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Ecliptic Altitude and Orientation

Consider a point on the ecliptic of ecliptic longitude $\lambda$. We wish to determine the altitude of this point, as well as the angle subtended there between the ecliptic and the vertical, $t$ hours before it culminates at the meridian.

Figure 13: Ecliptic altitude. $SCBE$ is the southern horizon, with $S$ and $E$ the south and east compass points, respectively. $DYB$ is the ecliptic. $ZDS$ the meridian, and $Z$ the zenith. $ZYC$ is an altitude circle.
\begin{figure} \epsfysize =3in \centerline{\epsffile{altitude1.eps}} \end{figure}

The situation is as shown in Fig. 13. Here, $Y$ is the point in question, and $ZYC$ an altitude circle (i.e., a great circle passing through the zenith) drawn through it. We wish to determine the altitude $a\equiv CY$ of point $Y$, as well as the angle $\mu\equiv ZYB$. Note that the angle, $\mu$, subtended between the ecliptic and an altitude circle is always defined such that it lies to the east of the altitude circle, and to the north of the ecliptic.

According to Eqs. (40) and (41), the declination and right ascension of point $Y$ are given by

$\displaystyle \sin\,\delta$ $\textstyle =$ $\displaystyle \sin\epsilon\,\sin\lambda,$ (69)
$\displaystyle \tan\,\alpha$ $\textstyle =$ $\displaystyle \cos\epsilon\,\tan\lambda,$ (70)

respectively. We can also write $\alpha_0=\alpha-t$, where $\alpha_0$ is the right ascension of the point on the ecliptic which is culminating (i.e., point $D$ in the diagram), and $t$ is measured in time-degrees. It follows from Eqs. (55) and (56) that the altitude and azimuth of point $Y$ are
$\displaystyle \sin a$ $\textstyle =$ $\displaystyle \sin L\,\sin\delta + \cos L\,\cos\delta\,\cos t,$ (71)
$\displaystyle \tan A$ $\textstyle =$ $\displaystyle \frac{\cos\delta\,\sin t}{\cos L\,\sin\delta - \sin L\,\cos\delta\,\cos t},$ (72)

respectively. According to Eq. (57), the ecliptic longitude, $\lambda_h$, of the intersection of the ecliptic and the eastern horizon (i.e., point $B$ in the diagram) is
\begin{displaymath} \tan\lambda_h = \frac{-\cos L\,\cos(\alpha-t)}{\cos L\,\cos\epsilon\,\sin(\alpha-t) + \sin\epsilon\,\sin L}. \end{displaymath} (73)

It follows from Eq. (68) that the azimuth of this point is
\begin{displaymath} \cos A_h = \frac{\sin\lambda_h\,\sin\epsilon}{\cos L}. \end{displaymath} (74)

Consider the right-angled spherical triangle $YBC$. According to a well-known theorem in spherical trigonometry,

\begin{displaymath} \tan (CY) = \tan (BY) \,\cos (CYB), \end{displaymath} (75)

where $CY$ and $CB$ are arcs, and $CYB$ an angle. In fact, $CY=a$, $BY = \lambda_h-\lambda$, and $CYB=180^\circ-\mu$. Hence,
\begin{displaymath} \cos\mu = -\frac{\tan a}{\tan(\lambda_h-\lambda)}. \end{displaymath} (76)

In the above analysis, we have tacitly assumed that the point on the ecliptic which is culiminating (i.e., point $D$) lies to the south of the zenith. The situation in which the point lies to the north of the zenith is shown in Fig. 14. In this case, analysis of spherical triangle $YBC$ yields

\begin{displaymath} \cos\mu = \frac{\tan a}{\tan(\lambda_h-\lambda)}. \end{displaymath} (77)

As is easily demonstrated, the point on the ecliptic which is culminating lies to the south of the zenith whenever $A-A_h>0$, and to the north whenever $A-A_h <0$.

Figure 14: Ecliptic altitude. $NCBE$ is the northern horizon, with $N$ and $E$ the south and east compass points, respectively. $DYB$ is the ecliptic. $ZDN$ the meridian, and $Z$ the zenith. $ZYC$ is an altitude circle.
\begin{figure} \epsfysize =3in \centerline{\epsffile{altitude2.eps}} \end{figure}

For times after the culmination of point $X$--i.e., for $t<0$--the above formulae remain valid, except that the point on the ecliptic which is culminating now lies to the south of the zenith whenever $A_h-A+180^\circ>0$, and to the north whenever $A_h-A+180^\circ < 0$.

According to Eq. (71), the critical value of $t$ at which point $Y$ reaches the horizon is given by

\begin{displaymath} \cos t_h = - \tan L\,\tan\delta. \end{displaymath} (78)

Of course, the above equation is only soluble if $\vert\tan L\,\tan\delta\vert<1$. However, it is easily demonstrated that if $\tan L\,\tan\delta < -1$ then point $Y$ never sets, whereas if $\tan L\,\tan\delta > 1$ then point $Y$ never rises.

Tables 21-29 show the altitudes of twelve equally spaced points on the ecliptic, as well as the angle subtended between the ecliptic and the vertical at these points, as functions of time, calculated for a series of observation sites in the northern hemisphere with equally spaced terrestrial latitudes. The twelve points correspond to the start of the twelve zodiacal signs, and are named accordingly. Thus, ``Aries" corresponds to ecliptic longitude $0^\circ $, ``Taurus" to ecliptic longitude $30^\circ$, etc. For each point, four columns of data are provided. The first column corresponds to the time (in hours and minutes) either before or after the culmination of the point, the second column gives the altitude of the point (which is the same in both cases), the third column gives the angle $\mu$ for the case in which the first column indicates time prior to the culmination of the point, and the fourth column gives the angle $\mu$ for the opposite case. The symbol N indicates that the ecliptic crosses the meridian to the north of the zenith. Data is only provided for cases in which the various points on the ecliptic lie on or above the horizon.

Figure 15: Map showing all stars of visual magnitude $<6$ lying within $15^\circ$ of the ecliptic plane. (a)
\begin{figure} \epsfysize =5.5in \centerline{\epsffile{ecliptic2.eps}} \end{figure}
Figure 16: Map showing all stars of visual magnitude $<6$ lying within $15^\circ$ of the ecliptic plane. (b)
\begin{figure} \epsfysize =5.5in \centerline{\epsffile{ecliptic1.eps}} \end{figure}


Table 4: Ecliptic longitudes and latitudes and visual magnitudes of selected bright stars lying within $10^\circ$ of the ecliptic plane.
Aries Libra
$\lambda$ $\beta$ Mag. Name $\lambda$ $\beta$ Mag. Name
$09^\circ 09'$ $+12^\circ 36'$ $+2.8$ $\gamma$ PEG $04^\circ 50'$ $+1^\circ 22'$ $+3.9$ $\eta$ VIR
$26^\circ 49'$ $+5^\circ 23'$ $+3.6$ $\eta$ PSC $10^\circ 08'$ $+2^\circ 46'$ $+3.5$ $\gamma$ VIR
$11^\circ 28'$ $+8^\circ 37'$ $+3.4$ $\delta$ VIR
$22^\circ 08'$ $+8^\circ 38'$ $+3.4$ $\zeta$ VIR
$23^\circ 51'$ $-2^\circ 03'$ $+1.0$ $\alpha$ VIR
Taurus Scorpio
$\lambda$ $\beta$ Mag. Name $\lambda$ $\beta$ Mag. Name
$03^\circ 58'$ $+8^\circ 29'$ $+2.6$ $\beta$ ARI $15^\circ 5'$ $+0^\circ 20'$ $+2.8$ $\alpha$ LIB
$07^\circ 39'$ $+9^\circ 58'$ $+2.0$ $\alpha$ ARI $19^\circ 22'$ $+8^\circ 30'$ $+2.6$ $\beta$ LIB
Gemini Saggitarius
$\lambda$ $\beta$ Mag. Name $\lambda$ $\beta$ Mag. Name
$00^\circ 00'$ $+4^\circ 03'$ $+2.9$ $\eta$ TAU $02^\circ 34'$ $-1^\circ 59'$ $+2.3$ $\delta$ SCO
$09^\circ 47'$ $-5^\circ 28'$ $+0.9$ $\alpha$ TAU $02^\circ 56'$ $ -5^\circ 29'$ $+2.9$ $\pi$ SCO
$22^\circ 34'$ $+5^\circ 23'$ $+1.7$ $\beta$ TAU $03^\circ 11'$ $+1^\circ 00'$ $+2.6$ $\beta$ SCO
$07^\circ 48'$ $-4^\circ 02'$ $+2.9$ $\sigma$ SCO
$09^\circ 46'$ $-4^\circ 34'$ $+1.0$ $\alpha$ SCO
$11^\circ 27'$ $-6^\circ 08'$ $+2.8$ $\tau$ SCO
$17^\circ 58'$ $+7^\circ 12'$ $+2.4$ $\eta$ OPH
Cancer Capricorn
$\lambda$ $\beta$ Mag. Name $\lambda$ $\beta$ Mag. Name
$05^\circ 18'$ $~-0^\circ 49'$ $+2.9$ $\mu$ GEM $01^\circ 16'$ $-7^\circ 00'$ $+3.0$ $\gamma$ SGR
$09^\circ 06'$ $~-6^\circ 44'$ $+1.9$ $\gamma$ GEM $04^\circ 34'$ $-6^\circ 28'$ $+2.7$ $\delta$ SGR
$09^\circ 56'$ $~+2^\circ 04'$ $+3.0$ $\epsilon$ GEM $06^\circ 19'$ $-2^\circ 08'$ $+2.8$ $\lambda$ SGR
$20^\circ 14'$ $+10^\circ 06'$ $+2.0$ $\alpha$ GEM $12^\circ 23'$ $-3^\circ 27'$ $+2.0$ $\sigma$ SGR
$23^\circ 13'$ $~+6^\circ 41'$ $+1.1$ $\beta$ GEM $13^\circ 38'$ $-7^\circ 11'$ $+2.6$ $\zeta$ SGR
$16^\circ 15'$ $+1^\circ 26'$ $+2.9$ $\pi$ SGR
Leo Aquarius
$\lambda$ $\beta$ Mag. Name $\lambda$ $\beta$ Mag. Name
$20^\circ 47'$ $+9^\circ 43'$ $+3.0$ $\epsilon$ LEO $23^\circ 24'$ $+8^\circ 37'$ $+2.9$ $\beta$ AQR
$29^\circ 37'$ $+8^\circ 49'$ $+2.6$ $\gamma$ LEO $23^\circ 33'$ $-2^\circ 36'$ $+2.9$ $\delta$ CAP
$29^\circ 50'$ $+0^\circ 28'$ $+1.4$ $\alpha$ LEO
Virgo Pisces
$\lambda$ $\beta$ Mag. Name $\lambda$ $\beta$ Mag. Name
$06^\circ 23'$ $+0^\circ 09'$ $+3.9$ $\rho$ LEO $06^\circ 43'$ $+8^\circ 14'$ $+3.8$ $\gamma$ AQR
$13^\circ 25'$ $+9^\circ 40'$ $+3.3$ $\theta$ LEO $08^\circ 52'$ $-8^\circ 12'$ $+3.3$ $\delta$ AQR
$17^\circ 34'$ $+6^\circ 06'$ $+3.9$ $\iota$ LEO $11^\circ 34'$ $-0^\circ 23'$ $+3.7$ $\lambda$ AQR
$27^\circ 10'$ $+0^\circ 42'$ $+3.6$ $\beta$ VIR $21^\circ 27'$ $+7^\circ 15'$ $+3.7$ $\gamma$ PSC



Table 5: Declinations and right ascensions of points on the ecliptic circle (a).
Aries Taurus Gemini
$\lambda$ $\delta$ $\alpha$ $\lambda$ $\delta$ $\alpha$ $\lambda$ $\delta$ $\alpha$
                 
00$^\circ$ +00$^\circ$$00'$ 000$^\circ$$00'$ 00$^\circ$ +11$^\circ$$28'$ 027$^\circ$$55'$ 00$^\circ$ +20$^\circ$$09'$ 057$^\circ$$49'$
02$^\circ$ +00$^\circ$$48'$ 001$^\circ$$50'$ 02$^\circ$ +12$^\circ$$10'$ 029$^\circ$$50'$ 02$^\circ$ +20$^\circ$$33'$ 059$^\circ$$54'$
04$^\circ$ +01$^\circ$$35'$ 003$^\circ$$40'$ 04$^\circ$ +12$^\circ$$51'$ 031$^\circ$$45'$ 04$^\circ$ +20$^\circ$$57'$ 062$^\circ$$00'$
06$^\circ$ +02$^\circ$$23'$ 005$^\circ$$30'$ 06$^\circ$ +13$^\circ$$31'$ 033$^\circ$$41'$ 06$^\circ$ +21$^\circ$$18'$ 064$^\circ$$07'$
08$^\circ$ +03$^\circ$$10'$ 007$^\circ$$21'$ 08$^\circ$ +14$^\circ$$10'$ 035$^\circ$$38'$ 08$^\circ$ +21$^\circ$$38'$ 066$^\circ$$14'$
10$^\circ$ +03$^\circ$$58'$ 009$^\circ$$11'$ 10$^\circ$ +14$^\circ$$49'$ 037$^\circ$$36'$ 10$^\circ$ +21$^\circ$$57'$ 068$^\circ$$22'$
12$^\circ$ +04$^\circ$$45'$ 011$^\circ$$02'$ 12$^\circ$ +15$^\circ$$26'$ 039$^\circ$$34'$ 12$^\circ$ +22$^\circ$$13'$ 070$^\circ$$30'$
14$^\circ$ +05$^\circ$$31'$ 012$^\circ$$53'$ 14$^\circ$ +16$^\circ$$02'$ 041$^\circ$$33'$ 14$^\circ$ +22$^\circ$$28'$ 072$^\circ$$39'$
16$^\circ$ +06$^\circ$$18'$ 014$^\circ$$44'$ 16$^\circ$ +16$^\circ$$37'$ 043$^\circ$$32'$ 16$^\circ$ +22$^\circ$$42'$ 074$^\circ$$48'$
18$^\circ$ +07$^\circ$$04'$ 016$^\circ$$36'$ 18$^\circ$ +17$^\circ$$11'$ 045$^\circ$$32'$ 18$^\circ$ +22$^\circ$$54'$ 076$^\circ$$57'$
20$^\circ$ +07$^\circ$$49'$ 018$^\circ$$28'$ 20$^\circ$ +17$^\circ$$44'$ 047$^\circ$$33'$ 20$^\circ$ +23$^\circ$$03'$ 079$^\circ$$07'$
22$^\circ$ +08$^\circ$$34'$ 020$^\circ$$20'$ 22$^\circ$ +18$^\circ$$16'$ 049$^\circ$$35'$ 22$^\circ$ +23$^\circ$$12'$ 081$^\circ$$17'$
24$^\circ$ +09$^\circ$$19'$ 022$^\circ$$13'$ 24$^\circ$ +18$^\circ$$46'$ 051$^\circ$$38'$ 24$^\circ$ +23$^\circ$$18'$ 083$^\circ$$28'$
26$^\circ$ +10$^\circ$$02'$ 024$^\circ$$07'$ 26$^\circ$ +19$^\circ$$15'$ 053$^\circ$$41'$ 26$^\circ$ +23$^\circ$$22'$ 085$^\circ$$39'$
28$^\circ$ +10$^\circ$$46'$ 026$^\circ$$00'$ 28$^\circ$ +19$^\circ$$43'$ 055$^\circ$$45'$ 28$^\circ$ +23$^\circ$$25'$ 087$^\circ$$49'$
30$^\circ$ +11$^\circ$$28'$ 027$^\circ$$55'$ 30$^\circ$ +20$^\circ$$09'$ 057$^\circ$$49'$ 30$^\circ$ +23$^\circ$$26'$ 090$^\circ$$00'$
 
Cancer Leo Virgo
$\lambda$ $\delta$ $\alpha$ $\lambda$ $\delta$ $\alpha$ $\lambda$ $\delta$ $\alpha$
                 
00$^\circ$ +23$^\circ$$26'$ 090$^\circ$$00'$ 00$^\circ$ +20$^\circ$$09'$ 122$^\circ$$11'$ 00$^\circ$ +11$^\circ$$28'$ 152$^\circ$$05'$
02$^\circ$ +23$^\circ$$25'$ 092$^\circ$$11'$ 02$^\circ$ +19$^\circ$$43'$ 124$^\circ$$15'$ 02$^\circ$ +10$^\circ$$46'$ 153$^\circ$$60'$
04$^\circ$ +23$^\circ$$22'$ 094$^\circ$$21'$ 04$^\circ$ +19$^\circ$$15'$ 126$^\circ$$19'$ 04$^\circ$ +10$^\circ$$02'$ 155$^\circ$$53'$
06$^\circ$ +23$^\circ$$18'$ 096$^\circ$$32'$ 06$^\circ$ +18$^\circ$$46'$ 128$^\circ$$22'$ 06$^\circ$ +09$^\circ$$19'$ 157$^\circ$$47'$
08$^\circ$ +23$^\circ$$12'$ 098$^\circ$$43'$ 08$^\circ$ +18$^\circ$$16'$ 130$^\circ$$25'$ 08$^\circ$ +08$^\circ$$34'$ 159$^\circ$$40'$
10$^\circ$ +23$^\circ$$03'$ 100$^\circ$$53'$ 10$^\circ$ +17$^\circ$$44'$ 132$^\circ$$27'$ 10$^\circ$ +07$^\circ$$49'$ 161$^\circ$$32'$
12$^\circ$ +22$^\circ$$54'$ 103$^\circ$$03'$ 12$^\circ$ +17$^\circ$$11'$ 134$^\circ$$28'$ 12$^\circ$ +07$^\circ$$04'$ 163$^\circ$$24'$
14$^\circ$ +22$^\circ$$42'$ 105$^\circ$$12'$ 14$^\circ$ +16$^\circ$$37'$ 136$^\circ$$28'$ 14$^\circ$ +06$^\circ$$18'$ 165$^\circ$$16'$
16$^\circ$ +22$^\circ$$28'$ 107$^\circ$$21'$ 16$^\circ$ +16$^\circ$$02'$ 138$^\circ$$27'$ 16$^\circ$ +05$^\circ$$31'$ 167$^\circ$$07'$
18$^\circ$ +22$^\circ$$13'$ 109$^\circ$$30'$ 18$^\circ$ +15$^\circ$$26'$ 140$^\circ$$26'$ 18$^\circ$ +04$^\circ$$45'$ 168$^\circ$$58'$
20$^\circ$ +21$^\circ$$57'$ 111$^\circ$$38'$ 20$^\circ$ +14$^\circ$$49'$ 142$^\circ$$24'$ 20$^\circ$ +03$^\circ$$58'$ 170$^\circ$$49'$
22$^\circ$ +21$^\circ$$38'$ 113$^\circ$$46'$ 22$^\circ$ +14$^\circ$$10'$ 144$^\circ$$22'$ 22$^\circ$ +03$^\circ$$10'$ 172$^\circ$$39'$
24$^\circ$ +21$^\circ$$18'$ 115$^\circ$$53'$ 24$^\circ$ +13$^\circ$$31'$ 146$^\circ$$19'$ 24$^\circ$ +02$^\circ$$23'$ 174$^\circ$$30'$
26$^\circ$ +20$^\circ$$57'$ 117$^\circ$$60'$ 26$^\circ$ +12$^\circ$$51'$ 148$^\circ$$15'$ 26$^\circ$ +01$^\circ$$35'$ 176$^\circ$$20'$
28$^\circ$ +20$^\circ$$33'$ 120$^\circ$$06'$ 28$^\circ$ +12$^\circ$$10'$ 150$^\circ$$10'$ 28$^\circ$ +00$^\circ$$48'$ 178$^\circ$$10'$
30$^\circ$ +20$^\circ$$09'$ 122$^\circ$$11'$ 30$^\circ$ +11$^\circ$$28'$ 152$^\circ$$05'$ 30$^\circ$ +00$^\circ$$00'$ 180$^\circ$$00'$



Table 6: Declinations and right ascensions of points on the ecliptic circle (b).
Libra Scorpio Saggitarius
$\lambda$ $\delta$ $\alpha$ $\lambda$ $\delta$ $\alpha$ $\lambda$ $\delta$ $\alpha$
                 
00$^\circ$ -00$^\circ$$00'$ 180$^\circ$$00'$ 00$^\circ$ -11$^\circ$$28'$ 207$^\circ$$55'$ 00$^\circ$ -20$^\circ$$09'$ 237$^\circ$$49'$
02$^\circ$ -00$^\circ$$48'$ 181$^\circ$$50'$ 02$^\circ$ -12$^\circ$$10'$ 209$^\circ$$50'$ 02$^\circ$ -20$^\circ$$33'$ 239$^\circ$$54'$
04$^\circ$ -01$^\circ$$35'$ 183$^\circ$$40'$ 04$^\circ$ -12$^\circ$$51'$ 211$^\circ$$45'$ 04$^\circ$ -20$^\circ$$57'$ 242$^\circ$$00'$
06$^\circ$ -02$^\circ$$23'$ 185$^\circ$$30'$ 06$^\circ$ -13$^\circ$$31'$ 213$^\circ$$41'$ 06$^\circ$ -21$^\circ$$18'$ 244$^\circ$$07'$
08$^\circ$ -03$^\circ$$10'$ 187$^\circ$$21'$ 08$^\circ$ -14$^\circ$$10'$ 215$^\circ$$38'$ 08$^\circ$ -21$^\circ$$38'$ 246$^\circ$$14'$
10$^\circ$ -03$^\circ$$58'$ 189$^\circ$$11'$ 10$^\circ$ -14$^\circ$$49'$ 217$^\circ$$36'$ 10$^\circ$ -21$^\circ$$57'$ 248$^\circ$$22'$
12$^\circ$ -04$^\circ$$45'$ 191$^\circ$$02'$ 12$^\circ$ -15$^\circ$$26'$ 219$^\circ$$34'$ 12$^\circ$ -22$^\circ$$13'$ 250$^\circ$$30'$
14$^\circ$ -05$^\circ$$31'$ 192$^\circ$$53'$ 14$^\circ$ -16$^\circ$$02'$ 221$^\circ$$33'$ 14$^\circ$ -22$^\circ$$28'$ 252$^\circ$$39'$
16$^\circ$ -06$^\circ$$18'$ 194$^\circ$$44'$ 16$^\circ$ -16$^\circ$$37'$ 223$^\circ$$32'$ 16$^\circ$ -22$^\circ$$42'$ 254$^\circ$$48'$
18$^\circ$ -07$^\circ$$04'$ 196$^\circ$$36'$ 18$^\circ$ -17$^\circ$$11'$ 225$^\circ$$32'$ 18$^\circ$ -22$^\circ$$54'$ 256$^\circ$$57'$
20$^\circ$ -07$^\circ$$49'$ 198$^\circ$$28'$ 20$^\circ$ -17$^\circ$$44'$ 227$^\circ$$33'$ 20$^\circ$ -23$^\circ$$03'$ 259$^\circ$$07'$
22$^\circ$ -08$^\circ$$34'$ 200$^\circ$$20'$ 22$^\circ$ -18$^\circ$$16'$ 229$^\circ$$35'$ 22$^\circ$ -23$^\circ$$12'$ 261$^\circ$$17'$
24$^\circ$ -09$^\circ$$19'$ 202$^\circ$$13'$ 24$^\circ$ -18$^\circ$$46'$ 231$^\circ$$38'$ 24$^\circ$ -23$^\circ$$18'$ 263$^\circ$$28'$
26$^\circ$ -10$^\circ$$02'$ 204$^\circ$$07'$ 26$^\circ$ -19$^\circ$$15'$</