Tidal Vectors?
If you live anywhere near the sea, you will be able to observe the tides. Although all sorts of complications arise when considering tidal flows for a given place on earth, in general there are two high tides and two low tides each day, separated on average by about 12 hours and 25 minutes.
While tides are commonly attributed to the Moon, the Sun also plays an important part in producing tides. Although the gravitational field of the Sun is very considerably stronger than that of the Moon in the region of the Earth, tidal force varies with the inverse cube of distance, and the Moon is much closer to Earth than the Sun. The net effect is that the Moon contributes about 70% to the tides, while the Sun contributes about 30% to the tides, on average.
To a close approximation, the Sun and Moon lie on the ecliptic. Suppose the Moon is just over two days past New Moon, and is 30 degrees after the Sun along the ecliptic, crossing the meridian about 2 hours later than the Sun.
Intuitively, high tide (on an ideal ocean) will occur sometime between the Sun being on the meridian and the Moon being on the meridian, probably nearer to the time when the Moon is on the meridian as the Moon is associated with the stronger tide-generating power. This often happens in practice, with high tide occurring earlier than Moon on the meridian being called "Priming" of the tide.
It is possible to draw a vector diagram, and to calculate the combined tidal vector, if you know the angle between the Sun and the Moon, and their distances from Earth. This vector will have magnitude and direction, and can be associated with a point on the ecliptic. Some of the references allude to this method. The difficulty is that the tidal vector obtained in this way bears no relation to the actual tides, unless the Moon is within 45 degrees or so of the Sun!
The problem, as I see it, is that if we are going to find the average vector for the Sun and Moon, we should be looking at the gravitational vector, rather than a tidal vector. This direction of this gravitational vector will always be well within a degree or so of the Sun, since the gravitational field of the Sun swamps that of the Moon.
So, how to work out the tidal vector? We need a vector that is going to align itself with high tide, or rather two high tides, on opposite sides of the Earth.
My own method, which may not be the simplest, is to work with the Real Projective Line (RP1), or, more simply, with what will be familiar to astrologers as the second harmonic.
This identifies opposite degrees of the ecliptic (and opposite bits of the earth under the ecliptic) in a "circle" of 180 degrees - with the advantage that there is now only one high and one low tide after this transformation. Now the original tidal vector approach works just fine. Of course the results have to be translated back into the original 360 degree ecliptic afterwards, and we end up with two tidal vectors that are opposite in the ecliptic, each corresponding with one of the two high tides.
I have produced tables or those who wish to experiment, first giving average values for the displacement of the tidal vector from the position of the Moon, then high values (Moon near apogee, Earth near perihelion) and low values (Moon near perigee, Earth near aphelion). High lunar latitude only changes the values very slightly (<1%)
Column 1 is Sun-Moon angle along the ecliptic (degrees Moon is after Sun)
Column 2 is the Sun-Moon Vector I have seen used elsewhere
Column 3 is the Sun-Moon Vector calculated as described above
The figures in Columns 2 and 3 refer to displacement. Positive values indicate the tidal vector is earlier in the zodiac than the Moon, negative values the reverse.
For example, for a Sun-Moon angle of 110 degrees, (Moon ahead of Sun by 110 degrees, waxing and gibbous), and Moon in (say) 15 degrees Sag, method 1 places the Vector at 18 Scorpio, while method 2 gives a vector at 27 Sag (and 27 Gem, as there are 2 vectors with this method)
Note that the values for 0 - 45 degrees (and 315 - 360 degrees) give similar results using either method.
Tidal Vector Tables
Average Values
Sun-Moon Angle Method 1 Method 2 (my preference)
0 0 0 5 1.570438244109924 1.568909118415464 10 3.137818236830929 3.125473494015772 15 4.69904462404468 4.656740865535964 20 6.250946988031544 6.148486675312091 25 7.79024014149906 7.58442702644628 30 9.313481731071928 8.945227898426229 35 10.81702611447563 10.20720223827347 40 12.29697335062418 11.34054780284484 45 13.74911197831142 12.30693051408653 50 15.16885405289256 13.05617786728267 55 16.55116065691376 13.52187805505172 60 17.89045579685246 13.61597669116266 65 19.18052624372539 13.22355010738297 70 20.41440447654693 12.20203667242668 75 21.5842314632701 10.39632056536122 80 22.68109560568968 7.691708423051655 85 23.6948438652362 4.123791942955503 90 24.61386102817307 0 95 25.4248135326384 -4.123791942955503 100 26.11235573456534 -7.691708423051655 105 26.65879972363539 -10.39632056536122 110 27.04375611010343 -12.20203667242668 115 27.24376465564846 -13.22355010738297 120 27.23195338232531 -13.61597669116266 125 26.97779741939964 -13.52187805505172 130 26.44710021476594 -13.05617786728267 135 25.60239579324122 -12.30693051408653 140 24.40407334485336 -11.34054780284484 145 22.81264227506687 -10.20720223827347 150 20.79264113072244 -8.945227898426229 155 18.31864074045383 -7.58442702644628 160 15.38341684610331 -6.148486675312091 165 12.00746460215217 -4.656740865535964 170 8.247583885911006 -3.125473494015772 175 4.200832191016906 -1.568909118415464 180 0 0 185 -4.200832191016906 1.568909118415464 190 -8.247583885911006 3.125473494015772 195 -12.00746460215217 4.656740865535964 200 -15.38341684610331 6.148486675312091 205 -18.31864074045383 7.58442702644628 210 -20.79264113072244 8.945227898426229 215 -22.81264227506687 10.20720223827347 220 -24.40407334485336 11.34054780284484 225 -25.60239579324122 12.30693051408653 230 -26.44710021476594 13.05617786728267 235 -26.97779741939964 13.52187805505172 240 -27.23195338232531 13.61597669116266 245 -27.24376465564846 13.22355010738297 250 -27.04375611010343 12.20203667242668 255 -26.65879972363539 10.39632056536122 260 -26.11235573456534 7.691708423051655 265 -25.4248135326384 4.123791942955503 270 -24.61386102817307 0 275 -23.6948438652362 -4.123791942955503 280 -22.68109560568968 -7.691708423051655 285 -21.5842314632701 -10.39632056536122 290 -20.41440447654693 -12.20203667242668 295 -19.18052624372539 -13.22355010738297 300 -17.89045579685246 -13.61597669116266 305 -16.55116065691376 -13.52187805505172 310 -15.16885405289256 -13.05617786728267 315 -13.74911197831142 -12.30693051408653 320 -12.29697335062418 -11.34054780284484 325 -10.81702611447563 -10.20720223827347 330 -9.313481731071928 -8.945227898426229 335 -7.79024014149906 -7.58442702644628 340 -6.250946988031544 -6.148486675312091 345 -4.69904462404468 -4.656740865535964 350 -3.137818236830929 -3.125473494015772 355 -1.570438244109924 -1.568909118415464 360 0 0
High Values
Sun-Moon Angle Method 1 Method 2 (my preference)
0 0 0 5 1.786527681174876 1.78527638205438 10 3.57055276410876 3.560440949244826 15 5.349538995660901 5.314828576344873 20 7.120881898489651 7.036605703164404 25 8.881872332866672 8.712020354099565 30 10.62965715268975 10.3244189565895 35 12.36119578871117 11.85288569204713 40 14.07321140632881 13.27028150658311 45 15.76213503229047 14.54032817637552 50 17.42404070819913 15.6131667213451 55 19.05456928703145 16.4184834823293 60 20.64883791317899 16.85485531874242 65 22.20133147911113 16.77373574149593 70 23.70577138409426 15.9581214855935 75 25.15495567119872 14.10573152138756 80 26.54056301316623 10.86235348549805 85 27.8529109690565 6.026390123352322 90 29.08065635275105 0 95 30.21042237389015 -6.026390123352322 100 31.2263334426902 -10.86235348549805 105 32.10943438509268 -14.10573152138756 110 32.8369669646586 -15.9581214855935 115 33.38147469938228 -16.77373574149593 120 33.70971063748484 -16.85485531874242 125 33.78133959997175 -16.4184834823293 130 33.54747148299187 -15.6131667213451 135 32.94916407614839 -14.54032817637552 140 31.91624297118699 -13.27028150658311 145 30.36718179634878 -11.85288569204713 150 28.21146304277513 -10.3244189565895 155 25.35681960432185 -8.712020354099565 160 21.72470697099611 -7.036605703164404 165 17.27701653031626 -5.314828576344873 170 12.05278024670464 -3.560440949244826 175 6.203093836591149 -1.78527638205438 180 0 0 185 -6.203093836591149 1.78527638205438 190 -12.05278024670464 3.560440949244826 195 -17.27701653031626 5.314828576344873 200 -21.72470697099611 7.036605703164404 205 -25.35681960432185 8.712020354099565 210 -28.21146304277513 10.3244189565895 215 -30.36718179634878 11.85288569204713 220 -31.91624297118699 13.27028150658311 225 -32.94916407614839 14.54032817637552 230 -33.54747148299187 15.6131667213451 235 -33.78133959997175 16.4184834823293 240 -33.70971063748484 16.85485531874242 245 -33.38147469938228 16.77373574149593 250 -32.8369669646586 15.9581214855935 255 -32.10943438509268 14.10573152138756 260 -31.2263334426902 10.86235348549805 265 -30.21042237389015 6.026390123352322 270 -29.08065635275105 0 275 -27.8529109690565 -6.026390123352322 280 -26.54056301316623 -10.86235348549805 285 -25.15495567119872 -14.10573152138756 290 -23.70577138409426 -15.9581214855935 295 -22.20133147911113 -16.77373574149593 300 -20.64883791317899 -16.85485531874242 305 -19.05456928703145 -16.4184834823293 310 -17.42404070819913 -15.6131667213451 315 -15.76213503229047 -14.54032817637552 320 -14.07321140632881 -13.27028150658311 325 -12.36119578871117 -11.85288569204713 330 -10.62965715268975 -10.3244189565895 335 -8.881872332866672 -8.712020354099565 340 -7.120881898489651 -7.036605703164404 345 -5.349538995660901 -5.314828576344873 350 -3.57055276410876 -3.560440949244826 355 -1.786527681174876 -1.78527638205438 360 0 0
Low Values
Sun-Moon Angle Method 1 Method 2 (my preference)
0 0 0 5 1.314767676709828 1.313014283398536 10 2.626028566797071 2.611894538596677 15 3.930240354950164 3.881928778481799 20 5.223789077193353 5.107210648722353 25 6.502951810395087 6.269944398894776 30 7.763857556963598 7.349628628733772 35 9.002445689627896 8.322075913008337 40 10.21442129744471 9.158234100062815 45 11.39520675185988 9.822808191332184 50 12.53988879778955 10.27277229168874 55 13.64316048013091 10.45607528325858 60 14.69925725746754 10.31129937335767 65 15.70188675706948 9.76988883078465 70 16.64415182601667 8.763882357003119 75 17.51846688757382 7.243235861717316 80 18.31646820012563 5.205319715203573 85 19.02891955315132 2.730357264684994 90 19.64561638266437 0 95 20.15529346882312 -2.730357264684994 100 20.54554458337748 -5.205319715203573 105 20.80276704875626 -7.243235861717316 110 20.91215056651717 -8.763882357003119 115 20.85773825246106 -9.76988883078465 120 20.62259874671535 -10.31129937335767 125 20.18916114064506 -10.45607528325858 130 19.5397776615693 -10.27277229168874 135 18.65758879854164 -9.822808191332184 140 17.52776471400624 -9.158234100062815 145 16.1391740384239 -8.322075913008337 150 14.48647172343463 -7.349628628733772 155 12.57248800908023 -6.269944398894776 160 10.41063943040715 -5.107210648722353 165 8.02689778401302 -3.881928778481799 170 5.460714529369988 -2.611894538596677 175 2.764311127694158 -1.313014283398536 180 0 0 185 -2.764311127694158 1.313014283398536 190 -5.460714529369988 2.611894538596677 195 -8.02689778401302 3.881928778481799 200 -10.41063943040715 5.107210648722353 205 -12.57248800908023 6.269944398894776 210 -14.48647172343463 7.349628628733772 215 -16.1391740384239 8.322075913008337 220 -17.52776471400624 9.158234100062815 225 -18.65758879854164 9.822808191332184 230 -19.5397776615693 10.27277229168874 235 -20.18916114064506 10.45607528325858 240 -20.62259874671535 10.31129937335767 245 -20.85773825246106 9.76988883078465 250 -20.91215056651717 8.763882357003119 255 -20.80276704875626 7.243235861717316 260 -20.54554458337748 5.205319715203573 265 -20.15529346882312 2.730357264684994 270 -19.64561638266437 0 275 -19.02891955315132 -2.730357264684994 280 -18.31646820012563 -5.205319715203573 285 -17.51846688757382 -7.243235861717316 290 -16.64415182601667 -8.763882357003119 295 -15.70188675706948 -9.76988883078465 300 -14.69925725746754 -10.31129937335767 305 -13.64316048013091 -10.45607528325858 310 -12.53988879778955 -10.27277229168874 315 -11.39520675185988 -9.822808191332184 320 -10.21442129744471 -9.158234100062815 325 -9.002445689627896 -8.322075913008337 330 -7.763857556963598 -7.349628628733772 335 -6.502951810395087 -6.269944398894776 340 -5.223789077193353 -5.107210648722353 345 -3.930240354950164 -3.881928778481799 350 -2.626028566797071 -2.611894538596677 355 -1.314767676709828 -1.313014283398536 360 0 0
I'll get a better picture up soon!
You can make out the double cycle of the green line through 360 (method two)
The fainter red line is method 1 - a single cycle, same shape, double amplitude
Links
Tides - excellent for a graphical look at tidal forces - choose a date and place!
Tidal Influences
Lunar Phenomena
Tidal Vector Forces
http--www.reedsalmanac.com-pdf-restless.pdf