CHAPTER 15
第十五章
AZIMUTHS AND AMPLITUDES
方位與振幅
尚作仁船長 編譯
「美國航海家實用指南」(American Practical Navigator Volume 1)詳述了所有現代和傳統的航海導航原理、方法和實務操作，是一本完整的航海百科全書。國內所有天文與地文航海教科書內容幾乎皆翻譯自此書，本章摘譯「振幅」(Amplitudes)內容如下，所謂"Amplitudes"照英文字面解釋為「振幅」，航海系學生往往難以理解何謂「振幅」? 其實它就是所謂的「太陽出沒方位角」，也就是日出或日沒(太陽位於水平線時)，太陽與正東(090°)或正西(270°)的夾角，由於此時太陽沒有高度(Altitude)，因此計算太陽方位的三角公式僅為單純的平面三角公式，航海員計算時不需藉助各種測算表(例如: Pub. No. 229表，Pub. No. 260表，HO. 211表，HO. 208表，Davis Table等)，在航海實務上實為一簡便計算船上羅經誤差的方法。
1503. Amplitudes
1503. 振幅
A celestial body’s amplitude angle is the complement of its azimuth angle. At the moment that a body rises or sets, the amplitude angle is the arc of the horizon between the body and the East/West point of the horizon where the observer’s prime vertical intersects the horizon (at 90°), which is also the point where the plane of the equator intersects the horizon (at an angle numerically equal to the observer’s co-latitude). See Figure 1503.
天體的振幅角是方位角的補角(兩角相加等於90°)。在天體升起或落下的瞬間，該方位角位於主垂直圈(經過東西兩點的垂直圈)與經過天體的垂直圈在天頂處的夾角。參見圖1503。
Figure 1503. The amplitude angle (A) subtends the arc of the horizon between the body and the point where the prime vertical and the equator intersect the horizon. Note that it is the compliment of the azimuth angle (Z).
圖1503. 振幅角(A)所對應的地平圈位於天體與主垂直圈和赤道與地平線相交點之間。請注意，它是方位角(Z)的補數。
In practical navigation, a bearing (psc or pgc) of a body can be observed when it is on either the celestial or the visible horizon. To determine compass error, simply convert the computed amplitude angle to true degrees and compare it with the observed compass bearing.
在實際導航中，當天體位於天球上或可見地平線上時，可以觀察到天體的方位（psc或pgc）。要確定羅經誤差，只需將計算出的振幅角轉換為真實角度，並與觀察到的羅經方位進行比較。
The angle is computed by the formula: sin A = sin Dec / cos Lat.
角度計算公式：sin A = sin Dec / cos Lat。
This formula gives the angle at the instant the body is on the celestial horizon. It does not contain an altitude term because the body’s computed altitude is zero at this instant.
這個公式可得出天體在天球地平線上那一刻的角度。但它不包含高度，因為此時天體計算的高度為零。
The angle is prefixed E if the body is rising and W if it is setting. This is the only angle in celestial navigation referenced FROM East or West, i.e. from the prime vertical. A body with northerly declination will rise and set
如果天體正在上升，則角度前綴為E；如果物體正在下降，則角度前綴為W。這是天文航海中唯一從東方或西方(即從主垂直圈)參考的角度。天體赤緯在北緯將會升起和落下
North of the prime vertical. Likewise, a body with southerly declination will rise and set South of the prime vertical. Therefore, the angle is suffixed N or S to agree with the name of the body’s declination. A body whose declination is zero rises and sets exactly on the prime vertical.
主垂直圈以北。同樣，天體赤緯在南緯的將在主垂直線以南升起和落下。因此，角度後綴為N或S，以與天體赤緯的名稱一致。赤緯為零的天體恰好在主垂直線上上升和落下。
Due largely to refraction, dip, and its disk size, the Sun is on the celestial horizon when its lower limb is approximately two thirds of a diameter above the visible horizon. The Moon is on the celestial horizon when its upper limb is on the visible horizon. Stars and planets are on the celestial horizon when they are approximately one Sun diameter above the visible horizon.
主要是因為折射、傾角及其太陽圓周大小的影響，當太陽直徑三分之二的下緣位於可見地平線上方時，太陽就位於天地平線上。當月亮的上緣位於可見地平線上時，月球就位於天地平線上。當恆星和行星在可見地平線上方大約一個太陽直徑時，它們就位於天地平線上。
When observing a body on the visible horizon, a correction from Table 23 Correction of Amplitude as Observed on the Visible Horizon must be applied. This correction accounts for the slight change in bearing as the body moves between the visible and celestial horizons. It reduces the bearing on the visible horizon to the celestial horizon, from which the table is computed.
當觀測可見地平線上的天體時，必須應用表23 可見地平線上觀測的振幅修正中的修正。這種修正說明了當物體在可見地平線和天體地平線之間移動時方位的輕微變化。它將可見地平線的方位修正到天體地平線，以作為查表計算的依據。
For the Sun, stars, and planets, apply this correction to the observed bearing in the direction away from the elevated pole. For the moon, apply one half of the correction toward the elevated pole. Note that the algebraic sign of the correction does not depend upon the body’s declination, but only on the observer’s latitude. Assuming the body is the Sun the rule for applying the correction can be outlined as follows:
對於太陽、恆星和行星，此修正值應「遠離」天仰極的方位。對於月亮，將一半的修正值「朝向」天仰極。請注意，修正的代數符號並不取決於天體的赤緯，僅取決於觀測者的緯度。假設物體是太陽，應用修正的規則可以概述如下：
Observer’s Lat. 觀測者緯度 | Rising/Setting 日出/日落 | How to Apply Correction 如何使用修正值 |
North 北緯 | Rising 日出 | Add correction to observed bearing 觀測方位加上修正值 |
North 北緯 | Setting 日落 | Subtract correction from observed bearing 觀測方位減去修正值 |
South 南緯 | Rising 日出 | Subtract correction from observed bearing 觀測方位減去修正值 |
South 南緯 | Setting 日落 | Add correction to observed bearing 觀測方位加上修正值 |
Table 1503. Amplitude correction for the Sun.
表1503. 太陽的振幅修正。
The following two articles demonstrate the procedure for obtaining the amplitude of the Sun on both the celestial and visible horizons.
以下兩篇文章解釋了取得太陽在天地平線與可見地平線上的振幅的過程。
1504. Amplitude of the Sun on the Celestial Horizon
1504. 太陽在天水平線上的振幅
Mariners may use Bowditch Table 22 (Amplitudes) to determine the Sun's computed amplitude. The procedure is similar to that done in Section 1501. Comparing the computed amplitude to the amplitude measured with the gyrocompass determines the gyro error. In computing an amplitude, interpolate the tabular amplitude angle for the difference between the table arguments and the actual values of declination and latitude.
航行員可以使用Bowditch表22 (振幅)來確定太陽的計算振幅。過程與第1501節中的過程類似。在計算振幅時，根據表列參數與赤緯和緯度的實際值之間的差異對表列幅度角進行內插。
Do this double interpolation of the amplitude angle as follows:
如下所示對振幅角進行雙重內插：
l Enter Bowditch Table 22 (Amplitudes) with the nearest integral values of declination and latitude. Extract a base amplitude angle.
輸入鮑氏表22(振幅)以及最接近的赤緯和緯度積分值。提取基本振幅角。
2 Reenter the table with the same declination argument but with the latitude to the next tabulated value (greater or less than the base latitude argument, depending upon whether the actual latitude is greater or less than the base argument). Record the amplitude and the difference between it and the base amplitude angle and label it Diff.
使用相同的赤緯參數重新查表，但使用下一個表列值的緯度(大於或小於基本緯度參數，取決於實際緯度是大於還是小於基本參數)。記錄振幅及其與基本振幅角之間的差異，並將其標記為Diff。
3 Reenter the table with the base latitude argument but with the declination to the next tabulated value (greater or less than the base declination argument, depending upon whether the actual declination is greater or less than the base argument). Record the amplitude and the difference between it and the base amplitude angle and label it Diff.
重新輸入表列基本緯度參數，但使用下一個表列值的赤緯(大於或小於基本赤緯參數，取決於實際赤緯是大於還是小於基本參數)。記錄振幅與基本振幅角之間的差值，並將其標記為Diff。
4 Compute the corrections due to latitude and declination not being exactly at a tabular value. Apply these corrections to obtain a final amplitude. The final amplitude is then converted to a true bearing. The difference between the true bearing and the gyro bearing gives the gyro error.
計算緯度和赤緯與表列值的內插修正。應用這些修正以獲得最終的振幅。然後將最終振幅轉換為真方位角。真方位和羅經方位之間的差異即為羅經差。
Example:
The DR latitude of a ship is 51° 24.6' N. The navigator observes the setting Sun on the celestial horizon. Its declination is N 19° 40.4'. Its observed bearing is 303° pgc.
船舶的DR緯度為北緯51°24.6'。其赤緯為北緯19° 40.4'。觀察到的方位角為303° pgc。
Required: Gyro error.
求電羅經差
Solution:
Interpolate in Table 22 for the Sun’s calculated amplitude as follows. See Figure 1504. The actual values for latitude and declination are L = 51.4° N and dec. = N 19.67°. Find the tabulated values of latitude and declination closest to these actual values. In this case, these tabulated values are L = 51° and dec. = 19.5°. Record the amplitude corresponding to these base values, 32.0°, as the base amplitude.
在表22中對計算出的太陽振幅進行內插，如下所示。請參閱圖1504。緯度和赤緯的實際值為L = 51.4° N和dec. = N 19.67°。找出最接近這些緯度和赤緯的表列值。在這種情況下，這些表格值為L = 51°和dec. = 19.5°。記錄與這些基準值32.0°對應的振幅值，作為基本振幅。
Next, holding the base declination value constant at 19.5°, increase the value of latitude to the next tabulated value: N 52°. Note that this value of latitude was increased because the actual latitude value was greater than the base value of latitude. Record the tabulated amplitude for L = 52° and dec. = 19.5°. Then, holding the base latitude value constant at 51°, increase the declination value to the next tabulated value: 20°. Record the tabulated amplitude for L = 51° and dec. = 20°: 32.9°.
接下來，將基本赤緯值保持在19.5°不變，將緯度值增加到下一個表列值：N 52°。請注意，該緯度值會增加，因為實際緯度值大於基本緯度值。記錄L = 52°和dec.的表列振幅。 = 19.5°: 32.8°。然後，將基本緯度值保持在51°不變，將赤緯值增加到下一個表列值：20°。記錄L = 51°和dec.= 20°32.9°的表列振幅。
The latitude’s actual value (51.4°) is 0.4 of the way between the base value (51°) and the value used to determine the tabulated amplitude (52°). The declination’s actual value (19.67°) is 0.3 of the way between the base value (19.5°) and the value used to determine the tabulated amplitude (20.0°). To determine the total correction to base amplitude, multiply these increments (0.4 and 0.3) by the respective difference between the base and tabulated values (+0.8 and + 0.9, respectively) and sum the products. The total correction is +0.6°. Add the total correction (+0.6°) to the base amplitude (32.0°) to determine the final amplitude (32.6°) which will be converted to a true bearing.
實際緯度(51.4°)的內差值是0.4，是表列振福值(51°)和(52°)之間的內差值。實際赤緯(19.67°)的內差值是0.3，是表列振幅值(19.5°)和(20.0°)之間的內差值。要確定基本振幅的總修正量，請將這些增量（0.4和0.3）分別計算基礎值和表格列值之間的差異(分別為+0.8和+0.9），並對乘積求和。總修正量為+0.6°。將總修正量(+0.6°)加到基本振幅(32.0°)獲得最終振幅(32.6°)將用於轉換為真方位角。
Because of its northerly declination (in this case), the Sun was 32.6° north of west when it was on the celestial horizon. Therefore its true bearing was 302.6° (270° + 32.6°) at this moment. Comparing this with the gyro bearing of 303° gives an error of 0.4°W, which can be rounded to 1/2°W.
由於赤緯是北緯(在本例中)，當太陽位於天地平線上時，太陽位於西偏北32.6°。因此真實方位為302.6°(270° + 32.6°)。與303°的羅經方位相比，誤差為0.4°W，可視為1/2°W。
Actual | Base | Base Amp. | Tab. Amp. | Diff. | Inc. | Correction |
L=51.4° N dec=19.67° N | 51° 19.5° | 32.0° 32.0° | 32.8° 32.9° | +0.8° +0.9° | 0.4 0.3 | +0.3° +0.3° |
|
|
|
|
| Total | +0.6° |
Figure 1504. Interpolation in Table 22 for Amplitude.
表1504 表22中的振幅內插值。
1505. Amplitude of the Sun on the Visible Horizon
1505. 太陽在可見地平線上的振幅
In higher latitudes, amplitude observations should be made when the body is on the visible horizon because the value of the correction is large enough to cause significant error if the observer misjudges the exact position of the celestial horizon. The observation will yield precise results whenever the visible horizon is clearly defined.
在高緯度地區，天體應在可見地平線上時觀測到振幅，因為如果觀測者錯誤判斷天體地平線的準確位置，將會導致明顯的誤差。只要能明確辨識可見地平線，就能產生精確的觀測結果。
Example:
Observer’s DR latitude is 59°47’N, Sun’s declination is 5°11.3’S. At sunrise the Sun is observed on the visible horizon bearing 098.5° pgc.
觀測者的DR緯度為北緯59°47’N，太陽赤緯為5°11.3’S。日出時，在可見地平線上觀測到太陽方位為098.5° pgc。
Required: Gyrocompass error.
求電羅經差
Solution:
解題：
Given this particular latitude and declination, the amplitude angle is 10.3°S, so that the Sun’s true bearing is 100.3° at the moment it is on the celestial horizon, that is, when its Hc is precisely 0°. Applying the Table 23 correction to the observed bearing of 098.5° pgc using the rules given in Section 1503, the Sun would have been bearing 099.7° pgc had the observation been made when the Sun was on the celestial horizon. Therefore, the gyro error is 0.6°E.
根據本題的緯度和赤緯，振幅角為10.3°S，因此太陽在天地平線上時的真實方位角為100.3°，即當其Hc恰好為0°時。使用第1503節的規則，將表23修正應用於觀測到的098.5° pgc方位，如果在太陽位於天地平線上時進行觀測，太陽的方位將是099.7° pgc。因此，羅經誤差為0.6°E。 (電羅經小，誤差東)
1506. Amplitude by Calculation
1506. 振幅計算
As an alternative to using the amplitude tables, if a calculator is available then the amplitudes can be computed using a slightly modified version of the altitude-azimuth formula. The modification is needed because azimuth (Z) and amplitude (A) angles are complimentary, and the cofunctions of complimentary angles are equal; i.e., cosine Z = sine A. In the following formulas, northerly latitudes and declinations are given positive values, and southerly latitudes and declinations are considered negative. If the resulting amplitude is positive, it is north of the prime vertical; conversely, a negative amplitude is south of the prime vertical.
作為使用幅度表的替代方法，如果有計算機可用，則可以使用高度-方位角公式的稍微修改版來計算振幅。需要修改是因為方位角（Z）和振幅角（A）是互補的，並且互補角的餘函數相等；即餘弦Z = 正弦A。若所得振幅為正，則位於主垂直圈以北；相反，負振幅位於主垂直圈以南。
a) The general case, when a body is not on the celestial horizon, the formula is:
一般情況，當天體不在天地平線上時，公式為：
where Dec is the celestial body's declination, Lat is the observer's latitude, and Hc is the object's computed altitude. For the Sun on the visible horizon, Hc = -0.7°.
其中Dec是天體的赤緯，Lat是觀測者的緯度，Hc是計算出的天體高度。對於可見地平線上的太陽，Hc = -0.7°。
b) When a body is on the celestial horizon (that is, its altitude, Hc = 0), the formula becomes:
當天體位於天體地平線上(即其高度，Hc = 0)時，公式變為：
Example:
例題：
Determine the gyrocompass error using the formulation instead of the tables, for the example in Section 1505.
使用公式而不是查表來確定羅經誤差，例如第1505節中的範例。
Required: Gyrocompass error.
求電羅經差
Solution:
解題：
The observed bearing of the Sun on the visible horizon is 098.5° pgc. The computed amplitude of the Sun when it is on the visible horizon (that is, Hc = -0.7°) is found by:
觀測太陽在可見地平線上的方位角為098.5° pgc。太陽位於可見地平線時(即Hc = -0.7°)的計算振幅可透過以下公式求得：
Amplitude = sin-1[(sin -5.19° − sin 59.78° sin -0.7°) / (cos 59.78° cos -0.7°)].
Evaluating, we find the amplitude is 9.1°. This is 9.1° degrees away from E, in the “negative” (or southerly) direction, so the calculated azimuth is 90° + 9.1° = 99.1°. The gyrocompass error is 99.1° 98.5° = 0.6° E. This value matches the answer obtained in Section 1505 using the tables.
經評估，我們發現振幅為9.1°。意即處於「負」(或向南)方向遠離E 9.1°，因此計算出的方位角為90° + 9.1° = 99.1°。羅經誤差為99.1° − 98.5° = 0.6° E。 (電羅經小，誤差東)
