Celestial Navigation with X-Plane 11 – Part 2 – The Sight Reduction Method

We continue our look at celestial navigation in X-Plane 11 with a breakdown of The Sight Reduction Method with Guest Contributor Cygon Parrot…

Part 2

The Sight Reduction Method that will be described in the following text does not depart from PaulRix’s description much, except in one major way; you must assume you do not have Google Earth upon which you can so graphically picture the intersection of three circles around the GP of the celestial bodies. You must still calculate the geographical positions of the bodies at the exact time of the sighting. You must still determine the Lines of Position from the bodies and establish their intersection point, which will be the navigator’s position. Where it differs is in the procedure. Before we dive in, however, there are a few precepts that must be acknowledged and understood, or the task will undoubtedly seem daunting and long winded.

First and foremost, the planning of the flight itself must be meticulous. Celestial Navigation, with this method, is a position confirmation exercise. You should already have a dead reckoning navigation plan, including as many en-route known or forecast conditions as possible, and know your aircraft’s performance.

We will need some resources. As before, we require a current Nautical Almanac. We will also need two specific documents, the Sight Reduction Form (SRF) and the Universal Plotting Sheet (UPS). For absolute purists, we will need the HO-229 publication. Finally, we must have at hand a ruler, dividers, and a pencil. A scientific calculator, although anachronistic, could also be helpful.

The Nautical Almanac has already been amply described, so let us move on to take an introductory look at the Sight Reduction Form.

This is the form where all the calculations take place. It is divided into four sections, the elements of each which will be described. It is worth noting that the last section, Determining a Line of Position, assumes use of the HO-229 publication for looking up the Computed Altitude of the body (Hc) and the Azimuth Angle (Z). There is a mathematical way to short cut the HO-229, it should be noted. We will see that, too, soon enough.

Before we start an explanation of the SRF I do want to mention, for those eager to get straight to navigating by the stars, that I developed a small application for Windows of a semi-automated SRF in Visual Basic. It will start the new celestial navigator on the road of rediscovery with relative ease, but I do advise becoming familiar with the calculations of the manual SRF, just the same. Find the instructions for that application after the description of the manual SRF procedure.

The Universal Plotting Sheet is nothing more than a blank segment of a chart, if you will, with no features, but with three evenly spaced parallels and one central meridian. Note that their value (Eg; N45°) is not included, and must be manually added, according to your dead reckoning position. It should now be obvious why you need to have a dead reckoning flight plan.

The Sheet omits including any additional meridians because of their tendency to converge towards the poles. The small legend in the lower right corner is used to determine the exact value of the distance between meridians at the corresponding parallel, using the dividers. As the navigator, it is your job to pencil in the two lateral meridians at the correct separation. An Assumed Position for each star sighting will then be penciled in, and from it the Line of Position drawn in. The intersection of these LOPs will pinpoint your location at the time of the sightings.

We will now take a quick look at the HO-229 publication. Here is one example page.

This publication comes in six large volumes. Its purpose is solely to provide the navigator with precomputed, tabulated spherical trigonometry calculation solutions. This was all from the time before there was even a pocket scientific calculator, of course. Wading through these sizable volumes to extract just three numbers was then more efficient than the navigator sitting at his table with sine and cosine charts and doing the formulae manually. Now, with a calculator, it is a breeze, and the HO-229 can all but be done away with. However, more for historical interest than anything else, I include the description of it.

In short, you select which volume you need of the HO-229 based on your Assumed Position latitude. Then, with reference to your Local Hour Angle (LHA), you find the applicable page (look at top and bottom right of the example). You would then establish if the sighting Declination is in the same or in the contrary hemisphere to your Assumed Position latitude. Finally, you use the value of your sighting’s Declination, on the column of your Assumed Position latitude, to find; the Computed Declination (Hc), the Computed Declination correction (d), and the Azimuth Angle (Z). These values then get written into the last section (IV) of the SRF, for some additional corrections. From that, you will have obtained the corrected distance from the Assumed Position, along a calculated Azimuth, which is then drawn onto the UPS to establish the LOP.

I am aware that the HO-229 is a very tall order for starting out. For that reason, I will base the description of filling in an SRF using the formulae found in the Nautical Almanac, for the main part, and using the VB SRF application (which makes it very easy).

That concludes the introduction. We can now move on to the description of how the SRF is filled in. Once again, for those impatient to start, skip forward to the Plotting Sheet section, to obtain an understanding of how it is used, and then on to the Sight Reduction Form Tutorial, for hands on navigating with the VB SRF application.

Filling in the Sight Reduction Form

Section I: Observations and Corrections

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Because we will be using Stellarium with Casper’s application in this example, some of the entries are redundant on the Form. We will look at them briefly, just the same.

Item 1. Simple, the name of the body you are using as reference.

Item 2. For our purposes, it suffices to write in the reading of the Altitude you obtain from Stellarium into the Sextant Altitude box. Do take note of the approximate direction we are looking, while we are doing this (in this example, NW). It will help us later.

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Note that I have converted it from (degrees/minutes/seconds) to (degrees/decimal minutes) format, and rounded to one decimal place (27’ 19.1” = 27.3’). This is the format of the Nautical Almanac.

The Index correction is a mechanical error that can affect sextants. It is particular to each particular sextant, and as its owner you would know it and add or subtract it from every reading you take with that sextant. It is not applicable to Stellarium.

Dip is a correction of eye height above the ground. Simply put, as the planet is spherical, the higher you are, the more you see over the horizon, and thus need to correct for in celestial calculations. Casper’s application eliminates the need to calculate this, as eye height is always zero. However, if it were applicable, you may calculate it with this formula…

Dip = 0.97 x Square root( eye height in feet )

The result should be subtracted from the sextant reading, so simply express that answer as a negative for the purposes of the Form.

Item 3. The main concern here, for the purposes of Stellarium, is the Altitude Correction. It is a correction for atmospheric refraction, which affects a sighting progressively more the closer to the horizon the celestial body is (because there is more slant distance through the atmosphere), from the observer’s view point. For it to work, in Stellarium, you must have the atmosphere set on. You can obtain the value from the Nautical Almanac, as described earlier by PaulRix, or you can use one of a couple of formulae to compute it. I will include them here.

For sightings that are higher than 8° above the horizon, a simpler formula can be used. Normally, this one will suffice for most sightings because you ideally want to select bodies that are between 20° to 75° degrees above the horizon for your sightings…

Altitude Correction = 0.96 x Tangent( Total Apparent Altitude )

Do note, you must convert the Apparent Altitude to decimal degrees for this to work. In our specific example, 33° 27.3 = 33.455°

If you simply must take a sighting of a body that is less than 8° above the horizon, then the following formula should be used (which incidentally calculates with slightly more precision for any sighting, including those above 8° from the horizon)…

Altitude Correction = Cotangent( Apparent Altitude + ( 7.31 / ( Apparent Altitude + 4.4 ) ) )

For those wondering, in order to express a cotangent on your calculator, consider that it is the reciprocal of a tangent (ie; 1 / Tangent).

The Additional Atmospheric correction is not significant for use with Stellarium. The Additional Corrections for Mars and Venus, which are very small and are seasonal, need to be obtained from the Nautical Almanac. They are almost insignificant, never exceeding 0.4’ for Mars (between August and September), and 0.2’ for Venus (between January and February), so they can safely be ignored for these exercises (I have not even been able to positively determine if the requirement for these corrections are even active in Stellarium, they are so small, so the process would probably be redundant even if you did perform it). Therefore, we will safely ignore it.

In the last box we write our final, Total Observed Altitude (Ho).

Section II: Time and Dead Reckoning

Here, we need to have accurately composed and maintained our navigation log. The best way to do this is to pre-plan a point at which you will take your sightings, calculate your estimated time at that point, and express the position of that point in latitude and longitude. When you get to the time when you would be at that expected point, according to your plan, take your sightings. Universal Time Coordinated (UTC/GMT) must be used for celestial navigation.

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Item 4. The date on the Prime Merdian at the time of taking the sighting.

Item 5. The latitude you expect to be upon at sighting time, according to your navigation log.

Item 6. The longitude you expect to be upon at sighting time, according to your navigation log.

Item 7. The exact time in UTC (GMT) of the sighting, HH:MM:SS.

Section III: Latitude and Longitude

This section starts out reasonably straight forward, but gets a little complicated at points. Some attention needs to be expended on getting it right. In this section, we will obtain our Assumed Position , which is a very important piece of information for the Universal Plotting Sheet. It is worth noting that the Assumed Position is not the same as the Dead Reckoning Position. It is not even a true “assumed” position, in the technical sense of the word. It is a calculated reference position that is derived from your Dead Reckoning position with a process. There is no guess work. Here is the completed section…

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I know that looks a little scary. However, each item is its own little process, and is not really as hard as it appears at first glance. From PaulRix’s earlier description, some of this is already familiar ground. Let us start.

Item 8. GHA (Greenwich Hour Angle)

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Note the date and the time. Locate the page in the Nautical Almanac for that date October 25th, in this case.

Move down the column for the UTC time, in hours, 16 for our example.

Then, move directly across to read the value of the GHA for Aries.

The result is 273° 45.0’. Write that into the first box.

Now, on the same page, you will find the SHA for the chosen star (Markab) on the given date.

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The result is 13° 33.8’. Write that into the third box for Item 8.

Now we need the minutes and seconds increment. We have 10 minutes with 50 seconds past the hour. Proceed in the Nautical Almanac to the Increments and Corrections pages, and find the table for minute 10.

This is a segment of that page. The first column represents the seconds, and the third column is the increment value for Aries. That is what we want.

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The result is 2° 42.9’. Write that into the second box.

Add all those values together, and write the answer into the fourth box.

273° 45.0’

2° 42.9’

+ 13° 33.8’

290° 01.7’

This represents the total meridian angle westward from the Prime Meridian for the star’s GP.

Item 9. LHA (Local Hour Angle)

Assumed Position Longitude

Now that we know how “far” west from the Prime Meridian the star’s GP is, we need to calculate how far we are, in terms of degrees, from the star’s GP, based on an Assumed Position. Again, this is measured west, from our Assumed Longitude to the star’s meridian. It is known as the Local Hour Angle.

The first thing we need to do, therefore, is establish our Assumed Latitude. It is essentially a “mix” of the Dead Reckoning Longitude degrees with the GHA minutes. Here is a simple example…

DR Lon = W 132° 41.5’

GHA = 87° 23.8’

Assumed Lon = 132° 23.8’

See how that works? Simple, right? But there is a small catch. Note that the DR Lon in the above example is in the western hemisphere, for which that calculation is correct. If your DR Lon is in the eastern hemisphere, then the minutes need to be “inverted”. This is because, going west about, minutes in the eastern hemisphere count down. So, you must subtract the GHA minutes from 60 when in the eastern hemisphere to get minutes for the Assumed Lon. Here is another example…

DR Lon = E 113° 30.0

GHA = 290° 01.7’

Minute Correction = 60’ – 01.7’ = 58.3’

Assumed Lon = 113° 58.3’

Whichever case applies, do the procedure and write the answer in the first box of Item 9.

Local Hour Angle

We may now start computing our Local Hour Angle. We have already stated that LHA is our angle to the star’s GP. How we calculate it depends on whether we are east or west of the Prime Meridian, as the GP is measured from this reference.

If the observer is in the western hemisphere , the value of the Assumed Longitude must be subtracted from the GHA to obtain LHA. If the observer is in the eastern hemisphere , the value of the Assumed Longitude must be added to the GHA to obtain LHA.

That is reasonably straight forward. Now we must consider another detail. LHA must be expressed in whole degrees, not considering minutes. Therefore, before we add or subtract the Assumed Lon to or from the GHA, as the case may be, we need to round both to the nearest degree, then perform the operation. For example…

GHA = 217° 47.2’ = 218°

Assumed Lon = W 28° 13.9’ = W 28° (note west, so subtract from GHA)

LHA = 218° – 28° = 90°

It is not ridiculously difficult. Here is a graphic to clarify LHA (consider all angles measured clockwise)…

Now, there may (often) be cases when LHA is greater than 360° or less than 0°, once the operation is done. When this happens, simply correct the LHA value by subtracting 360° from it or adding 360° to it, respectively. Write the result into the last box of Item 9.

That is all there is to it, and there is no need to expand anymore about LHA. Applying all that has been outlined above, with some additional self-practice, should answer any questions.

Item 10. Declination.

As we should already be familiar with from PaulRix’s discussion, Declination is simply the latitude of the celestial body’s Ground Point. It is found in the Nautical Almanac, on the daily pages. In fact, you will find it right next to the place where you obtained the SHA, earlier.

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See it there, on the Markab row? 15° 18.8’. As it is positive, that means it is north latitude. Negative would be south. Write that value into the first box of item 10. For stars, that would be all you need to do. There is no “d” correction to be applied for stars, so a zero would go into the next box, and the Declination value would be repeated in the last box.

So, you might be asking, what is the “d” correction? Well, it is applied for planets, the Moon, and the Sun. As the declination of these bodies can change by small amounts, within the hour, the “d” correction refines it, and should be added to the base declination to give Total Declination. There is a bit of a process for it. Here is an example for the Sun.

This is part of a daily Nautical Almanac page, in the Sun columns.

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The first column is the UTC hour. The second column is the base GHA at that hour. The third column is the Declination at that hour. At the bottom, under the Declination column, you see d=0.7. That is the “d” correction factor, and you would write on the corresponding line on the SRF, to the right of the Declination (not in the box).

Now, you would go to the increments and corrections pages of the Nautical Almanac. Let us say we are at minute 27, for example. Here is a section of the table…

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Under “v and d corr”, find 0.7. Right next to it, you will see the number 0.3. That is what goes in the box under Declination on the form. Now, to know if you add this correction or subtract it, look at the trend of the Declination for the Sun on the daily pages. Note, the value is increasing as hours progress, which means we add it.

Here is the result, for hour 21, minute 27…

Declination: N 16° 04.2’ d corr factor: 0.7

D Correction: 0.3’

Total Declination: N 16° 04.5’

Item 11. Assumed Latitude

Assumed Latitude, like Assumed Longitude, is derived from your DR Latitude. Very simply, round to the nearest whole degree, and write that in the box.

S 21° 49.5’ rounds up to S 22°

Now look at the Declination again. If it is in the same hemisphere as your Assumed Latitude, circle SAME. If it is in the opposite hemisphere, circle CONTRARY.

We are done with this section.

Section IV: Determining a Line of Position

This is the main operation of the Sight reduction. Here, we will obtain two pieces of key information;

The “Distance”, called “Intercept” on the Form, expressed in degrees and minutes, to correct our Assumed Position, and…

The “Azimuth” (Zn), and angle along which we will apply the correction distance.

How exactly we apply them once we have them is not important, yet. At the moment, we must concentrate on obtaining them. As mentioned before, there are two ways to accomplish this goal; look up tables in the HO-229 publication, and a mathematical (trigonometrical) way. We will do the latter way. Here is the completed section of the Form…

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Item 12. Computed Altitude

The Computed Altitude (Hc) is essentially similar to the reading you took with your sextant, with one major difference. Instead of physically looking at the celestial body with an instrument, you use the positional information you have to determine where (ie; at what altitude) that body would appear, by rights, if that positional information were correct. You compare this to your real sighting (Ho) to determine an “error”, which is used to correct your assumed position. Look up on your Form and gather the following information for this step:

Declination, Assumed Latitude, LHA , and whether the hemisphere is SAME or CONTRARY .

For this example, we have the following (covert Declination to decimal)…

Declination = N 15° 18.8’ = N 15.313°

Assumed Latitude = S 22° CONTRARY

LHA = 44°

Note, if CONTRARY , express the Declination as a negative number. In this case it is, so Declination’s final value for the calculation is -15.313° .

Now, here is the formula, followed by the solution for our example (use your scientific calculator with angles set to degrees, not radians, or you will have to do yet another conversion!):

Hc = ArcSin( Sin( Dec ) x Sin( A Lat ) + Cos( A Lat ) x Cos( Dec ) x Cos( LHA ) )

Hc = ArcSin( Sin( -15.313 ) x Sin( 22 ) + Cos( 22 ) x Cos( Dec ) x Cos( 44 ) )

Hc = 32.98° (make a note of this decimal value, we will need it again in a moment)

Convert that decimal to minutes ( 0.98 x 60 = 58.8 ), and we have our final Hc…

Hc = 32° 58.8’

Is that not thoroughly cool? Write that into the first box of Item 12.

As the calculation with the formula is exact for its purposes, we do not need to do any further corrections. If we were using the HO-229 lookup tables, we would, but we will see that later. For now, just write zero into the second box, and repeat our result in the third box, for Total Hc.

Item 13: Intercept Altitude

Take a look at your Ho (Total Observed Altitude) and Hc (Total Computed Altitude). The lesser one must be subtracted from the greater one, to obtain our correction (Intercept Altitude). Write the greater one into the first box, the lesser in the second, perform the operation, and write the result into the third box of Item 13.

0° 27’ is the value we obtained from that operation.

Now, consider this: HoMoTo. If Ho is “more” than Hc, then the correction is TOWARDS. If not, then the correction is AWAY. Circle the appropriate T or A, and keep that in mind.

Item 14: Azimuth Angle

Azimuth Angle (Z) is another piece of information we would normally obtain from the HO-229 publication look up tables. Like in Item 12, there is an alternate, mathematical method to obtain the required value, which we will be employing here. Apart from the values we used in the previous formula, we will also need our newly computed Hc, in decimal. Here is the formula, and solution…

Azimuth Angle = ArcCos( ( Sin( Dec ) – Sin( A Lat ) x Sin( Hc ) ) / ( Cos( A Lat ) x Cos( Hc ) ) )

Azimuth Angle = ArcCos( ( Sin( -15.313 ) – Sin( 22 ) x Sin( 32.98 ) ) / ( Cos( 22 ) x Cos( 32.98 ) ) )

Azimuth Angle = 127.0°

Item 15: Azimuth

Unfortunately, Z is not the end of the story, and some corrections for hemisphere and LHA must be performed before we can use the Azimuth. Not going too deep into the matter, it has to do with trigonometry’s tendency for only dealing effectively with one quadrant in positive values, and leaving the reasoning of the other three to the particular case, which is what we are doing here.

You need to look at your LHA again, and your Assumed Latitude hemisphere. Here are the rules (you can see them on the Form, too)…

North Hemisphere Assumed Latitude:

LHA > 180, then no modification of the Azimuth Angle (Z) is needed to obtain Azimuth (Zn)

LHA < 180, subtract Z from 360.

South Hemisphere Assumed Latitude:

LHA > 180, subtract Z from 180.

LHA < 180, add Z to 180.

Write the result into the Azimuth box. In our case, our Assumed Latitude is SOUTH 22, and the LHA is 44. Therefore, the solution is;

Zn = 180 + 127 = 307°

Phew, we are done with the Form. No doubt there is a feeling of great accomplishment. This is good, because now you need to go through it again. Twice. This gives you the chance to get that wonderful feeling again, and yet again! Yes, you need to perform the sight reduction for at least three celestial bodies, as PaulRix has already described, to obtain an accurate fix. We now understand why there was a specific requirement for a navigator crew member on flight crews of old, right?

In any case, we can now get down to the real objective; plotting. Print out a Universal Plotting Sheet, get your ruler and dividers, and let us get to it!

Plotting a Line of Position on the Universal Plotting Sheet

Before anything, we must prepare our UPS. The three represented parallels are not identified. It does not include any meridians, except the central one, also not identified. Meridians converge onto the Geographic Poles, so depending on which parallel we are upon, the distance between them will vary, up to a maximum of 60 nautical miles on the equator. For those interested, the distance between meridians at a given latitude is equal to the cosine of that particular latitude. But it need not be that academic; the little legend in the lower right corner is there to help us.

Start by identifying the parallels on the Sheet. Our rounded DR Latitude rounds to S 22°. Make that the central parallel. The parallel above it, closer to the equator, will be S 21°, and the lower one will be S 23°.

Write that in on the Sheet, as seen in the image.

On the legend in the lower right corner, draw a line across it at 22°. Then use your dividers to set the total span distance of the scale along that line. This is the scale spacing between meridians, at that latitude.

With your dividers set, mark a point at this span along the parallels, either side, from the central meridian, both top and bottom. Join these marks vertically drawing a line with a ruler.

The meridian on the left will be E 113°, and the one on the right will be E 115°. Remember to take into account hemispheres to ascertain in which sense the numbers ascend.

Now we must draw in our computed Assumed Position. It will help to gather the information you will need for plotting on the Sheet, somewhere where it is unobtrusive. You will want; the Celestial Body’s name, the Assumed Position, the Azimuth (Zn), the Intercept Altitude, and the direction of intercept.

This is our Assumed Position, from the Sight Reduction Form: S 22° 0.0’, E 113° 58.3’.

We want to mark that on the plotting sheet. Use the scale again, on our line, to set the dividers to 58.3’, as in the first image below. Then move the dividers to the 113° meridian, and mark the point 58.3’ along our reference S 22° parallel. Draw a small circle at this point, and label it AP57. 57 is the number of the particular star we are using, Markab. That is it. Our reference Assumed Position is established.

Using whatever method you prefer (protractor, or the scale around the circle), centered from the AP, draw a line through the AP on a bearing of 307°, which is our Azimuth (Zn), from the SRF.

Note, it is important that this angle be from the center of the AP, not the middle of the chart. If you look closely at the picture, you will see that I did draw a faint line through the intersection of the central parallel and meridian, out on a 307° bearing. This was only to help me draw a line through the AP (the heavier line), keeping it equidistant from the reference.

At the end of this “Intercept Line” (Azimuth) that points towards the celestial body, draw an arrow head and the ID of the body, to disambiguate it.

Finally, we can draw our Line of Position. Take note of our Intercept Altitude, 0° 27.0’. Set your dividers to this value on the index marks on the central meridian. Consider if the Intercept was Towards or Away from the celestial body, as noted on your SRF. In this example, it is Towards, so from the center of the AP, mark a point 27’ long up the Azimuth line, towards the body.

Perpendicular to the Azimuth line, at the marked point, draw another line, and label it LOP, and identify it with the body and the time of the sighting.

Essentially, this LOP is the periphery of the circle PaulRix showed us in his description, the only difference here being that we calculated it with a procedure on the SRF, instead of using Google Earth.

We must now repeat that process, on the same UPS, for the other two sightings.

Where the three LOPs are at their closest to intersection point will be our celestially computed position, at the time of the sightings. If the process was done correctly, it should be quite accurate. We could then add our DR position on the Sheet, to see how far off course we were.

Sight Reduction Form Application Tutorial

The SRF application will do a large amount of the work described above automatically, while still not detracting from the experience of reading the Almanac or taking the sighting of celestial bodies. It will not eliminate errors of readings taken either from Stellarium or the Almanac, nor will it advise or correct you if you are making a mistake, so the same care must be taken in these procedures as would be if you were using the manual form. This way, it still leaves open the challenge of celestial navigation, but reduces time with the mathematical computations. You will still also have the joy of plotting your position on the UPS, with the data that this form provides from a successful sighting.

Here is the application interface, and a brief description of its features.

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As can be appreciated, the format is almost identical to the SRF form described in the previous chapter. There is some color coding to take note of. White boxes are user inputs. Yellow boxes are computed information. Green boxes are computed information which is specifically used when plotting the LOP on the UPS.

In Section I, the celestial body and its altitude are set. The body name is selected from the drop down list. Selection of the body will configure the rest of the form to accept the required parameters from the Nautical Almanac. The selection of the Apparent Altitude is in degrees, minutes, and decimals of minutes. The yellow box below is a repeater in degrees, minutes and seconds, to better equate the value to the Apparent Altitude value in Stellarium.

In Section II, the most important information to enter correctly is the DR position. Be sure to click the appropriate radio buttons for N/S, W/E. Date and time do not have any function in computations, but are included for your reference.

In Section III, data from the Nautical Almanac must be input. Total GHA, Assumed Longitude, and LHA are computed values. The Declination must be input for the appropriate body, from the Nautical Almanac, and again, be sure to click the correct N/S radio button.

In Section IV, the data for calculation of the LOP will be presented automatically, once the “Execute Calculation” button is clicked. Data can be edited in the user input boxes, and the calculation re-executed, or the form can be reset to default values for a fresh start by clicking “Reset Form”.

There are a few points of caution to be observed, when using Stellarium along with the SRF Application for XP flights, to avoid some gross errors.

In Stellarium, always make sure the Atmosphere option is set to ON. The SRF corrects the Ho (Observed Altitude) for the effects of atmospheric refraction. Stellarium simulates this effect with reasonable accuracy. The closer a body is to the horizon, the more distortive effect refraction will have on the sighting. Sensibly, one would normally try to take sightings of bodies that are more than 20° above the horizon to try and reduce this distortion to a minimum (despite the correction), but if it becomes necessary to take a low sighting, having the Atmosphere option OFF will quite seriously affect results of the final LOP.

Set Stellarium’s Time Zone to UTC (GMT), so that it actually displays UTC as the current time on the Date Time bar. It is not essential, but it certainly helps to have the correct reference for the Nautical Almanac already at hand.

The SRF Application can be opened as multiple, independent instances. For practical purposes, while navigating, open three separate SRF apps, and perform the sightings for the separate celestial bodies in these. It is preferable to do this than to just open one, and sequentially perform the sightings on that single form.

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Determining a Three Star Celestial Fix with the SRF Application

The following example shows how three stars will be used to establish position. For reference, the date is the 23 February, 2020, and we are on our way, eastbound, to Madagascar. The DR Navigation Log should establish way points as positions in latitude and longitude, and have an estimated time in UTC for that point.

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Fairly self-explanatory. We estimate to be at position S 22° 03.4’ E 040° 04.7’ at GMT time 20:48. We have no other way of establishing our true position in relation to our planned position except by time. Therefore, at 20:48 GMT we will start taking our sightings, which will consist of measuring and recording the altitude of three stars. These stars will be; Regulus (26), Gacrux (31), and Sirius (18).

The sighting and recording will take several seconds each (20 seconds, for instance), so for the Almanac reading, we will use the times in this example as; Regulus – 20:48:20 GMT, Gacrux – 20:48:40 GMT, and Sirius – 20:49:00 GMT.

Here they are…

Regulus, at 20:48:20. Apparent Altitude 55° 23’ 03.2”

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Armed with this information, we are done with using Stellarium for the rest of the procedure. Now, we may now open three instances of the SRF Application, and enter what data we already have. It might come as a surprise that you already have a good portion of the information to complete the Form, without having looked at the Almanac, yet. Let us take stock.

For starters, you have the names of the stars, and each one’s Ha (Apparent Altitude). You have the date and time. And you have the DR position, from your navigation log. Here is a shot of the three Forms, completed up to this point…

Note how you can use the Apparent Altitude Ha (D:M:S) automatic box to get as close to the Stellarium reading as possible. The Nautical Almanac uses the format degrees, minutes and decimal minutes, so this feature helps you “round to the nearest second” without having to resort to using a calculator. The DR position should be the same for all the Forms. Again, make sure the hemisphere radio buttons for latitude and longitude are correctly selected, S and E in this case, on all the Forms. Sections I and II are complete.

Now we may turn to the Nautical Almanac, making sure it is the correct publication for the year 2020, and fill in Section III. As we are dealing with the same hour (20 GMT), the Tabulated GHA will be identical for all the Forms. Let us find it…

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We are interested in the Aries column, for Sunday 23 February, hour 20 GMT. The result is: 93° 10.7’ . That value should be input to the Tabulated GHA boxes on all three Forms.

We may now look up the SHA and the Declination for each of our stars, which is on the same page of the Nautical Almanac, in the STARS information box. Obtain the following information from it, and enter it on each of the Forms, for the respective stars. For Declination, please note, positive numbers are North Hemisphere, negative numbers South Hemisphere.

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Regulus:

SHA 207° 38.2’ , DEC (N) 11° 52.1’

Gacrux:

SHA 171° 55.4’ , DEC (S) 57° 13.3’

Sirius:

SHA 258° 29.3’ , DEC (S) 16° 44.7’

Your Forms should now look like this…

There is but one last piece of information to input, now, before we can hit Execute Calculation; the increments. In the Nautical Almanac, turn to the page corresponding to the tables for Increments and Corrections, for Minutes 48 and 49. Here it is…

We are interested in the Aries column, again, for the minutes and seconds of our sightings, as follows;

Regulus at 48 minutes, 20 seconds = 12° 07.0’

Gacrux at 48 minutes, 40 seconds = 12° 12.0’

Sirius at 49 minutes, 0 seconds = 12° 17.0’

Input those values into the GHA Increment boxes on the Forms, respectively, for each star.

You may now hit Execute Calculation on each Form, to compute all the data. They should look like this…

Note that once all the data has been computed, the Execute Calculation button becomes grey. This is to show you that the calculation for the currently entered data is valid. If you change any value in any of the user input boxes after a calculation has been performed, the Execute Calculation button will turn yellow again, to indicate that a recalculation is necessary.

You now have all you will need to do the fun part; plotting your LOPs onto the UPS. The data you need for plotting is in the green boxes: Assumed Longitude and Assumed Latitude, The Intercept Distance, and the Azimuth (Zn). Write the data down for each star, mark up your UPS, and plot away!

Once done, you should have something like this…

For clarity, I did some coding on this image so it can be better interpreted.

Dotted Lines are the Azimuth (Zn), with arrows showing the TOWARD direction. Solid lines are the LOPs, at 90° to the Azimuth line. The star color coding is self-explanatory. Assumed Positions are solid circles. The orange and black square is the navigation log DR position. The black diamond is the Celestial Fix, at the intersection of the three LOPs. Note that in this example all the distance directions were AWAY, so all LOPs intersect the Azimuth line the far side (away from the arrow) of the Assumed Position, for each star.

If you take this a bit further, you can interpret that the aircraft was slightly ahead of plan, and slightly south of desired track.

The real position of the aircraft in X-Plane, at the time of this sighting set, was S 22° 05.3’ E 40° 9.8’, so the Celestial Fix, in this case, was accurate to approximately 1 nautical mile.

Determining Position from the Sun

Now, something easy. We are going to use Stellarium to establish our approximate position using an adaptation of the age old method of the Noon Sighting. This method can only be used at a very specific moment, and can only be performed once in each 24 hour cycle for the given celestial body (so do not miss the opportunity!).

First, a brief description of the original method, used by sailors before the invention of the accurate Marine Chronometer, in 1761. Without an accurate time piece, it is impossible to establish a position relative to longitude, but it is possible to determine latitude. Here is how.

As the Sun travels through the sky (from our point of view), we see its angle (altitude) relative to the horizon change. It starts close to the eastern horizon in the morning, rises to its highest elevation at midday, and settles to the western horizon in the evening. The navigator must catch it, and take the altitude reading with the sextant, at precisely the highest elevation. It stands to reason that this moment coincides with the exact moment the Sun crosses your meridian.

Then, a simple formula is used to determine the observer’s latitude. As the Sun’s Declination is known (and published in the Nautical Almanac), and its angular distance from the point directly above our head (Zenith) can be computed, we will have enough information to determine our own latitude.

Let us say we are in the northern hemisphere, the date is the 27 February, 2020, and we observe the Sun’s highest (Noon) elevation at 47°, with the sextant. Here is a shot of that situation…

Note, in this relative position the Sun will coincide with your own specific meridian. Exactly which one it is in relation to the globe’s lines of longitude, we do not know, for now. As we are in the Northern Hemisphere, probably north of the Tropic of Cancer, we have to look south to measure the altitude of the Sun at an angle less than our Zenith. This consideration is important.

Now, how far away is the Sun from our Zenith? Well, as our Zenith is 90° from the horizon (of course), and the Sun is 47° above the horizon, the additional angle the Sun would need to be over our directly over our heads would be;

90° – 47° = 43°

43°. That is called our Zenith Distance. Keep that number in mind. Now, break out the Nautical Almanac and find the page for the 27 February, 2020, Sun columns.

Here it is.

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Now, note that the Sun has a variable Declination through the 24 hour cycle. This is reasonably obvious, or else we would not have seasons. To get precise data, we would need our timepiece, once again. There will be more, regarding this “feature” of the Sun.

For now, we are going to pretend we are in 1760, or prior to. There would have been a “daily average” Declination (if sailor-navigators of olde could not establish an exact time, they could, at least, establish a date). Let us say the daily average for the 27 February might be S 08° 27’.

So, we now know the approximate latitude the sun is over (Declination), and we know how far from being over the tops of our heads it is (Zenith), too.

The Sun is 8° 27’ into the opposite hemisphere, so if we subtract that from our Zenith Distance, we will have our latitude.

N 43° – S 8° 27’ = N 34° 33’

And there it is. Our approximate latitude, give or take 10 minutes of arc. Very simple.

Let us get a little more in depth, now, as we abhor the concept of “very simple”. Flash forwards to the year 2020 again. We have an excellent, accurate time piece available. We can now determine that the time at the precise moment of the image of the Sun at its highest altitude is 20:38:43 GMT. We can also determine the exact Declination and the longitude of the Sun at that time. Here is how.

Look at the table above once more, and extract the GHA and Declination for hour 20;

GHA 116° 48.8’

Dec S 08° 18.7’

Also, take note of the value for “d” at the foot of the table, where it states d=0.9. Finally, take note if the Declination is increasing or decreasing after hour 20. As the minute value changes from 18.7 to 17.7 from 20 to 21 hours, we conclude that the Declination is decreasing .

In the Nautical Almanac, find the Increments and Corrections table for minute 38.

Scan down the first column to the 43 second. Under the next column (Sun Plan.), extract the GHA Increment of 9° 40.8’. Add this to the original, tabulated GHA for the hour.

9° 40.8 + 116° 48.8’ = 126° 29.6’

As that is less than 180°, it presents us with the longitude of the Sun, west, on the globe at that precise time. As we are right on the same meridian as the Sun, it also happens to be our own position in longitude.

Now, move across to the v and d corr column. Remembering the d value obtained from the previous table, which was 0.9, we locate 0.9 in the table, and index the number alongside it. In this case, that number is 0.6.

Previously, we had determined that the Declination was decreasing, so that 0.6 needs to be subtracted from the table Declination, which was S 08° 18.7’.

S 08° 18.7’ – 0.6’ = S 08° 18.1’

This is the exact Declination of the Sun at the given time, which is slightly more accurate than the “daily average” method. We can now do the same procedure of subtracting it from our Zenith Distance of 43° 00.0’, to obtain our precise latitude position.

N 43° 00.0’ – S 8° 18.1’ = N 34° 41.9’

So, our position according to this method is:

N 34 41.9’ W 126° 29.6’

Let us see where Stellarium says we are…

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…which is not all that bad, for a quick and dirty determination of position off a single body. It has nowhere near the accuracy of a proper, triangulated Celestial Fix, determined with a Sight Reduction, but it will certainly put you in the ball park of where you are, if you are lost.

Now that we have seen that it works, it is time to bear in mind a couple of additional points regarding the Zenith Distance to latitude calculation. In the example, the Sun was in the opposite hemisphere from us. There are two other cases. We can be in the same hemisphere as the Sun, but closer to the equator than the Declination of the Sun, or we can also be in the same hemisphere as the Sun, but with the Sun closer to the equator. Here are the pertinent operations, straight from the Nautical Almanac.

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Indeed, you may use this exact same method of single body sighting with stars, the Moon, or planets, as well. Provided that you wait for the selected body to be at the highest altitude, you may reference it in the Nautical Almanac and establish your approximate position. This gives you 63 opportunities (discount Polaris, it is the only one that is useless for a single body determination of longitude) in a 24 hour cycle.

Finally, Planets!

Now that we have studied all of this, let us have a look at using planets and the Moon as valid navigational celestial bodies, using the SRF Application. They have a slightly different method of calculation to stars, though it is similar.

The scenario. We are using Mars as one of our bodies, on 3 March, 2020, at 23:32:15 GMT. Our DR position is N 15° 04.9’ E 089° 30.5’. Here is the sighting…

The Apparent Altitude of Mars is 36° 45.6’. Let us fill in the Form with this data. Then, reference the Nautical Almanac as per normal, to obtain the Tabulated GHA and Declination for Mars. Note the values of the v and d, at the bottom of the table. Also note, Declination increasing or decreasing.

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The Form will look like this…

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All our preliminary information is in there. We now need to go to the Increments and Corrections section of the Nautical Almanac, for minute 32.

But before we do, let us look closely at the Form. When Mars (or any of the four planets, or the Moon, is selected), the SHA is replaced by the V Correction, and under Declination appears the D correction.

The D Correction we have already seen in action while performing the Sun Noon sighting. To recap, we will look at it again. The V Correction is essentially the same procedure, but bear the following in mind.

D correction corrects Declination, and whether it is subtracted or added to the Declination depends on the trend of change. If The Declination is decreasing, the D Correction will be subtracted, and vice versa. In this case, it was decreasing .

The V Correction, on the other hand, is always added to the GHA.

Now, let us proceed to the respective Increments and Corrections page, for minute 32.

Here is the table…

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For minute 32, move down to second 15, and obtain the Increment for Sun Planets, in this case:

8° 03.8’

That will go into the GHA Increment box on the Form.

Now, remember the v and d values from the table:

v = 0.6

d = 0.1

Look up both in the v and d corr column in the table. Retrieve the number next to the values, in this case v (red) is 0.3, and d (blue) is 0.1.

Enter all of these values into the respective boxes on the SRF Form. Note that as Declination was decreasing, d correction must be subtracted, so enter a negative value (-0.1). Hit Execute Calculation to get your plotting data for the UPS. Here is what we get…

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Plot it the exact same way you would a star on the UPS, as from this point.

The procedure for the Moon is identical to the planets, so there is no need to provide a separate example.

That would be all, basically.

Happy Celestial Navigating!

Cygon Parrot