Latitue epenence of the axiu uration of a total olar eclipe Author: Jen Buu, with aitance fro Jean Meeu Contact: 6 Baker Street, Gayton, Northant, NN7 3EZ, UK jbuu@btinternet.co Introuction It i well known that the axiu uration of a total eclipe (in the current epoch) i jut over 7 3 (45), in Morel III Jean Meeu give the value a 7 32 (452). It i obviou that the longet uration will occur when the Earth i near aphelion (currently aroun 5 July), an that it will therefore happen lightly north of equator. A latitue of 5 north wa entione by Danjon on page 318 in Atronoie Générale (Pari, 1952). However, the ore general quetion: What i the axiu uration of a total olar eclipe a a function of the latitue? oe not appear to have been aree. A a etaile anwer to thi quetion woul be an aruou tak, we are intea uing ipler etho to erive a reaonably accurate etiate. The firt approach i to look at the ata fro the Five Millenniu Canon. The econ i to evie a highly iplifie atheatical oel to provie an approxiate reult. Finally, the thir approach i to look at the uration of artificial eclipe; thee are bae on real eclipe, but their central line have been iplace. Data fro the Five Millenniu Canon Figure 1 how uration veru latitue for all the total eclipe in the Five Millenniu Canon. We notice in particular the axiu uration of 449. Thi happen on 17 th July 2186 at a latitue of 7.4 north. Thi uration i very cloe to the theoretical axiu of 452. 5 45 4 35 3 25 2 15 1 5-9. -75. -6. -45. -3. -15.. 15. 3. 45. 6. 75. 9. Figure 1. Scatter plot of eclipe uration in econ v latitue in egree (-9 correpon to the South Pole, +9 to the North Pole). Data fro the Five Millenniu Canon. 1
Although figure 1 qualitatively agree with the expectation, an quantitatively with the abolute axiu uration, we cannot be ure that thi finite ata et capture the axiu uration for all value of the latitue. Thi qualification i particularly iportant for the polar region where there are far fewer ata point. Siple atheatical oel The firt tep i to fin an expreion for the eclipe agnitue. We tart by writing the angular ize of the Sun a - δ co( ϕ - ϕ ) (1) = In thi expreion i the average angular ize, δ i the aplitue of the variation in the angular ize, an ϕ ecribe the poition of the Earth in it orbit (i.e. the ate) with ϕ = ϕ when the Earth i at aphelion. The angular ize of the Moon at perigee, a een fro the centre of the Earth i given by Moon = (2) where Moon i the iaeter of the Moon an i the iniu itance between the centre of the Earth an the centre of the oon at perigee. We can now write the eclipe agnitue with the Sun at it average itance an the Moon at perigee a M = (3) with a ate epenent correction for the variation of the angular ize of the Sun given by δ M1 = co( ϕ - ϕ) (4) Next we expre the eclination of the axi of the Earth with repect to the Sun a = ax co( ϕ - ϕ ) (5) Here ax i the axiu eclination (23.5 ), an again ϕ ecribe the poition of the Earth with ϕ = ϕ at Suer oltice The iniu itance D fro a point at latitue on the urface of the Earth to the centre of the Moon, with the Moon at perigee, a eclination, an the Earth raiu r E i given by D = - (6) 2 2 2rE co( - )+ re 2
which can be approxiate a re D ( 1- co( - )) (7) a D iffer fro ue in (2), we get an aitional agnitue correction given by re M2 = co( - ) (8) We finally fin the agnitue a a function of the latitue ( ) an the ate (through ϕ ) by cobining the above expreion M(, ϕ) = M (9) +M1 + M2 The axiu agnitue M ax ( ) at a given latitue can now be foun by axiiing M with repect to ϕ. Note that in thi operation we nee to exclue value of ϕ which correpon to the Sun being below the horizon. On the outhern heiphere thi happen for > 9 +. The following value have been ue for the variou contant: M = = 1.4835, δ re =.179, =.167, ϕ = 3.2 ra (correponing to 5 July) an ϕ = 2.96 ra (correponing to 21 June). The econ tep i to tranlate the eclipe agnitue into an eclipe uration, uing the axiu agnitue for each latitue. Firt we ue a iple etiate of the average angular velocity of the Moon (een fro the Earth) relative to that of the Sun va' = 29.5 36 24h / 36 / h =1.412 1-4 / (1) However, we nee to correct for the fact that for a axiu uration eclipe the Earth i near aphelion an the Moon i near perigee. Thi ean that the angular velocity of the Sun i lower, an the angular velocity of the Moon i higher. Accoring to Keppler econ law, velocity tie itance i contant, but ince angular velocity equal velocity ivie by itance it follow that angular velocity equal a contant ivie by the quare of the itance. The cobine effect of a higher itance to the Sun (an hence a lower angular velocity of the Sun), an a lower itance to the Moon (an hence a higher angular velocity of the Moon), an the epenence of the quare of the itance ean that the relevant relative angular velocity of the Moon i higher than the value given by equation (1). A ore accurate value i given by va = va' 1.18 (11) The final tep i to inclue the influence of the rotation of the Earth. The uration of an eclipe for a tatic Earth (or at the pole) i 3
Where ' Du' ( ) = (Max ( ) -1) = (Max ( ) -1) 3144 (12) v a ' =.524 i the iniu value of the angular ize of the Sun. With v E being the urface velocity of the Earth at equator (4,k/24h) an velocity of the Moon, we finally get the eclipe uration at the latitue Du( 1 v E -1-1 ) = Du'( )( - co( )) = Du'( )( -.4287 co( )) (13) vm 1 v M being the Nuerical ipleentation an reult The require nuerical calculation are all traight forwar an conveniently carrie out in a iple preaheet. The firt tep i to generate a 2-D table of the agnitue M a a function of the ate an the latitue bae on equation (1)-(9). For each value of the latitue the axiu agnitue can then be foun by interpolation. A entione above, we nee to exclue perio where the Sun i below the horizon. For the northern heiphere thi i not an iue ince the axiu agnitue happen in July. However, cloe to the South Pole the Sun i below the horizon at the tie of axiu agnitue, o the axiu viible agnitue will occur when the Sun i jut on the horizon at i-ay. The ot extree cae i the South Pole itelf, where the axiu viible agnitue will occur at the autunal equinox. Uing equation (1)-(13) we fin the axiu uration a a function of the latitue hown in figure 2. 5 45 4 35 3 25 2 15 1 5-9. -75. -6. -45. -3. -15.. 15. 3. 45. 6. 75. 9. Figure 2. Calculate axiu total eclipe uration a a function of latitue (re curve with all blue arker) uperipoe on the catter plot fro figure 1. There i a very light kink near -7 becaue we have to ue the highet value for the viible agnitue. 4
It i een that the calculate reult agree extreely well with the Five Millenniu ata near equator, an that there i a reaonable agreeent for ot of the northern heiphere. However, in the polar region, in particular near the South Pole, the agreeent i le goo. At thi point of the invetigation it i not obviou whether thi apparent iagreeent for latitue near the South Pole i caue by eficiencie in the oel, or whether it i becaue eclipe with a uration cloe to the axiu at thee latitue are o rare that they o not occur in the perio covere by the Five Millenniu Canon. Artificial eclipe, circutance at the South Pole In orer to clarify the ituation near the South Pole we conier eclipe atifying the following conition: (1) the Sun i above the horizon, (2) the eclipe i total in the funaental plane (3) Gaa i between -.6 an -1.2. For each of thee eclipe the central line i hifte o it pae over the South Pole an the uration i calculate. The reult are uarie in the following table. Perio Year an ate of eclipe Maxiu uration Year 1 to 1 45, 16 March 157 11-2 1913, 3 Septeber 148 21-3 2285, 29 Septeber 158 We ee that the lat two cae occur jut after the autunal equinox, wherea the firt cae occur jut before the vernal equinox. The reaon for thi i that in the year 1246 aphelion coincie with the Suer oltice. Thi correpon to the value of ϕ being equal to the value of ϕ. Before 1246 aphelion wa cloer to the vernal equinox, after 1246 it ha been cloer to the autunal equinox. The effect of thi can eaily be tuie with the iple oel, the reult are hown in figure 3. The reult fro the tuy of the artificial eclipe are reaonably cloe to thoe preicte by the iple oel, in particular if we take into account that the uration of the artificial eclipe coul be longer by up to 5 ha they occurre even cloer to the equinox. Thi inicate that the preicte 164 axiu uration (in the current epoch) for a total olar eclipe at the South Pole preicte by the iple oel, i a reaonable etiate. For the North Pole the ituation i uch ipler ince the axiu agnitue an uration will happen near Suer oltice. A iilar tuy of artificial eclipe, but with Gaa now retricte to the interval +1.1 to +.6, gave a axiu uration of 218 in the perio 11 to 2, an 216 for the perio 21 to 3. Thi i in quite goo agreeent with the reult of 229 fro the iple oel (ee figure 2). 5
Figure 3. Maxiu uration in econ of a total olar eclipe at the South Pole a function of the year. A) at vernal equinox. B) at autunal equinox. The three black quare point inicate the reult fro the artificial eclipe. Concluion The fining can be uarie a follow: We have evelope a iple oel for the calculation of the axiu uration of a total olar eclipe a a function of the latitue. The reult fro thi oel are in reaonable agreeent with ata fro the Five Millenniu Canon, except in the polar region. However, a tuy bae on artificial eclipe inicate that the apparent icrepancie (in particular in the outhern heiphere) are iply ue to the fact that at thee latitue eclipe with a uration cloe to the axiu are very rare, an have not occurre in the perio covere by the Canon. We note that the circutance near the South Pole are ignificantly ore coplicate than at other latitue, with the axiu poible uration varying ignificantly over tie (about 1.5 per century). For other latitue the eclipe agnitue a a function of the ate ha a fairly flat axiu; conequently there i little change in the axiu uration when the eparation between aphelion an Suer oltice change. All in all, the iple oel ee to be correct to within a few econ, an we have thu etablihe the eire approxiate reult for the uration of a total eclipe a a function of the latitue. Acknowlegeent The author (JB) ha benefite ignificantly fro icuion about thi topic with Jean Meeu, without thee thi paper coul not have been written. Jean Meeu contribute the reult bae on artificial eclipe. 6