Navigation Calculator Workbook - · PDF fileGreat Circle Distance . 24. Rule of 1/12 for Tide...

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28 February 2014

User Guide for Celestial Navigation Calculator Version 8_3_1

Table of Contents (Bookmarks) Introduction This workbook contains the following Worksheets 1. 60 D ST ( Speed, Time & Distance calculations ) 2. Set & Drift 3. Interpolation & Data Conversions 4. Arrival Coordinates 5. Course & Distance 6. Great Circle Route 7. Nav Bodies 8. 24 Hc & Zn 9. Analemma 10. Meridian Transit 11. Sight Averaging 12. Calculating Ho from Hs 13. Sight Reduction Using Law of Cosines Method 14. Yellow Pages ( Nautical Almanac Increments & Corrections ) 15. USPS Form SR96a Sight Checker 16. Sight Reduction by Direct Computation 17. Two Body Fix, From a DR Position 18. Two Body Fix, From an Assumed Position 19. Three Body Fix, From a DR Position 20. Three Body Fix, From an Assumed Position 21. Sight Planning 22. Latitude by Sight on Polaris 23. Rhumb Line Distance vs. Great Circle Distance 24. Rule of 1/12 for Tide Prediction Appendix A: Installing the Excel "Solver Add-In" Appendix B: Enabling Macros Appendix C: Accuracy ________________________________________________________________ Several Worksheets provides Hyperlinks to the following:

http://aa.usno.navy.mil/data/

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Introduction

This Excel Workbook, which contains 24 worksheets, was developed over the past ten years as a Teaching Aid for USPS Advanced Grade classes. This Excel Workbook may be used at no charge as a Teaching Aid for Celestial Navigation classes and as an aid in checking JN and N sight folders & exams. This Teaching Aid won "BEST OF SHOW" at the United States Power Squadrons District 16 Fall Conference in 2009. Note to Instructors: Excerpts from The Nautical Almanac provided with the 2008 Edition of The Navigation Course Student Manuals were taken from the 2004 Nautical Almanac. Whereas the 2007 Edition of the Junior Navigation Course Student Manuals provided Excerpts from the 1972 Nautical Almanac, which was used by all previous editions of JN & N Student Manuals since the 1973 Edition of JN and the 1975 Edition of N. I want to thank P/D/Lt/C Stanley L. Klein, SN for his suggestions & help in testing this workbook and P/C Richard Michels, SN for posting the this Excel Workbook on http://calmseas.org All cells of the workbook are protected except for the yellow user data entry cells. This prevents the inadvertent modification of cells containing formulas use to calculate the results. Most of the Worksheets have an associated Macro for clearing the user input cells of previously entered data. These Worksheet Macros are provided to prevent data from a previous problem being overlooked when entering data for a new problem. Units associated with cells use abbreviations & acronyms from The American Practical Navigator "Bowditch" Pub No.9 2002 Bicentennial Edition. The Nav Bodies worksheet is the core of this workbook. The date, time & DR position specified in Nav Bodies is used by many of the other worksheets. When opening the Celestial Navigation Calculator workbook you may, depending on the security level you have set for Excel, get a Security Warning Message which will allow you to “Enable” or “Disable” Macros. Select “Enable Macros” (See Appendix B for additional information on enabling macros).

For best performance while using the Sight Averaging worksheet change the Excel Formula Calculations Options to Manual and use the Shift F9 key or the Calculate Sheet Command after entering or deleting any data values. For all other worksheets set the Excel Formula Calculations Options to Automatic. The worksheets in this workbook were formatted for viewing my HP w2558hc screen. You can resize the worksheet to fit on your computer screen by selecting the range of cells to view, then click on “View” then select “Zoom” from the drop down menu and select the appropriate size for your display screen. The print area has been pre-set, just click on “Print Pre-view” to see what will be on the printed page.

Note: Although this workbook provides accurate results its primary purpose is for classroom instruction and is not to be used for on the water navigation purposes. Please send an E-Mail to [email protected] if you have a question, comment or suggestion concerning any worksheet. P/C Ron Jones, SN North Olympic Sail & Power Squadron Return to Table of Contents

http://calmseas.org/

mailto:[email protected]

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Worksheet 1

60 D ST Speed, time & distance calculations. Time is entered or calculated in hours minutes & seconds. Speed is entered or calculated in knots. Distance is entered or calculated in nautical miles. ■ Given speed & time, calculates distance. ■ Given time & distance, calculates speed ■ Given distance & speed, calculates time In addition to the speed, time & distance calculations, this worksheet also calculates the following: ■ Given compass heading, deviation & variation, calculates true heading ■ Given true heading variation & deviation, calculates compass heading Return to Table of Contents

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Worksheet 2 Set & Drift

Given the DR latitude & DR longitude specified in Nav Bodies, the fix latitude & fix longitude and the elapsed time from the previous fix, this worksheet calculates set & drift. Return to Table of Contents

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Worksheet 3 Interpolation & Data Conversions

Linear Interpolation: Given (X0, Y0), (X2, Y2) & X where X is between X0 and X1, using linear interpolation, this worksheet calculates the value of Y associated with the specified value of X from: Y = { (X − X1)/(X0 − X1) }Y0 + { (X − X0)/(X1 − X0) }Y1 Double Linear Interpolation: Enter a set tabular values: (X1, Y1, Z11), (X2, Y1, Z21), (X1, Y2, Z12) and (X2, Y2, Z22) into the corresponding yellow cells. Then enter the value of (X & Y) for which the value of Z is to be calculated where: ( X1 < X < X2) & (Y1 < Y < Y2) The worksheet calculates the coefficients Xc and Yc for the interpolation from: Xc = (X - X2)/(X1 - X2) Yc = (Y - Y2)/(Y1 - Y2) Next the worksheet calculates the value of the parameters A & B from: A = Z21 + Xc (Z11 - Z21) B = Z22 + Xc (Z12 - Z22) Then the worksheet calculates the value of Z corresponding to X & Y from: Z = B + Yc (A - B) Second Order Lagrange Interpolation: Given (X0, Y0), (X1, Y1), (X2, Y2) & X, the worksheet calculates the value of Y associated with the specified value of X using 2nd order Lagrange's interpolation formula. Third Order Lagrange Interpolation: Given (X0, Y0), (X1, Y1), (X2, Y2), (X3, Y3) & X, the worksheet calculates the value of Y associated with the specified value of X using 3rd order Lagrange's interpolation formula. Data Conversion: This worksheet also provides for converting time to arc, arc to time, decimal (Base 10) to Sexagesimal (Base 60), temperature conversions between Fahrenheit & Celsius, pressure conversions between inches Hg and Milibars, conversion of Local Mean Time to Zone Time and conversion of Zone Time to Local Mean Time. Return to Table of Contents

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Worksheet 4

Arrival Coordinates Given departure latitude (L1),departure longitude (Lo1), true course heading (C), distance (D) in nautical miles this worksheet calculates the arrival latitude (L2), & arrival longitude (Lo2). ∆L= D x COS(C) L2 = L1 + ∆L Lm = (L1 + L2) / 2 ΔLo = D x SIN(C) / COS(Lm) Lo2 = Lo1 + ΔLo Given variation & deviation this worksheet also calculates compass course. If given speed in knots, then time in route is calculated. If given time in route then speed in knots is calculated. Return to Table of Contents

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Worksheet 5

Course & Distance Given departure latitude (L1), departure longitude (Lo1), arrival latitude (L2), arrival longitude (Lo2) & speed in knots this worksheet calculates true course heading (C), distance (D) in nautical miles. Where rhumb line distance & course heading are calculated as shown below: ΔLat = L2− L1 ΔLo = Lo2− Lo1 Δφ = ln[tan(L2/2+π/4)/tan(L1/2+π/4)] IF Δφ = 0 then β = cos(L1) else β = ΔL/Δφ Distance D = √[ΔL² + β².ΔLo²].R Course Heading C = atan2(Δφ, ΔLo) Given variation & deviation this worksheet also calculates compass course. If given speed in knots, then time in route is calculated. If given time in route then speed in knots is calculated. Return to Table of Contents

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Worksheet 6

Great Circle Route Given departure latitude (L1), departure longitude (Lo1), arrival latitude (L2) & arrival longitude (Lo2), this worksheet calculates great circle distance & initial true course heading, using Law of Cosines method. Return to Table of Contents

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Worksheet 7 Nav Bodies

Given latitude (L), longitude (Lo), date & zone time, this worksheet calculates the following data for the Sun: Local Mean Time (LMT), Local Apparent Time (LAT), Equation of Time, Time of Meridian Passage, Declination (DEC) of Sun, Greenwich Hour Angle (GHA) of the Sun, True Altitude (Hc) of Sun, Azimuth (Zn) of Sun, the semi-diameter SD of the Sun. Also Refraction (R0) at a pressure of 1010 mb and a temperature of 10° C is calculated along with the Parallax in altitude (PA). R0 and PA are then used with the semi-diameter of the Sun (SD) to calculate the apparent altitude (ha) of the Sun Upper Limb & the apparent altitude (ha) of the Sun Lower Limb. Times for Sunrise, Sunset, Civil Twilight, Nautical Twilight & Astronomical Twilight are also shown as they would occur at the DR position for the specified date. The time and date of the Equinoxes & Solstices are also shown for the year. In addition to the above Sun data, this worksheet also calculates GHA of Aries (γ), the Greenwich Hour Angle (GHA), Declination (Dec), Altitude (Hc), Azimuth (Zn) and the rise, transit and set times for the Planets† & Moon†, also the Sidereal Hour Angle (SHA), Declination (Dec), Altitude (Hc) and Azimuth (Zn) of the 57 Navigation Stars listed in the Daily Pages of the Nautical Almanac. The listed data is calculated based on the DR position, Date and Time. This worksheet also determines if a Sun ~ Moon two body fix is available at the specified time and DR position. Also the date & time of New Moon, First Quarter, Full Moon & Last Quarter are calculated and displayed. Notes attached to the cells showing First Quarter & Last Quarter dates provide suggested dates and times when a Moon ~ Sun two body fix may be available. Calculations are based Astronomical Algorithms 2nd Edition by Jean Meeus and those shown on pages 280 & 281 of the Nautical Almanac. Note: For the years 1800 through 2100 the calculated values of GHA of Aries (γ), GHA & DEC for the Sun, Moon & planets have a maximum error of approximately ±0.1 arc minutes. Cells with a small red triangle in the upper right hand corner have an attached note. To view the note place the cursor on the cell. This worksheet is useful for sight planning purposes and as a quick check on sight reductions when used in conjunction with the Sight Reduction Using Law of Cosines Method, SR96a & Sight Planning worksheets. † The Planet data is based on the most important periodic terms from the French planetary theory VSOP87. For the Earth the full set of 2,425 periodic terms from the French planetary theory VSOP87 is used. †The Moon's position is calculated using the most important periodic terms from the ELP-2000/82 theory and have an accuracy of 0.17' for GHA & 0.07' for Dec. See Appendix C for additional information on accuracy. Return to Table of Contents

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Worksheet 8 24 Hc & Zn

This worksheet will display a graph of the Sun's altitude (Hc) & azimuth (Zn) over a 24 hour period for the date, DR latitude & DR longitude specified in the Nav Bodies worksheet. The graph of Hc is useful for visualizing how the duration of daylight & twilight periods change with latitude and time of year. Return to Table of Contents

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Worksheet 9 Analemma

The Analemma is a graph of Sun declination vs. equation of time for every day of the year. The equation of time is defined as the difference between universal mean time and local apparent time at the Greenwich meridian. Local apparent time is based on the exact position of the Sun in the sky. A sundial for example indicates local apparent time. Mean & apparent time are different because of two main reasons: ■ The eccentricity of the Earth's orbit causes the Earth to speed up or slow down in different parts its annual elliptical orbit around the Sun (Kepler's Second Law†), so we get ahead of or behind where we would be if the Earth's orbit was a perfect circle. ■ The Earth's axis is tilted to its orbit, and so the Sun's apparent motion along the (tilted) Ecliptic has a varying effect when viewed along the Equatorial plane (which clocks use). The rate of change in the value of the equation of time varies throughout the year from a maximum of around 40 seconds per day to a minimum of less than 1 second per day. This worksheet also displays the actual values for the equation of time & declination of the Sun based on the date & time specified in the Nav Bodies worksheet. † Kepler's Second Law: A line joining the Earth to the Sun sweeps out equal areas in equal times as the Earth moves in its elliptical orbit. A hyperlink is also provided to NOAAs Solar calculator to find sunrise, sunset, solar noon & solar position for any place on Earth..

A hyperlink is also provided to the U.S. Naval Observatory's Celestial Navigation Data Services, which provides computed altitudes and azimuths and other data for an assumed position and time.

Return to Table of Contents


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Worksheet 10 Meridian Transit

Given declination of Sun (Dec) & observed altitude (Ho) taken at meridian transit. This worksheet calculates the latitude of the observer using the DR latitude (L) specified in the Nav Bodies worksheet. A hyperlink is also provided to the U.S. Naval Observatory's Celestial Navigation Data Services, which provides computed altitudes and azimuths and other data for an assumed position and time.



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Worksheet 11

Sight Averaging Given time & sextant altitude (hs) for up to10 sights taken over a short period of time, this worksheet calculates the average time & average sextant altitude (hs). A plot of sextant altitude (hs) vs. time is provided to help identify any bad sights in the string of sights. To remove a bad sight from the list of sights click on the yellow square associated with the sight to be removed. The time of each sight must be increasing with sight number. Sight number 1 must contain a valid time & sextant altitude (hs). For best performance while using this worksheet change the Formula Calculations Options to Manual and use the Shift F9 key or the Calculate Sheet Command after entering or deleting any data values. Return to Table of Contents

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Worksheet 12

Calculating Ho from Hs Provides an analytic solution for calculating observed altitude (Ho) based on sextant altitude (hs). ■ Calculate dip (D) for a natural horizon using height of eye (h) in feet D = 0.97 * sqrt(h) ■ Calculate apparent altitude (Ha) where (IC) is the sextant index error & (Hs) is the altitude measured with the sextant from: Ha = Hs +IC -D ■ Calculate atmospheric refraction (Ro) at a standard temperature of 10° Celsius (C) and a pressure of 1010 milibars (mb) from: Ro = 0.0167/tan(Ha +7.32/(Ha+4.32)) If the temperature T° C and pressure P mb are known calculate the atmospheric refraction from: R= λ * Ro Where λ ≈ 0.28P/(T+273) or λ = (283/1010)P/(T+273) ■ Calculate the parallax in altitude (PA) from the horizontal parallax (HP) and the apparent altitude (Ha) for the Sun, Moon, Venus and Mars as follows: PA = HP * cos(Ha) For Jupiter, Saturn and the navigational stars HP = 0.0 For Sun HP = 0.0024 For Venus & Mars the HP is taken from the critical table at the bottom of page 259 of a current Nautical Almanac and converted to degrees. ■ Obtain the semi-diameter for the Sun & Moon as follows: Sun SD taken from bottom of appropriate daily page of Nautical Almanac Moon SD = 0.2724*HP Where HP is taken for the nearest hour from appropriate daily page of Nautical Almanac ■ Calculate Ho as Ho= Ha -R +PA ± SD ( For Upper Limb subtract SD & for Lower Limb add SD ) This worksheet is primarily intended to show the JN/N students how the data on pages A2, A3 & A4 of the "Nautical Almanac" were derived. See pages 280 & 281 of the "Nautical Almanac" for additional information. See Appendix C for an extract from page 261 of the Nautical Almanac discussing accuracy of Nautical Almanac data. Return to Table of Contents

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Worksheet 13

Sight Reduction Using Law of Cosines method Using the date, time, and position specified in the Nav Bodies worksheet along with the name of the body, declination of the body & local hour angle (LHA) of the body this worksheet calculates Hc, Zn & intercept (a), by the Law of Cosines method. This worksheet also produces the Time Diagram and the Drawing on the Plane of the Observer's Celestial Meridian associated with the above sight data. Also included in this worksheet are the calculation of the parameters associated with determining the intercept and azimuth by the Nautical Almanac sight reduction tables (NASR) method. The values for H, corr1, corr2, & Hc shown in this worksheet are rounded to the nearest tenth of an arc minute, whereas the NASR tabulated values are rounded to the nearest whole arc minute. A detailed, step by step, procedure for manually constructing the meridian diagram on the plane of the observer's celestial meridian is described in columns F through Z rows 55 through 85 of this worksheet. Return to Table of Contents

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Worksheet 14

Nautical Almanac Increments & Corrections (Yellow Pages) Given UT minute this worksheet calculates increments and corrections for 0 through 60 seconds following the specified UT minute. The increments and corrections calculated by this worksheet replicate those found in Nautical Almanac on pages ii -xxxi. Note: The values shown by this worksheet are rounded to the nearest 0.01' whereas those in the Nautical Almanac are rounded to the nearest 0.1' The following formulas are used to calculate the values shown by the Yellow Pages worksheet: Calculation of GHA increment (∆GHA °) Sun & Planets ∆GHA °= 15*(UTm +UTs/60)/60 Aries ∆GHA° = (15 +(36000/36525)/24)*(UTm +UTs/60)/60 Moon ∆GHA ° = (14 +19/60)*(UTm +UTs/60)/60 Calculation of corr′ d corr′ = d*(UTm +0.5)/60 v corr′ = v*(UTm +0.5)/60 A hyperlink "Text Box" is also provided to the U.S. Naval Observatory's Celestial Navigation Data Services, which provides computed altitudes and azimuths and other data for an assumed position and time. A hyperlink "Text Box" is also provided to "The Online Nautical Almanac", an online service that generates daily pages like those in the "Nautical Almanac". Return to Table of Contents

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Worksheet 15 SR96a & SR96b

This worksheet provides for the calculations needed to checking the data on the front page of SR 96a & SR 96b Sight Data Reduction Forms. This worksheet is primarily intended to aid JN/N Instructors & Sight Checkers when reviewing their student’s sight folders. Use the Nav Bodies worksheet to specify date, time & location of the sight you are checking. Next clear the yellow user data input cells, this is very important to avoid using data from a previous sight. After clearing the user data input cells, enter the appropriate data from the Nautical Almanac and other data as required from the front of the Sight Reduction Form. This worksheet will automatically inserts the correct GHA Increments & Corrections from the “Nautical Almanac Yellow Pages” based on the GMT minutes and seconds associated with the sight. All cells that require user data input are color coded yellow. If a user input cell is blank the numeric value associated with that cell is treated as though a numeric value of zero had been entered. This worksheet also calculates the EP latitude & EP longitude. Note: The Nautical Almanac GHA Increments used to calculate Total GHA by this worksheet depends on the following: 1.) If the Body Name specified is Moon then the Moon Increments will be used. For Moon sights the Limb (UL or LL) is also required for the worksheet to correctly insert the -30.0’ correction for Moon UL sights. 2.) If the SHA specified is greater than 0° 0.0' then the ARIES Increments will be used. 3.) If the name of the body is other than Moon & SHA is zero, then the Sun & Planets Increments will be used. Return to Table of Contents

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Worksheet 16

Sight Reduction by Direct Computation (SR by DC) This worksheet was added to illustrate how by using the information contained on pages 277 through 285 you can find ha, Ho, Hc, intercept (a) , Zn, and estimated position (EP) by direct computation. The only data you need from a current Nautical Almanac are the following: 1. For all bodies The GHA from the daily page for the whole UT hour prior to the exact time of the sight (GHA @ UT hr) and the GHA for the next whole hour after the time of the sight (GHA @ UT hr +1). For stars also enter the SHA in addition to the GHA for Aries. 2. The declination of the body from the daily page for the whole UT hour prior to the exact time of the sight (Dec @ UT hr) and the declination of the body for the next whole UT hour after the time of the sight (Dec @ UT hr +1). 3. For the Moon the HP from the daily page for the whole UT hour prior to the exact time of the sight (HP @ UT hr) and the HP of the Moon for the next whole UT hour after the time of the sight (HP @ UT hr +1). Note for Venus and Mars enter the HP of these bodies from the data found at the bottom of page 259 of the Nautical Almanac. 4. For the Sun enter the semi diameter (SD) of the body from the bottom of the appropriate Nautical Almanac daily page. Input temperature in degrees Celsius (°C) and the atmospheric pressure in milibars (mb) Enter height of eye, dip short distance, sextant altitude (hs) and the sextant index correction (IC). Note for a natural horizon enter dip short distance as 0. The DR latitude & longitude used by this worksheet is the same as the DR latitude & DR longitude specified in the Nav Bodies worksheet. The worksheet will then calculate ha, Ho, Hc, intercept (a) and Zn, EP latitude & EP longitude. Note: That the Increments & Corrections table values are not required as the GHA & Dec of the body for the time of the sight are determined from the data supplied at the whole hour prior to and after the time of the sight. This worksheet provides an alternative method for a quick check of a sight. See Appendix C for an extract from page 261 of the Nautical Almanac discussing accuracy of Nautical Almanac data. Return to Table of Contents

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Worksheet 17

Two Body Fix, From a DR Position This worksheet provides the calculations needed to establish the latitude & longitude of a fix from the intercepts, and Azimuths from two celestial sights. This worksheet also calculates the set & drift when the time from the previous fix is provided. To calculate a fix click on the box containing the text "Click To Solve For Fix". The parameter "ψ" indicates the approximate combined accuracy (latitude + longitude) of convergence to the fix location in minutes of arc. The parameter "Xing θ" is the crossing angle between LOPs for bodies 1 & 2. The ideal crossing angle of the LOPs for a two body fix is 90°. Note: You must install the Solver "Add-In" to your copy of Excel in order to use this worksheet and you also must enable macros to run. (See Appendix A "Installing the Excel Solver Add-In" for details) The DR latitude & longitude used by this worksheet is the same as the DR latitude & longitude specified in the Nav Bodies worksheet. Return to Table of Contents

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Worksheet 18

Two Body Fix, From an Assumed Position This worksheet provides the calculations needed to establish the latitude & longitude of a fix from the intercepts, and azimuths from two celestial sights when using the Nautical Almanac Sight Reduction Tables or the HO229 Sight Reduction Tables. Booth of these methods require an assumed position (AP). To calculate the fix click on the box containing the text "Click Here To Solve For Fix". The parameter "ψ" indicates the approximate combined accuracy (latitude + longitude) of convergence to the fix location in minutes of arc. The parameter "Xing θ" is the crossing angle between LOPs of the two bodies. The ideal crossing angle of the LOPs for a two body fix is 90°. Note: You must install the Solver "Add-In" to your copy of Excel in order to use this worksheet and you also must enable macros to run. (See Appendix A "Installing the Excel Solver Add-In" for details) Return to Table of Contents

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Worksheet 19

Three Body Fix, From a DR Position This worksheet provides the calculations needed to establish the latitude & longitude of a fix from the intercepts (a), and azimuths (Zn) from three celestial sights. . This worksheet also calculates the set & drift when the time from the previous fix is provided. To calculate a fix click on the box containing the text "Click To Solve For Fix". The parameter "ψ" indicates the approximate accuracy (latitude + longitude) of convergence to the coordinates of the fix location in minutes of arc. The parameter "Xing θ" is the crossing angle of each set of LOPs (e.g. The crossing angle between LOPs for bodies 1 & 2, bodies 1 & 3 and for bodies 2 & 3). The ideal crossing angle for each set of LOPs for a three body fix is 60°. The set of parameters "X L" & "X Lo" specify the latitude and longitude of the intersection for each set of LOPs. This set of "X L" & "X Lo" coordinates define the vertices of a triangle. The three body fix latitude & fix longitude calculated by this worksheet approximates the center of an inscribed circle within this triangle. Clicking in sequence each of the individual boxes contain the text "Intersection between LOPs for Bodies 1&2", " Intersection between LOPs for Bodies 2&3", and Intersection between LOPs for Bodies 1&3" to solve for the positions where each set of LOPs intersect is equivalent to clicking on the box containing the text "Click To Solve For Fix". The DR latitude & longitude used by this worksheet is the same as the DR latitude & longitude specified in the Nav Bodies worksheet. Note: You must install the Solver "Add-In" to your copy of Excel in order to use this worksheet and you also must enable macros to run. (See Appendix A "Installing the Excel Solver Add-In" for details) Return to Table of Contents

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Worksheet 20

Three Body Fix, From an Assumed Position This worksheet is used to calculate the latitude & longitude of a celestial fix when using the Nautical Almanac or HO 229 Sight Reduction Tables where an assumed position (AP) is required. To calculate a fix click on the box containing the text "Click Here To Solve For Fix". The parameter "ψ" indicates the approximate accuracy (latitude + longitude) of convergence to the coordinates of the fix location in minutes of arc. The parameter "Xing Angle" is the crossing angle of each set of LOPs (e.g. The crossing angle between LOPs for bodies 1 & 2, bodies 1 & 3 and for bodies 2 & 3). The ideal crossing angle for each set of LOPs for a three body fix is 60°. The three body fix latitude & fix longitude calculated by this worksheet approximates the center of an inscribed circle within the triangle formed by the three LOPs.. Note: You must install the Solver "Add-In" to your copy of Excel in order to use this worksheet and you also must enable macros to run. (See Appendix A "Installing the Excel Solver Add-In" for details) Return to Table of Contents

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Worksheet 21

Sight Planning Use the Nav Bodies worksheet to specify location & time of planned sight taking and also the altitude range & magnitude of bodies to flag as "Valid for Sight Taking". Bodies flagged as "Valid for Sight Taking" by the Nav Bodies worksheet will then be shown in columns O, P, Q & R of this worksheet. Star data is shown in rows 2 through 58. Sun, Moon & Planet data is shown in rows 60 through 65. Column O will contain the "Valid for Sight Taking" flag "♦", column P will contain the name of the body & column Q will contain the Zn of the body. If no value for Zn is shown, the body is below the horizon. If a value is shown for Zn but the body is not flagged as "Valid for Sight Taking" then the body did not meet the altitude range and/or visual magnitude you specified in the Nav Bodies Worksheet. Column R will contain the Hc of the body. Time of meridian transit is shown for the Sun, Moon & Planets in column S rows 60 through 65. Using the Two Body Fix LOP Crossing Angle Calculator, select the Zn of the first body for a two body fix, the "Two Body Fix LOP Crossing Angle Calculator" then calculates the valid ranges of Zn values for the second body that would provide a crossing angle of 45 degrees or grater of the two LOPs. Given the Zn of the second body the "Two Body Fix LOP Crossing Angle Calculator" calculates the crossing angle of the two LOPs. Using the Three Body Fix LOP Crossing Angle Calculator, select the Zn of the first body the for a three body fix, the "Three Body Fix LOP Crossing Angle Calculator" then calculates the valid ranges of Zn values for a second body that would provide a crossing angle of 30 degrees or grater. Given the Zn of the second body, enable macros and run the "ValidZnRanges" macro to calculates the valid ranges of Zn for the third body. The cells labeled "Valid Zn Ranges for 3rd Body" are not valid after entering new values for "1st Body Zn" & "2nd Body Zn" until running the "ValidZnRanges" macro. Given the Zn of the third body the "Three Body Fix LOP Crossing Angle Calculator" calculates the crossing angles of the LOPs from Body 1 & Body 2, from Body 1 & Body 3 and the crossing angle of the LOPs from Body 2 & Body 3. Return to Table of Contents

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Worksheet 22

Latitude by Sight on Polaris Use the Nav Bodies worksheet to specify date, time & DR Position for the Polaris sight you are using to determine Latitude. User inputs required are the Sextant altitude of Polaris (hs), Sextant IC, height of eye & dip short distance if applicable. For non-standard atmospheric pressure & temperature, this spreadsheet also provides for the input of atmospheric pressure in mili-bars temperature in degrees centigrade. This worksheet calculates the observed altitude (Ho), the Polaris Correction and the observers latitude based on the Polaris sight data. Return to Table of Contents

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Worksheet 23

Rhumb Line Distance vs. Great Circle Distance Given departure latitude (L1), departure longitude (Lo1), arrival latitude (L2) & arrival longitude (Lo2), and using spherical trigonometry equations this worksheet calculates the rhumb line course & distance and also the great circle distance & initial course heading. Note: The calculations for Rhumb Line distance and heading used in this work sheet agree with results obtained by Digital Wave's Visual Passage Planner (VPP2) used by the USPS Navigation 2009 Course. This worksheet also calculates arrival coordinates given departure latitude (L1), departure longitude (Lo1), rhumb line distance, true course heading. Return to Table of Contents

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Worksheet 24

Rule of 1/12 for Tide Prediction Given values for high and low tides this worksheet calculates intermediate tide heights for the six hours between low to high for incoming tides & intermediate tide heights for the six hours between high to low for outgoing tides. Return to Table of Contents

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Appendix A

Installing the Excel Solver Add-In

Steps to add the Solver add-in

1. Open Excel and click the Microsoft Office Button , and then click Excel Options.

2. Next from the pop-up window click Add-Ins, and then in the "Manage" drop down box, select "Solver Add-ins".

3. Then click on Go, which will cause the Add-Ins available box to appear.

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4. In the Add-Ins available box, select the Solver Add-in, Analysis ToolPak & Analysis ToolPak - VBA check boxes, and then click OK.

Tip: If Solver Add-in is not listed in the Add-Ins available box, click Browse to locate the Solver-add-in. If you get prompted that the Solver Add-in is not currently installed on your computer, click Yes to install it.

5. After you load the Solver Add-in, the Solver command is available in the Analysis group on the Excel Data tab.

You may also need to perform the following steps to reference the Microsoft Excel Solver add-in for macros: 1. Open your workbook. 2. On the Tools menu, point to Macro, and then click Visual Basic Editor. 3. On the Tools menu, click References. 4. In the Available References list, click to select the Solver.xls check box, and then click OK. Note If you do not see Solver.xls in the Available References list, click Browse. In the Add Reference dialog box, locate and select the

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Solver.xla file, and then click Open. The Solver.xla file is typically found in the C:\Program Files\Microsoft Office\Office\Library\Solver subfolder. Google "Excel Solver add-in" to find additional information about the Excel Solver add-in. Return to Table of Contents

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Appendix B Enabling Macros

Click "Options" in the "Security Warning bar"

This will pop up a "Microsoft Office Security Options" window with the following options: ʘ Help Protect me from unknown Content (Recommended) ʘ Enable this content If you select the first option the macros will remain disabled. If you select the second option the macros will run but you will need to enable them each time you open the workbook. Next select "Enable this content" and click "OK"


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Appendix C Accuracy

The values of GHA for Aries (γ), the SHA of stars, the GHA and Dec for the Sun, Moon & Planets calculated by the Nav Bodies worksheet are in general agreement with the values tabulated in the Nautical Almanac (with a difference of approximately ±0.1 arc minutes for values @ whole hour & ±0.2 arc minutes where Nautical Almanac increments & correction are applied this is due to the rounding errors of the Nautical Almanac values). Page 166 of Meeus, Astronomical Algorithms, states that a very high accuracy (0.01 arc seconds) is obtained when use is made of the complete VSOP87 theory. Worksheet Nav Bodies uses the complete set of 2425 terms of the VSOP87 theory for the Earth & uses the most important periodic terms of the VSOP87 theory for the planets Venus, Mars, Jupiter & Saturn. Data for the Sun from Exmple 25.a of Meeus are shown below in the column labeled VSOP87 and compared with the results obtained by the Nav Bodies worksheet. The maximum difference is less than 0.01 arc seconds.

UT date 13 October 1992 UT time 00:00:00

Meeus Example 25.a page 166 Sun VOSP87

Nav Bodies

Difference geocentric longitude ( Θ+∆Θ ) 199.9072722° 199.90727215° 0.000180 arc sec

geocentric latitude ( β+∆β ) 0.0002° 0.00020015° 0.000540 arc sec

Apparent longitude ( λ ) 199.9059889° 199.90598809° 0.002916 arc sec

radius vector ( R ) 0.99760853 AU 0.99760852 AU 0.00000001 AU

right ascension (α) 198.3781209° 198.37812344° 0.009144 arc sec

declination (δ) -7.78381667° -7.78381766° 0.003564 arc sec The difference in the values for the Moon's right ascension (α) and declination (δ) calculated by the Nav Bodies worksheet and the values shown in example 47.a test case on page 342 of Meeus, which were obtained by using the complete ELP-2000/82 theory are the shown in the following table. Moon ELP-2000/82 Nav Bodies Difference right ascension (α) 8h58m45s.1 8h58m45s.2 0h0m0s.1

declination (δ) 13.7683680 degrees 13.76836666 degrees 0.000084 arc minutes For additional information on accuracy see chapter 2 "About Accuracy" in the second edition of "Astronomical Algorithms" by Jean Meeus. ISBN 0-943396-61-1

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Extract from Page 261 of the Nautical Almanac stating the accuracy of its data. Return to Table of Contents

ACCURACY 24. Main data. The quantities tabulated in this Almanac are generally correct to the nearest 0'.1; the exception is the sun's GHA which is deliberately adjusted by up to 0'.15 to reduce the error due to ignoring the v-correction. The total GHA and Dec at intermediate times cannot be obtained to this precision, since at least two quantities must be added; moreover, the v - and d -corrections are based on mean values of v and d and are taken from tables for the whole minute only. The largest error that can occur in the GHA or Dec of any body other than the Sun or Moon is less than 0'.2; it may reach 0'.25 for the GHA of the Sun and 0'.3 for that of the Moon. In practice it may be expected that only one third of the values of GHA and Dec taken out will have errors larger than 0'.05 and less than one tenth will have errors larger than 0'.1 25. Altitude Corrections. The errors in the altitude corrections are nominally of the same order as those in GHA and Dec, as they result from addition of several quantities each correctly rounded off to 0'.1. But the actual values of the dip and the refraction at low altitudes may, in extreme atmospheric conditions , differ considerably from the mean values used in the tables.

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