, 2006 of 5.7 arcsecs. That calculates ( 218 x (1-81.5%)) to about 40 pixels for a 18.5% defect of illumination.
Mark this linear size of the defect of illumination on the bright limb line. Measure from point d in Figure 17. The terminator will pass through this point and end at the cusp points c and c'.
Now use lookup Table 5 at the 81% or 82% illuminated fraction row and select a y value. For illuminated fractions between 100% to 85%, I like to use Sidgwick's values, for lower than 85% the sagitta method value. Here we will use Sidgwick's value of 1.36 for 82%.
Calculate the radius of a circle that will approximate the elliptical terminator - r' using the formula in Table 4 - r' = y * r where r is the linear radius of the planetary body. In this case, y * r = 1.36 * 218 pixels = 146.8 pixels = r'.
Next, measure along the bright limb line p - dfrom the defect of illumination marker at point i. Measure off the distance r'. This is the radius of a circle that we will use to approximate the terminator. Mark the center of this new construction circle as point f.
Place your compass on points f and i and approximate the terminator through the cusps at points c and c'. Sidgwick suggestions drawing 2/3 of the terminator with the compass and then completing the line to the cusps by hand. If you have used the sagitta method value, the compass by definition will go through points c, i and c'. With the sagitta method, the terminator can be drawn entirely with your compass.
Using your protractor mark the position angle of the North Planetary Pole (NPP). In the worked example it is 15°. In this example the NPP is 2.8 arcsecs from the center of Mercury's 5.7 apparent diameter disk. The radius of Mercury's apparent disk is about 2.8 arcsecs. Mercury's NPP is sitting on the disk's apparent edge on August 20 at 12 UTC.
The final blank, Figure 20, is similar in its orientation to the corresponding NASA JPL Planetica image. Minor differences are attributable to using Sidgwick's value for y instead of the sagitta method.
Table of radii for drawing the terminator
Table 5 shows values for y - the radius of a circle used to approximate an elliptical terminator - using Sidgwick's method and using an alternative sagitta method. This author was unable to replicate Sidgwick's method to find y, which is related to the formula of an ellipsis. Curve fitting indicates Sidgwick's values are approximately y=x^-1.5. Because it was apparent that Sidgwick's y values did not replicate the terminator at illuminated fractions near 50%, an alternative sagitta method was developed. Sidgwick's method is probably more accurate for illuminated fractions between 85%-90%, but his y values fail to draw accurate curves below about 75% illuminated fraction. The relative y values yielded by both approximations are shown in the following two figures. The sketcher is left to choose which best approximates the terminator for their specific application.
Because drafting techniques have been discussed in depth in prior examples, only brief instructions are provided in this example.
Materials assumed are MS-Paint, MS-Picture Manager, an MS-Office component, or any other drawing product that will allow you to rotate an image by a flexible number of degrees, and drafting data from the worked example ephemeris sample data file.
The key to constructing a Saturn blank using MS-Paint is understanding how MS-Paint draws ellipses. As noted above, the ellipsis tool in MS-Paint works by drawing the ellipsis that would be subscribed inside a square or rectangle. If the lengths of the major and minor axes of the required ellipsis are known, a square or rectangle can be constructed with sides equal to the major and minor axes's lengths. Registering the MS-Paint ellipsis tool on the corners of the constructed rectangle draws the desired ellipsis.
Compute the major and minor axes lengths for the parts of the Saturn figure
Let's use the oblate sphere of Saturn as an example. From the working data file, Saturn on August 20, 2006 has a tilt of about 15.8° away from the Earth and the NPP is -6.4° from the North Celestial Pole. This ephemeris file differs from the others that we have reviewed. It includes a new data item - the observer's sub latitude and longitude a.k.a. the sub-Earth latitude and longitude. The sub-Earth latitude and longitude are the Saturn topocentric latitude and longitude looking back at the Earth. The sub-latitude defines the apparent tilt of the rings. North (1997) provides a series of equations to approximate the various of rings and the body of Saturn at various angles of inclination. The following table lists the pertinent equations and works sample data for Saturn on October 20, 2006 at 10 UTC:
Table 6 - Figuring the size of Saturn ellipses
Measurement
Equation
Apparent size arcsec
Image units
ImageScale
Axis units
Tilt (beta)
-
-15.8
-
-
-
NPP position angle
-
-6.4
-
-
-
Equatorial diameter:
D
17.4
8.7
5.8
50.5
Polar diameter:
D-(2.1*D*cos(beta))
13.9
6.9
5.8
40.3
Ring C inner diameter major axis:
1.21*D
21.1
10.5
5.8
61.1
Ring C inner diameter minor axis:
1.21*D*sin(beta)
-5.7
-2.9
5.8
-16.6
Ring A outer diameter major axis:
2.26*D
39.3
19.7
5.8
114.1
Ring A outer diameter minor axis:
2.26*D*sin(beta)
-10.7
-5.4
5.8
-31.1
Equation source: North, G. 1997. Advanced Amateur Astronomy. Cambridge Univ. Press. At pp. 215-216.
Construct the registration marks in preparation for drawing the ellipses that make up the Saturn blank
In the Figure 27, two perpendicular lines have been drawn representing the North Planetary Pole (NPP) and E-W equatorial line of Saturn. A length on the East-West line equal to the apparent diameter of Saturn's body arbitrarily was chosen, resulting in an image scale of 5.8 pixels per arcsec. Next, the corresponding number of linear units for the computed polar diameter of Saturn, 13.8 arcsecs or 40.2 drawing units per semi-major axis is drawn on the North-South line. In Figure 28, perpendicular lines are extended from this units to find the corners of the subscribing ellipsis rectangle. In Figure 29, the ellipsis tool is used between these registration points to draw the appropriately sized and proportioned oblate sphere.
In the Saturn blank construction example, we are drawing three ellipses: 1) The body of Saturn, 2) the outer edge of Ring A - the outer ring, and 3) the inner edge of Ring C - the inner ring.
In practice, the registration rectangle corners and each ellipsis are not drawn sequentially. Rather, as shown in Figures 30 and 31, all the registration points are located first, and then all construction marks used find the registration points are erased.
Open the drawing in MS-Picture manager and rotate it in an amount equal to the north planetary pole position angle. Close and reopen the drawing in MS-Paint. Add two perpendicular lines representing the North Celestial Pole and the E-W line of the celestial sphere. Label the NCP
Open the drawing in MS-Picture manager and rotate it in an amount equal to the position angle of the Sun. Close and reopen the drawing in MS-Paint. Add one perpendicular line representing the bright limb line to the Sun. Label the Sun line.
Finalize the drawing per the usual conventions. The final drawing approximates the NASA JPL Planetica rendering except that this construction is rotated an additional 6.4° per the data in Table 6.
Because drafting techniques have been discussed in depth in prior examples, only brief instructions are provided in this example.
This method to draw the equator is generic and can be used to add an equator line to your planetary sketch blanks.
Materials assumed are MS-Paint, MS-Picture Manager, an MS-Office component, or any other drawing product that will allow you to rotate an image by a flexible number of degrees, and drafting data from the worked example ephemeris sample data file.
The key to constructing a Sun blank with the equator correctly oriented extends the methods that you study for drawing a Saturn blank. MS-Paint's feature is used to draw an ellipsis equal to the obs-Earth latitude of the Sun at the time of observation. The solar sub-Earth latitude represented angle of tilt relative to the Earth based observering and the Sun.
As noted above, the ellipsis tool in MS-Paint works by drawing the ellipsis that would be subscribed inside a square or rectangle. If the lengths of the major and minor axes of the required ellipsis are known, a square or rectangle can be constructed with sides equal to the major and minor axes's lengths. Registering the MS-Paint ellipsis tool on the corners of the constructed rectangle draws the desired ellipsis that represents the equator.
Compute the minor axis length for the north-south major axis of the Sun's equator
The example data file lists the Sun's tilt ("Obs-lat" field) as 6.76°. We'll use beta as the variable to designate this tilt relative to an Earth-based observer. Since the equator touches the east-west line of the Sun, we only need find the length of the north-south semi-major axis. Use the formula:
- to find the drawing units of this length. The image scale is the D" / pixels of the drawn diameter of the Sun. For the working example and data in the sample file, the calculation is:
Add the registration marks for the equator ellipsis and then draw the equator.
Add the two position marks on the north-south Sun polar axis that represent the semi-minor axis of the Sun's equator, tilted to 6.76°. See Figure 39, above. Then add the four perpendicular construction lines to find the rectangle corners for registering the ellipsis tool. Figure 40. Then draw the equator ellipsis using the ellipsis tool and the registration marks a and b. Figure 41.
This note is amateur astronomer product. Corrections to any errors are welcomed and appreciated.
References:
Duffett-Smith, P. 1988 (3ed). Sections 58, Phases of the planets, and 59, The position-angle of the bright limb. In Practical Astronomy with Your Calculator. Cambridge.
North, G. 1997. Advanced Amateur Astronomy. Cambridge Univ. Press. ISBN 0-521-57407-2.
Schlosser, W. Kaler, T.S., Milone, E.F. 1991. Challenges of Astronomy. Springer-Verlag.
Sidgwick, J.B. 1971. Observational Astronomy for Amateurs. Dover. pp. 121-124
