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    Re: Celestial up in the air
    From: Gary LaPook
    Date: 2008 Aug 6, 01:52 -0700

    Gary LaPook writes:
    
    
    Now that we have discussed the technique of flight navigation we can
    now look at an example of how it is actually done. I will use the
    Polhemus computer to illustrate this process which will also show how
    handy this device is but it can also be done on a plotting sheet
    though not so conveniently.
    
    We are ferrying a 1978 Cessna Skyhawk (Cessna 172) from Casa Blanca
    Morocco to Porto Santo Island in the Madeira Islands on October 4,
    2008 (1.jpg and 2. jpg.)
    http://www.geocities.com/glapook@pacbell.net/1.jpg
    http://www.geocities.com/glapook@pacbell.net/2.jpg
    
    The coordinates of the departure airport are 33� 33' north, 7� 40'
    west. The destination airport is at 33� 04' north, 16� 21 west. Using
    the 35� latitude disk we label the graticle and we then plot the
    locations of both airports and draw a course line between them (3.jpg
    and 4.jpg.) Now, by rotating the disk so that the destination is
    directly above the departure, we can read  out the distance by
    counting the grid lines, departure is at  minus 215 (visually
    interpolating between 210 and 220) and destination is at plus 219 for
    a total distance of 434 NM (5.jpg.) We can also read out the true
    course at the "true Index" which is 266.5�, which we will round to
    266� (6.jpg.)
    
    http://www.geocities.com/glapook@pacbell.net/3.jpg
    http://www.geocities.com/glapook@pacbell.net/4.jpg
    http://www.geocities.com/glapook@pacbell.net/5.jpg
    http://www.geocities.com/glapook@pacbell.net/6.jpg
    
    
    Looking at the aircraft flight manual climb schedule (7.jpg), we see
    that we will climb at an airspeed of about 70 knots and that it will
    take 21 minutes to climb to our planned cruising altitude of 10,000
    feet (Flight Level 100, FL100) using 3.7 gallons of fuel in the climb
    and covering 27 NM. Since we also used 1.1 gallons taxiing we will
    have used up 4.8 gallons by the time we level off FL100. We adjust our
    throttle to make 2500 rpm. Since the air temperature is about standard
    (15 � C at sea level and cooling off about  2� C per thousand feet)
    this power setting will produce 61% power which will give us a true
    airspeed of 114 knots and a fuel flow rate of 6.8 gallons per hour
    (8.jpg.) Since we will cover 27 NM in the climb we will have an
    additional 407 NM to cover in cruise after we reach the top of climb
    (TOC). Winds are forecast "light and variable" so using our true
    airspeed of 114 knots we compute that it will take an additional 3+34
    for the enroute phase of the flight for a total flight time of 3+55.
    We will burn 24.3 gallons in cruise plus the climb and taxi fuel means
    we will burn a total of  29.1 gallons out of our total fuel on board
    of 50 gallons leaving us a comfortable 20.9 gallons fuel reserve.
    
    http://www.geocities.com/glapook@pacbell.net/7.jpg
    http://www.geocities.com/glapook@pacbell.net/8.jpg
    
    We take off at 1745Z and climb on course with our true heading of 266�
    and level off 21 minutes later at FL100 at 1806Z and set the power and
    auto pilot. Our ETA is 2140Z, 3+55 after takeoff. We plan on taking a
    celestial fix at 1920Z to allow for an enroute leg longer than one
    hour so as to allow for the determination of an accurate wind vector.
    We will cover 140 NM during the 1+14 minute cruise from TOC at 1806Z
    to 1720Z fix time. Since we covered  27 NM in the climb we will be 167
    NM from departure at the planned fix time. Since the departure was at
    minus 215 on the Polhemus grid we simply subtract 167 from 215 and
    place a mark on our course line at the minus 48 grid line, visually
    interpolating between 40 and 50. Rotating the grid to north up we can
    read out our 1920Z DR of 33 � 27' north, 11� 00' west (9.jpg.)
    
    http://www.geocities.com/glapook@pacbell.net/9.jpg
    
    
    About a half hour before the fix time we start planning our fix. We
    will be using H.O 249 Volume 1, Selected Stars, since it is the most
    convenient. First we look in the Air Almanac for 1920Z, October 4,
    2008 and take out the GHA of Aries without interpolation, 303�
    51' (10.pdf.) We select an AP of 33� 00' north and 10� 51' west so
    that the LHA Aries will be 293� exactly. We use only one AP since we
    are using H.O. 249 vol. 1 and we are accounting for MOO
    mathematically, not advancing the earlier LOPs to the fix time. We
    then look at the 33� north page of H.O 249, vol. 1 and looking at the
    LHA Aries we see that the selected stars are Alpheratz, Enif, Altair,
    Antares, Arcturus, Alkaid and Kochab. Because the visibility is
    limited in a Cessna 172 by its high wing we must choose, not the three
    recommended stars, but Kochab, Arctures and Antares (11.pdf.) We will
    shoot Kochab first since it is nearly on the wing tip and so advancing
    its LOP the most to the fix time will have little effect on its
    accuracy. We plan our shooting schedule, Kochab at 1912Z (eight
    minutes before fix time), Arcturus at 1916Z (four minutes early) and
    Antares at 1920Z. We enter this data our the Celestial Precomputation
    form (12.jpg, if using the Polhemus computer or the 1 minute
    adjustment tables from H.O 249 or 13.jpg if using the 4 minute
    adjustment tables from H.O. 249.) We only compute LHA for the the fix
    time shot but we use the same LHA, 293� to take out the Hcs for all
    three bodies and enter this data on the form in the row labeled "HA HO
    249" and enter the Zns in the same columns and also on the left side
    of the form for computation of the motion adjustments. We also take
    out the "Precession and Nutation" correction for 2008 for LHA Aries of
    293� and for a latitude of 33� north, either visually interpolating or
    simply taking the nearest tabulated value since they are all small,
    we'll use .6 NM at 241� and enter it on the appropriate form (14.pdf.)
    We will use this to adjust the fix position. We plot the AP on the
    Polhemus grid at 33� 00' north, 10� 51' west, visually interpolating
    (15.jpg.)
    
    
    http://www.geocities.com/glapook@pacbell.net/10.pdf
    http://www.geocities.com/glapook@pacbell.net/11.pdf
    http://www.geocities.com/glapook@pacbell.net/12.jpg
    http://www.geocities.com/glapook@pacbell.net/13.jpg
    http://www.geocities.com/glapook@pacbell.net/14.jpg
    http://www.geocities.com/glapook@pacbell.net/15.jpg
    
    
    Now set the front of the Polhemus "SET TRACK" pointer to 266, the "SET
    GS" (ground speed) to 114 and the "SET LAT" to 33 and then tighten the
    nut to keep the settings from changing (16.jpg.) (We could also do the
    same adjustments using the MOB and MOO tables from H.O 249.) Looking
    around the outside edge for the Zns of each body we find the relative
    Zns (ZN-TR) and enter them on the form, 75� for Kochab, 15� for
    Arcturus and 44� for Antares. Using each body's Zn look in the
    "CORRECTION FOR MOTION OF THE BODY" window and take out the MOB one
    minute correction and enter it in the appropriate blank (17.jpg or
    18.pdf or 19.pdf.) Since all three Zns are to the west they are all
    found on the black scale so the signs are all plus. Next, using the
    relative Zns  (ZN-TR) look in the "CORRECTION FOR MOTION OF THE
    OBSERVER" window and take out the one minute corrections for MOO
    (20.jpg or 21.pdf or 22.pdf.) Since all of the relative Zns are ahead
    of the plane they are all found on the white scale making all of their
    signs negative. We sum the MOB and MOO adjustments to the "ONE MINUTE
    ADJ." line keeping track of the signs. Since we are planning the first
    shot 8 minutes early, the second shot 4 minutes early and the last
    shot on time, we multiply the one minute adjustments by the delta time
    to produce the total motions adjustments. (We get identical values if
    we use the one minute MOB and MOO tables.  If we are using the 4
    minute adjustment tables we multiply by 2 and 1 adjustment periods
    respectively and get the same values.) We look at the refraction table
    (23.pdf) in the 10,000 foot altitude column and take out the
    refraction for each body and enter it on the form with a plus sign and
    carry them to the "MISCEL" line. Add the total motion adjustment to
    the MISCEL line to arrive at "TOTAL ADJ." and carry to the right side
    of the form into the appropriate columns. Combine the "HA" from H.O.
    249 with the total adj. to arrive at Hc.
    
    http://www.geocities.com/glapook@pacbell.net/16.jpg
    http://www.geocities.com/glapook@pacbell.net/17.jpg
    http://www.geocities.com/glapook@pacbell.net/18.pdf
    http://www.geocities.com/glapook@pacbell.net/19.pdf
    http://www.geocities.com/glapook@pacbell.net/20.jpg
    http://www.geocities.com/glapook@pacbell.net/21.pdf
    http://www.geocities.com/glapook@pacbell.net/22.pdf
    http://www.geocities.com/glapook@pacbell.net/23.pdf
    
    
    
    The last bit of information we take from the Polhemus is the Coriolis
    correction which is found in the "CORIOLIS & WANDER CORR." window.
    Look at the latitude, 33�, and take out the coriolis correction of 1.7
    NM (20.jpg), ( 2 NM if taken from the H.O. 249 table 23.pdf.) We will
    use this to move the plotted fix 1.7 NM in direction 356� , 90� to the
    right of the track to account for coriolis. (Alternatively we could
    make the same adjustment to to the AP prior to plotting the LOPs,
    dealers choice. We could also use the Polhemus to derive a coriolis
    correction to be applied to each Hc mathematically but that is a
    needless complication especially at low air speeds.)
    
    http://www.geocities.com/glapook@pacbell.net/20.jpg
    http://www.geocities.com/glapook@pacbell.net/23.pdf
    
    
    We are now done with the precomputations and can relax until time to
    shoot Kochab. About 1908Z we get the sextant ready, illumination on,
    bubble formed, averager set and altitude set to about 37�. We also
    make sure that the directional gyro is set and that the autopilot is
    set to heading mode. We look out the window, locate Kochab and bring
    it into the center of the bubble. At 1911Z we trigger the averager and
    continually adjust the altitude knob to keep Kochab centered in the
    bubble. Two minutes later the shutter on the sextant automatically
    closes ending the two minute shooting period and the average time of
    the shot is 1912Z. (If using an A-10A and some other sextants you must
    keep track of the progress of the shot and stop at the two minute
    mark.) The sextant altitude of Kochab is 37� 35'. We enter this in the
    form, compare it to the already adjusted and corrected Hc and
    determine that the intercept is 7 NM toward Kochab, Zn 341�. (No need
    to correct Hs for refraction as this was already taken of by applying
    the refraction correction with reversed sign to adjust Hc.)
    
    
    
    We complete the same steps with Arcturus and Antares and get an Hs of
    19� 51' for Arcturus and an Hs of 16� 20' for Antares producing
    intercepts of  9 NM away and 11 NM away respectively. Using the
    Polhemus we plot the three LOPs. First we set the Zn of Kochab, 341�,
    at the TRUE INDEX and then measure up 7 NM from the A.P. and draw the
    Kochab LOP parallel to the right-left grid lines on the Polhemus base
    (24.jpg and 25.jpg.) We do the same for the Arcturus and Antares LOPs
    remembering to measure down since these are AWAY intercepts (26.jpg
    through 29.jpg.)  We move the fix from the center of the cocked hat
    1.7 (or 2) NM in direction 356� for coriolis and then .6 NM in
    direction 241� for precession and nutation. We do this with visual
    interpolation since these are small values (30.jpg.) If larger, we
    would set the respective Zns under the TRUE INDEX and measure up the
    appropriate amount for each of these corrections. The fix is 33� 13'
    north, 10� 41' west.
    
    http://www.geocities.com/glapook@pacbell.net/24.jpg
    http://www.geocities.com/glapook@pacbell.net/25.jpg
    http://www.geocities.com/glapook@pacbell.net/26.jpg
    http://www.geocities.com/glapook@pacbell.net/27.jpg
    http://www.geocities.com/glapook@pacbell.net/27.jpg
    http://www.geocities.com/glapook@pacbell.net/28.jpg
    http://www.geocities.com/glapook@pacbell.net/29.jpg
    http://www.geocities.com/glapook@pacbell.net/30.jpg
    
    Showing the convenience of the Polhemus even more, we find the wind
    encountered in flight and the new course and distance to the
    destination. Since our DR in this case is also our "no wind position"
    or "air position" where we would be if there were no wind. Any
    difference between the DR and the fix must be caused by the wind. We
    now rotate the disk to place the fix directly below the DR and read
    out the distance between the DR and the fix which shows how far the
    wind pushed the plane, in this case 21 NM (31.jpg)in the 1+14 the
    plane flew in cruise so we divide this 21 NM by this amount of time
    and find the wind speed of  17 knots. The direction of true wind is
    also now aligned with the "TRUE INDEX" which shows 311� (32.jpg.)
    
    http://www.geocities.com/glapook@pacbell.net/31.jpg
    http://www.geocities.com/glapook@pacbell.net/32.jpg
    
    Now we can rotate the disk to place the destination directly above the
    fix and find the distance and true course to the destination of  282
    NM (33.jpg), course 270� (34.jpg.) Using this new course and the
    measured winds on our E-6B or MB-2A we calculate a new wind correction
    angle, new heading, new ground speed, new ETA and new fuel required.
    Wind correction angle will be  6� RIGHT making the new heading of
    276� . The new ground speed will be 101 knots for the remaining
    distance of 282 NM  which means it will take an additional 2+48 to
    arrive, making the new ETA of 2208Z. This means that  we will arrive
    28 minutes later than planned, using  an extra 3.2 gallons reducing
    our fuel reserve to 17.7 gallons which is still a comfortable safety
    margin, more than two hours of extra fuel.
    
    http://www.geocities.com/glapook@pacbell.net/33.jpg
    http://www.geocities.com/glapook@pacbell.net/34.jpg
    
    Although this is a fairly short flight it is still very useful to get
    the celestial fix so that we can be sure we are not running into a
    strong headwind or getting blown far off course.
    
    Celnav is done the same way in faster aircraft. Since most jets are
    flight planned at about .7 mach, about 450 knots, this just makes the
    adjustment for MOO and coriolis larger but the same methods are used.
    Using the Polhemus it takes only 40 seconds to plot the three LOPs,
    about 13 seconds each, and just 30 seconds total to measure the wind
    drift and direction and the course and distance to destination. Then
    25 seconds on the MB-2A gives you wind correction angle and ground
    speed and another 30 seconds gives you time to destination and fuel
    required. So by doing precomputations and by using the Polhemus you
    can have the fix and the new heading, ETA and fuel required only two
    minutes after finishing the last shot. Try that with other computation
    and plotting method!
    
    
    
    gl
    
    On Jul 28, 5:24�am, glap...@pacbell.net wrote:
    > One more thing to discuss before giving an example of in flight celnav
    > is corrections to sights taken in flight. We discussed this back on
    > December 14, 2007 in the thread "additional corrections... (just
    > search "additional corrections") which include an excerpt from AFPAM
    > 11-216. You should download the entire manual 
    here:http://www.e-publishing.af.mil/shared/media/epubs/AFPAM11-216.pdf
    >
    > Review chapters 10 through 13.
    >
    > I want to add to the manual on this.
    >
    > Coriolis can be handled in a number of ways. You can move the A.P. to
    > the right (northern hemisphere) 90� to the course (track) prior to
    > plotting the LOPs by the amount of coriolis correction shown in the
    > table in the Air Almanac and in H.O. 249 (previously posted). Or you
    > can move the final fix the same way. Or, the most complicated way, is
    > to make a correction to each Hc by multiplying the coriolis correction
    > by the sine of the relative Zn, the Polhemus makes this relatively
    > painless.
    >
    > Rhumb line correction is avoided by steering by directional gyro
    > during the two minute shooting period and this is what is normally
    > done anyway.
    >
    > Wander correction is small at low airspeeds and it can be avoided by
    > making sure the heading is the same at the end of the shot as it was
    > at the beginning of the shot. It doesn't matter how the heading
    > changes during the shot (within reason) as the errors will average
    > out.
    >
    > Ground speed correction can also be avoided by making sure the
    > airspeed is the same at the end as at the beginning, any changes in
    > between will also average out.
    >
    > Auto pilots do a good job of maintaining airspeed and heading for the
    > two minute shooting period so eliminating the need for the above
    > corrections.
    >
    > The AFPAM states you must figure the refraction correction based on
    > the actual Hs as opposed to using the refraction correction based upon
    > the Hc but this is a needless refinement and keeps you from completing
    > the pre computation prior to the shot. Look at the refraction table in
    > H.O. 249 (previously posted) and you will see for altitudes exceeding
    > 10� that the brackets are at least two degrees wide. So only in the
    > rare cases where the altitude is almost exactly at the break point
    > could you come up with a different refraction correction using Hc
    > rather than Hs and even then it could only be a difference of one
    > minute of altitude. For example the break point between a 5'
    > correction and a 4' refraction correction is 12� so if Hs were 11� 50'
    > and Hc were 12� 15' then using Hc would get you a 4' correction and
    > using Hs would get you a 5' correction. This is actually only 1/2 of a
    > minute error because the corrections are rounded to the nearest full
    > minute.
    >
    > The parallax in altitude correction for the moon is printed on each
    > page of the Air Almanac based upon the horizontal parallax (H.P.) for
    > the moon on that particular day. This parallax varies with the
    > distance to the moon and moves in lock step with the S.D. since they
    > are both related to the distance to the moon. The H.P varies from 54'
    > to 61' during the year. For example, using the page from the Air
    > Almanac found on page 206, a day when the H.P is 60', and an altitude
    > of 36� we find the parallax in altitude correction to be 48' and this
    > would be the correction to use with a bubble sextant. If using a
    > marine sextant and shooting the lower limb we would add the S.D. of
    > 16' to produce a total correction (but not including refraction yet)
    > of 64'. Subtract the refraction correction of 1' gives the total
    > correction of 63'. Using the correction table in the Nautical almanac
    > for the identical parameters you get 63.5'. The Nautical Almanac moon
    > correction table includes a procedure for using it with a bubble
    > sextant and what this does is just backs out the S.D. correction which
    > is included in the correction table and not needed for a bubble
    > observation. Using this procedure produces a correction for a bubble
    > observation of 47.2' which compares with the 48' from the Air Almanac.
    >
    > Remember to reverse the signs of these corrections and apply them to
    > Hc to produce Hp (pre computed altitude) which you then compare
    > directly with Hs to compute intercept.
    >
    > gl
    >
    > On Jul 25, 7:48 pm, Gary LaPook  wrote:
    >
    > > We can also use the Polhemus computer to calculate the MOO adjustment.
    > > We do this by setting the ground speed �in the setting window and read
    > > out the MOO in the "ZN-TR" window adjacent �to the relative Zn. (See
    > > Pol1.jpg) (Zn-TR is another way of saying "relative Zn" since you
    > > calculate relative Zn by subtracting Track from Zn.) Looking at the top
    > > of the TR-ZN window where the relative Zn of 000� is adjacent to "5" in
    > > the MOO window showing that the aircraft moves 5NM per minute which
    > > causes the altitude to also change 5' every minute when the body is
    > > directly ahead of or directly behind the aircraft. This MOO is
    > > equivalent to the MOO table at page 6 of the original PDF which
    > > tabulates the MOO adjustment per minute. Multiplying this 5' times the
    > > same eight minute period gives the same 40' adjustment we got from the
    > > MOO table on page 4 of the PDF. You will also find that the adjustment
    > > is 2.5' adjacent to the relative Zn of 60� which multiplied by eight
    > > minutes gives the 20' adjustment we found in the table on page 4.
    >
    > > The Polhemus makes it easy to figure the relative Zn. You place the "SET
    > > TRACK" pointer on the track of the aircraft ,130� �as shown in the
    > > attached image. (see Pol2.jpg) Look at the next image (Pol3.jpg) for the
    > > second case, a track of 70� and you find the relative Zn, 60� on the
    > > inner scale.
    >
    > > The Polhemus also makes it easy to figure the sign to use for the
    > > adjustment, if the relative Zn is on the white scale, meaning the body
    > > is ahead, then the sign is minus and if found on the black scale (the
    > > body is behind) then the sign is plus when these adjustments are made to
    > > Hc, the normal method. This same pattern is revealed in the two MOO
    > > tables, the top of the tables show the body ahead and the bottom has the
    > > body behind.
    >
    > > gl
    >
    > > glap...@pacbell.net wrote:
    > > > Now let's talk about the "motion of the observer" (MOO) adjustment.
    > > > Every fix in the air is a running fix because the aircraft moves a
    > > > considerable distance between the first and last sight. Assuming the
    > > > normal eight minute spacing between the first and last shot, a slow
    > > > airplane, say 100 knots, will have traveled 14 NM while a 450 knot
    > > > plane will have traveled 60 NM. In marine practice the navigator will
    > > > advance the earlier LOPs to cross them with the last shot. The MOO
    > > > adjustment accomplishes the same thing.
    >
    > > > As an example of how this works consider a sun shot taken at 1000Z
    > > > resulting in an observed altitude, Ho, of 35� 55'. After doing the
    > > > normal sight reduction you end up with an Hc of 35� 45' at the chosen
    > > > A.P and a Zn of 130�. This results in an intercept of 10 NM toward the
    > > > body, 130�. To plot this LOP you draw the azimuth line from the A.P
    > > > and measure off the 10 NM intercept toward the sun and plot the LOP
    > > > perpendicular to the Zn.
    >
    > > > Then, two hours later at 1200Z you take another altitude of the sun
    > > > and to obtain a 1200Z running fix you must advance the 1000Z sun line
    > > > to cross the 1200Z line. There are three ways to advance the LOP.
    > > > First, you can pick any spot on the LOP and lay off a line in the
    > > > direction of travel of the vessel, measure off the distance traveled
    > > > along that line, make a mark there and then draw a line through that
    > > > mark that is parallel �to the existing LOP and label the advanced LOP
    > > > "1000-1200Z SUN." A second way is to advance each end of the LOP and
    > > > then just draw a line through these two points, this avoids having to
    > > > measure the azimuth when laying down the advanced line. The third way
    > > > is to advance the original A.P and then from the ADVANCED A.P. plot
    > > > the LOP using the ORIGINAL intercept and Zn. Any of these methods will
    > > > produce the same advanced LOP.
    >
    > > > Now let's consider a simple case. Suppose the vessel's course is the
    > > > same as the Zn, in this case, 130� and the vessel's speed is 20 knots
    > > > meaning it has traveled 40 NM in the two hour period. In this simple
    > > > case we can just extend the Zn line an additional 40 NM and then plot
    > > > the advanced LOP at that point. So, �the LOP is now 50 NM from the
    > > > original A.P., the original 10 NM intercept plus the additional 40 NM
    > > > that the vessel has traveled on the same course as the azimuth. Since
    > > > we have no interest in actually plotting the 1000Z LOP, as we are just
    > > > planning on having the 1200Z running fix, we can skip drawing the
    > > > earlier LOP and just plot the advanced LOP by adding the distance
    > > > traveled to the original intercept to get a total intercept now of 50
    > > > NM and using that adjusted intercept to plot the advanced LOP using
    > > > the ORIGINAL A.P. This method also creates the exact same advanced LOP
    > > > as the other three methods. This last described procedure is how the
    > > > MOO table is used.
    >
    > > > Look now at the MOO table, page 4 of the PDF in my original post.
    > > > Assume now we are in 300 knot airplane and the first sight is taken at
    > > > 1152Z, eight minutes prior to the planned fix time. At the top of the
    > > > column marked "300" knots ground speed you find the number "20"
    > > > showing that the plane will travel 20 NM (and so the altitude of the
    > > > body should change by 20 minutes of arc) in a 4 minute period. Also
    > > > notice that the top row of values are marked for a relative Zn of 000�
    > > > meaning the body is directly ahead, as in our example. The plane will
    > > > obviously travel 40M in the normal 8 minute period from the first to
    > > > the last shot of a three star fix. The sign convention is the same as
    > > > that for the MOB table so simply draw a horizontal line
    >
    > ...
    >
    > read more �
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