22 February 2015 conjunctions

A recent thread on NavList pointed out that a number of celestial objects appear in close proximity during the second half of February 2015. A quick application of the relevant spreadsheets show Venus and Mars separated by 24.7’, which is less than the apparent Moon disk size. The spreadsheet sailings.xls can be used in this case, since the angle of 1 minute of arc corresponds to the distance of 1 nautical mile on the surface of the Earth.

venus_mars

 

With a Moon crescent in the vicinity of the two planets, it has been mentioned that Uranus is also in the area, separated from the Moon limb by about 2 degrees. While in the first example the parallaxes were essentially negligible (and hence the sailings.xls spreadsheet could be used to calculate the Venus-Mars angular separation), this is not the case for the Moon. Therefore, to compute the lunar distance, there is a dedicated spreadsheet (ld_prec.xls) which takes the viewing location on the surface of the Earth (“AP”) as additional input to account for the parallax effect.

uranus_lunar

Uranus appears slightly to the west of the (almost new) Moon crescent, so the interesting quantity is its distance from Moon’s illuminated near limb. Subtracting the Moon semidiameter (16.4′) from the topocentric centered lunar distance of 2° 15.6′ yields a value that is very close to 2 degrees.

 

Ephemerides for the planet Uranus (along with Neptune and Mercury) are not listed in celestial navigation almanacs, as these objects are not suitable for astronavigation purposes. Nevertheless, we provide almanac spreadsheets for those three planets as well, since their data can be computed from the same VSOP87 planetary theory that we use for the other planets.

Venus-Jupiter conjunction of August 2014

As Frank Reed pointed in a recent NavList posting, Venus and Jupiter appeared close to each other in the morning sky of August 18, 2014, separated by about half a degree (30′, or by about the Moon apparent diameter). A quick use of spreadsheets venus.xls, jupiter.xls, and sailings.xls confirms this fact. The first two spreadsheets provide the planets’ ephemerides. The third one calculates the great circle distance of the bodies’ subpoints (geographical positions) in nautical miles, which is numerically very close to their angular separation in the sky in minutes of arc.

venus_jupiter_Aug2014

 

Additional details can be found on Steve Owens’s blog.

Example of a great-circle route

In a recent NavList thread titled “Great Circle Puzzle” it is revealed that a great-circle (i.e., straight, or, direct) sailing path exists between Pakistan and Russia.  This may seem impossible based on a quick look at the world map.  However, several NavList contributors established the end points and provided a general description of such a path.  Using these results it is possible to calculate this path in detail with sailings.xls and waypoints.xls spreadsheets:

waypoints.xlswaypoints

 

If you have a globe handy, check it out!

gc12

gc34

For additional details see this post and the entire thread.

World map credits:
http://www.mapsfordesign.com
http://www.bjdesign.com

(first published November 3, 2013)

Set and drift

The Sailings calculations determine the course for a vessel to follow in order to get from the point of Departure to the Destination.  This is course with respect to ground which can differ from the course to steer if the vessel is deflected sideways by currents and/or winds.  This leads to a class of “set and drift” problems described, for example, in the Dead Reckoning chapter in Bowditch.  These problems are often solved graphically by plotting procedures.  As for the equivalent numerical solution we can do the following.

The essence of these problems is the relationship between three vectors in which the ground speed is the vessel’s speed relative to the water plus the set and drift vector.  The relevant geometry occurs in 2-D, so this relationship translates into two equations for the vectors’ components.  Therefore, given four pieces of information on input we can solve for the remaining two.

If these two numbers both pertain to the same unknown vector (e.g. its magnitude and direction), then two Cartesian-component equations provide the solution to the problem by direct addition or subtraction of the other two (fully specified) vectors.  This is the case for spreadsheets:

http://www.navigation-spreadsheets.com/dead_reckoning.html#set_and_drift

ground_speed.xls:
Calculation of the ground speed from the current’s speed and direction (i.e. set and drift) and the vessel speed relative to the water.

course_and_speed.xls:
Calculation of the required vessel speed and course from the set and drift and the desired ground speed and track.

The situation is a little bit more complicated for:

course_to_steer.xls:
Given the set and drift, the vessel’s speed and the intended direction relative to ground, this spreadsheet calculates the required vessel course and the resulting ground speed.  If the vessel’s speed is too small to counteract the current, an error message is displayed in row 4.

in which the two unknowns are distributed between two partially known vectors.  This problem can be solved in the following steps encoded into the spreadsheet:
1) Decompose the current’s vector into components parallel and perpendicular to the prescribed “Sailings” (ground) direction.
2) Reverse the sign of the perpendicular component; this becomes the perpendicular component of the vessel’s speed with respect to water.  That way the deflecting effect of the set and drift is neutralized.
3) Use the Pythagorean theorem to determine the component of the vessel’s speed (w.r.t. water) parallel to the ground direction.  The known vessel’s speed w.r.t water is the hypotenuse and the result of step 2) is one of the sides.
4) Convert the vessel’s speed’s now known two components (along with the ground direction) into course to steer.
5) Add the two parallel speeds’ components in order to obtain the ground speed.  If this problem has two mathematically good solutions, this picks the “faster” one.

coursetosteerxls

This problem does not always have a solution.  If the vessel’s speed w.r.t. water is less than the current’s perpendicular component, the vessel is not fast enough to compensate for the deflection off course.  If the current’s parallel component is negative (e.g. a strong headwind in case of an aircraft) and larger in absolute value than the parallel component of the vessel’s speed w.r.t. water, then the vessel is not fast enough to progress toward its destination.  In either case, the spreadsheet zeroes out the output and displays an error message.

(first published on November 11, 2011)

Composite sailing

If the computed great-circle route were to reach into higher than desired latitudes, it is possible to eliminate that problematic section of the voyage with a composite sailing calculation.  Such a path consists of three sections:

1) great-circle route from Departure to the chosen Limiting parallel,
2) parallel sailing along the Limiting parallel,
3) great-circle route from the Limiting parallel to Destination.

The two great circles from parts 1) and 3) are chosen in a way that places both of their vertices on the Limiting parallel.

The example below uses the same San Francisco-Yokohama trip from our earlier SAILINGS blog entry.  Here the Limiting parallel is set at N 40º, below the original great-circle vertex latitude of N 48º 03.7′.  The spreadsheet is composite.xls.

compositexls

 

(first published on February 2, 2011)

Sailings

The term “celestial navigation” (sometimes called “astronavigation”) typically evokes in our minds an image of a sextant and comes along with terms like Universal Time, line of position, nautical almanac, intercept, etc. Indeed, those are among the topics discussed in our spreadsheet project so far:

http://www.navigation-spreadsheets.com/

With these concepts, techniques, and gadgets you can establish your position from astronomical observations.  You can also keep track of your changing position during a trip, for example by bringing the method of dead reckoning into consideration.  Until now, however, an important aspect of navigation (which is relevant whether you use celestial or not) has not been addressed by our suite: trip planning, also known as sailings calculations.

We are pleased to announce that we now provide this additional capability in the latest extension to our suite.  Determining the direction (course) in which to sail (and knowing in advance the length of the journey) is an essential skill that any navigator must have.  For short trips one may directly measure the constant rhumb-line course on a Mercator chart for the path that connects the point of departure with the destination.  However, the bigger the separation between departure and destination, the more extra distance is associated with the rhumb-line path compared to the shorter great-circle path, especially in higher latitudes.  The problem is that attempting to follow the requisite great circle is very difficult since it requires a continuous adjustment of heading.

Thus, on the one hand, the great-circle route (orthodrome) is shortest but it is difficult to steer.  On the other hand, the rhumb-line route (loxodrome) can be well followed along its constant course but it is longer.  Each choice thus has a strength that is a weakness in the other one.  A solution to this dilemma is outlined, for example, in Bowditch which recommends a hybrid path combining the advantages of the two sailing possibilities.  Here the starting point in developing the sailing plan is the shorter great-circle route from departure to destination, but then along that path one identifies waypoints (separated, for example, by 5 degrees of longitude) between which the vessel is to follow easier-to-steer rhumb lines.

We illustrate such a calculation using as an example a trip from San Francisco (USA) to Yokohama (Japan) borrowed from Bowditch.  This classic publication demonstrates the idea graphically using the chart of the North Pacific Ocean shown in two different projections.

First, the great-circle path from San Francisco to Yokohama is found as the straight line connecting the two cities on the gnomonic projection chart.  We mark the waypoints as this path crosses meridians separated by 5 degrees of longitude.  Second, these waypoints are translated onto the Mercator projection chart on which they are connected by straight-line segments representing rhumb lines.  The constant course headings within each successive pair of waypoints can be directly measured on this chart.  Our new spreadsheets perform this exact same function (plus the distance calculations) with even higher accuracy, because they are not affected by the inaccuracies of physical plotting on a chart.

The problem to solve is fully specified by the coordinates of the departure and destination locations.  We have:

San Francisco:
Lat: N 37º 48.0′
Lon: W 122º 33.0′

Yokohama:
Lat: N 34º 42.0′
Lon: E 140º 06.0′

These coordinates enter the spreadsheet sailings.xls in row 2.

sailingsxls

The use of this spreadsheet is shown in this YouTube demo video.

The differences between the calculated great-circle and rhumb-line paths are substantial.  The rhumb line is longer by over 200 miles and its (constant) heading is south of west, while the initial great-circle course is north of west sailing into higher latitudes first.  Row 11 displays the coordinates of the vertex, which is the point along the great circle closest to the Pole.  The spreadsheet also shows the (relatively minor) differences arising from the use of a perfectly spherical or slightly flattened ellipsoidal model of the Earth surface in the calculation.

For our purposes the main result of this spreadsheet is the initial great-circle course displayed in the yellow cell C6.  This number, combined with the departure coordinates, completely defines the great circle.  The value is shown with three decimal digits (302.240) in order to cut down numerical round-off errors once it is copied as input into the next spreadsheet: waypoints.xls.

waypoints1

Here, the coordinates of San Francisco and this initial course from sailings.xls enter in row 2.  Then, in column A starting in row 11 we begin entering the longitudes of each waypoint.  All great circles (except those running across both Poles) intersect every meridian exactly once.  The calculated latitude of each waypoint is displayed in columns C, D, E.  The rhumb-line distance and course from the previous waypoint (using the flattened Earth model) is shown in columns F and G.

The initial course is north of west so the latitudes of the subsequent waypoints increase at first.  The courses between them, however, progressively turn away from north and at longitude W 170º (close to the vertex, row 7) it is essentially due west.  The course then heads ever more southward as the path descends back to lower latitudes toward Yokohama.  Figure 2404 in Bowditch shows the E 150º waypoint at latitude N 40º, for which waypoints.xls calculates N 39º 45.5′.  The last waypoint is the destination itself (Lon: E 140º 06.0′) with the correctly reproduced latitude (N 34º 42.0′).  This hybrid path is still longer (cell F2) than the pure great-circle route, but not by much, and it is easier to steer.

(first published on January 15, 2011)