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Terrestrial Magnetism ttmospheric Electricity

VOLUXtE37 MARCH, 1932 No. 1

TERRESTRIAL-MAGNETIC ACTIVITY AND ITS RELATIONS TO BY J. BARTELS A bstractmA homogeneousseries of monthly means of terrestrial-magnetic activity for the years 1872 to 1930 is derived and extendedbackward, in annual means,to 1835. The annual variation of magnetic activity and of the relative -numbersis dis- cussedby means of new tests for periods. Only the semi-annual wave in magnetic activity is recognizedas physicallysignificant. Its maxima prefer the times whenthe is in the celestialequator, and not, as has beensuggested, the times when the Sun's axis is most inclined towards the ecliptic. This view is supportedby tests usingthe harmonic dial and the probable-error circle, and several independent considerations. The close relations between sunspot-numbersand terrestrial-magnetic activity in the annual and monthly means are discussed. Some general statistical aspectsare given for the treatment of the correlationbetween such series with after-effects,for 'which both solar activ.ity and terrestrial-magneticactivity are typical. The homo- geneity of the whole availableseries for relativesunspot-numbers and for areasof sunspotsand faculaeis tested;some inhomogeneities are found, apart from a general lag of terrestrial-magneticactivity that hasoccurred in somesunspot-cycles. A break in the homogeneityof the internationalmagnetic character-figures in recentyears is discovered. The individual 27-day recurrencesin terrestrial-magneticactivity during 1906-31, and their relationsto solar activity are discussedwith the help of a graphicalday-by- day record. They indicatethe existenceof persistentactive areas on the Sun'ssurface, calledM-regions, which, in manycases, cannot be coordinatedto suchsolar phenomena as are observableby directastrophysical methods. This holdsin particularfor the new solar indiceswhich are available for the years 1928-30,and which are found so closely correlatedto sunspot-numbers,that they fail to improvethe correlationbetween solar activityand terrestrial-magnetic activity. Observationsof terrestrial-magneticactivity yield thereforenot only informationabout geophysicalinfluences of suchsolar phe- nomenathat may be tracedin astrophysicalobservations, but supplementthese direct observations themselves.

CONTENTS 1--Introduction 2•Magnetic activity, as expressedby the internationalcharacter- figures and other measures 3•The u-measureof activity; monthlymeans 1872 to 1930 4--Annua! means of activity since 1834 5•Activity and energyof disturbance 6--The u•-measure 7--The annual variation of magneticactivity 8---The annualvariation of sunspot-numbers("-effect?") 9--Testsfor explanationsof the equinoctialmaxima of activity 2 J. BARTELS Iv(•. 37,N•,. 51

10---Testsusing harmonic dial and probable-errorcircle 11---Thequestion of annualrecurrences (influence of cometsand meteors) 12--Relations between annual means of solar activity and terrestrial- magneticactivity 13--Tests of homogeneityfor measuresof solar and terrestrial activity 14--The lag of the annualmeans of terrestrial-magneticactivity behind those of solar activity 15---Relationsbetween monthly means of solar activity and terrestrial- magneticactivity 16•General remarks on correlation in serieswith after-effect (monthly and annual means) 17.... The individual 27-day recurrences,!906-31, and their relation to 18...... The solar indices, 1928-30, compared with terrestrial-magnetic activity 19...... Summary ! ...... I•lrod•tction Among lhe geophysicalphenomena which have i>eeninvestigated with regardto solarinfluences of other kind than the regulartliurnal and seasonalvariations, terrestrial magnetism stands out as yielding the most consistentantl reliable re!ationshiI)S•. The derivation of a new long and lmmogeneousseries for terrestrial-magneticactivity, which will comprisethe firstpart of this paper,justifies the reconsiderationof someof thesi•relationships that httve often linen treated with lesssatis- factorymaterial. The statisticalasl)ects will l•e empl•asized,since the relationshipis of a sta!isli(':alnalure inasmuchas eachobserved solar phenomenonseems to t)roducea geoi)hysicaleffect truly with a certain probability,and since all geophysic. al t.he{•ries either gi'•>w rmt r•f statistical results,or mustbe subjectedt{• s!atisticaltests. The meth{•rlsdescribed in this papermay be usedto test r•therrelationshiI•s, such as those supposedin meteorologyor in wirelesstransmission-phenomena. It will appearthat cautionis necessaryin dealingwith short seriesof observations,taken over intervals of a few years only. This holds especiallyfor the recentyears 1928-30, in whichthe trendin the direct solar observationsdiffers const•i{'u(•uslyfrom the trend in terrestrial- magnetit:activity, in contrastl{• the fairly high ('r•rrelationsfound in the greatestpart of the seriessi•ce i872. 2...... Magnetic activity, as expressedby theinternational magnetic character- figures and other measures Terrestrial-magneticactivity at a given station, and in a certain interval,may be definedas an expressionfor the frequencyand intensity of magneticdisturbances in that interval. There are many ways in whichthis generaldefinition may be expressedas a numericalmeasure. Characterization,the simplest, is now widely used. !n this measure everyobservatory assigns, from the character of its photographicrecords, to each interval of 24 hours,between successiveGreenwich midnights, a character-figure,"0" for quiet,"1" for moderatelydisturbed, and "2" for greatlydisturbed days. The averagefor all collaboratingobserva- • Thegeophysical effectsof solar phenomena havebeen summarized bythe author inErgebnisse der exakten Naturw., 9, 37-78 (Berlin 1930). RELATIOiVS MAGNETIC ACTIVITY TO SOLAR PHENOMENA 3 tories (the number of which increased from 30 to about 45, since this measure was begun in 1906) is the international magnetic character- figure• C. The values of C representthe fluctuationsof activity from day to day very satisfactorilyand have been usedwith advantagein selectingthe international quiet and disturbed days. They also have been used in several investigationssuch as those of C. Chree and J. M. Stagg3 on the 27-day recurrencephenomenon, which is due to the rotation of the Sun, and in L. W. Pollak's periodogram-analysis4;they will be used for similar purposesin õõ 17 and 18. Becauseof the complexityof the disturbance-phenomenonand its different appearanceat equatorialand polar stations,Ad. Schmidt• advocatesthe practiceof leavingthe assignmentof the character-figures more or lessto the subjectivejudgment of the observerin chargeat the observatory(though, of course,certain model curves might be selected at eachstation as guidingexamples) and not to give strict prescriptions based on actual measurements. He agreesthat the characteristicsof magneticdisturbance, such as shift in the meanvalue, the magnitude, derivation, number of the single oscillations, etc., are so numerous, that the establishmentof an objectivevalue for the activity of each singleday couldbe gainedonly by very elaborateinvestigations. The time and laborspent in this work wouldbe quite out of proportionto the result, since this characterizationof a single day, or of another interval of time, would be, in any case,only statisticallysignificant, without furtheringessentially a physicalexplanation. On the other hand, just becauseof this limited importanceof the characterization, the simpleprocedure adopted for the internationalcharacter-figures servesits purpose,namely, the relative comparison of the magnetic activity on successivedays within, say, threemonths to a year. That on so many days practicallyall observatoriesreport character "0", and, during greatmagnetic storms, all report"2", provesnot onlythe well-known world-widenature of magneticactivity, but also the satisfactorinessof this method to derive C. There are, however,as Schmidtpoints out, certainlimitations in the useof the averageinternational character-figures C. As statedwhen comparingactivities on twodays close together, we mayuse the values of C with confidence.Thus, we may acceptas about equallydisturbed April I and 18, 1910,since on boththese days C= 1.3,and may regard April 12, 1910,with C= 1.0,as lessdisturbed than the othertwo days, but we shouldhesitate to say that the magneticactivities on April 12, !930, with C= !..3, or April 15, 1930,with C=I.0, wereequivalent to thosefor the daysin April 1910. For, althoughthe arrangedinter- nationalvalues of C are the same,there is no safeguardthat the standard of characterizationmay not haveshifted within the 20 yearsfrom 1910 to 1930. There are severalreasons for suchshifts: (a) Changeof ob- • Publishedby the Royal MeteorologicalInstitute of the Netherlands,and reprintedannually in this JOUKNAL. • Phil. Trans. R. Sot., A, •2•, 21-62 (1927). ' PragerGeophysikalische Studien, 3 [[2echoslovakischeStatist•k Bd. 64 (1930)1;reviewed in Terr. Mag.,36, 110(1931); also Naturw., 18, 343-349 (1930). The results of thisvaluable paper are outside thescope of thispaper and will thereforenot be discussedhere, though an applicationof themethods described in õõ 9 and 10 would seem promising. * ReportInternat. Met. Committee,Berlin Meeting 1910, p. 93 (LondonMet. Off., 1912). See Schmidt'sdiscussion of the character-figures,Met. Zs., 33. 481-492 (1916). 4 J. BART.ELS [v,',•..37, No. il server...... for instance,an observerwho is usedto polar recordswill judge tropical recordsmostly as quiet; (b) shift of standard for the same observer--for instance, an observer during a sunspot-minimum (as 1923) and therefore accustomedto quiet curves may assign the char- acter-figure2 to recordsof a type to which in a disturbedyear (as !91.7) during a sunspot-maximum,when he is, so to say, hardened by the frequencyof disturbances,he would assignthe figure 1; (c) systematic change in the method of characterization-•this has actually occurred in recent years, since several observatorieshave introduced numerical measures for estimating the character, as described in the De Bilt circulars for 1923 and 1924; and (d) establishmentor discontinuation of collaborating observatories. As will be seen later (õ 13), such shifts have occurred (though, for- tunate!y, lessfrequently than might have !)eenexpected). ,,1•,•objective meas•ireof activity is thereforeneeded .]'or establishinga homogerteous series.for all thetime sinceconsistent terrestrial-m(zg•zetic observations were begun. It will be sufficient,fr)r the time t)eing, to devisesuch a measure ()nly as averagesfor intervalsof monthsor years, since C is available for the relative characterization of days within a month.

3...... 2'he u-meas•tre of activity; monthlymeans 1872 to 1930 There has been chosenas such a measure the interdiurnal variability, U, that is, the averagedifference, regardless ()f sign, t•etweensuccessive daily meansof the magnetichorizontal intensity. Referring to former papers•.,,s in which U and its relationsto other measuresof activitity have been discussed,a short summary of its main propertieswill be sufficient. Each magneticdisturlmnce affects systematically the daily averageof the -vector•. This phen•)me•on,commonly called post- perturbation (Nachsti3rung),most I)ron(mncedin the h()rizontalcom- ponent II or in the north componentX, is one of the most regular phenomenain terrestrialmagnetism •ø. FigureI sh()ws,for an interval chosen at random, crmsecutiveaverages over 24-hour!y intervals, centeredat ()•, 6 h, 12•, and 18h, Greenwichmean time, of X or of H, for threewidely separated observat(Mes, namely, Seddin (near Potsdam, Germany), Huancayo (Peru), Watheroo (Western Australia). The three curvesare strikinglysimilar, showing the typical depressi(m()f ]I or of X duringa disturbanceand the gradualrecovery t.o normal wdue afterwards. Obviously,a measure,such as U, derived from these sys- tematicchanges of H or of X, at st)meobservatories, must representthe world-wide part of magneticactivities in a suitable way. Figure 1 hasalso a bearingon field-ot)servations.Judging from the curvefor Huancayo,measurements in tropicalSouth Americathrough- out 24 hourson the quiet day November5, 1928,would be expectedto give a daily meanvalue for horizontalintensity that is about 50T lower than the value, at the samestation, on the quiet day November9. Post- •Ad. Schmidt, Ergebnisseder erdmagnetischenBeobachtungen in Potsdamund Seddin im Jahre 1921, 6 if. (1924). *J. Bartels,Met. Zs., 40, 301-305(1923); 42, 147-152(1925); Archer des Erdmagnetismus,Heft 5 [Berlin, Abh. Met. Inst., 8, No. 2 (1925)]. • W. van Bemmelen, Met. Zs., 42, t43-147 (1925). • See stereogram 7, Terr. Mag., 36, 194 (1931). • J. A. Brown,Trans. R. Soe. Edinburgh,2z, Part 3 (1861). W. van Bemmelen,Met. Zs., !2, 321- 329 (1895). Ad. Schmidt, Zs. Geophysik, 1, 9-13 (1924-25). RELATIONS MAGNETIC A CTIVITI• TO SOLAR PHENOMENA 5 perturbationsshould, therefore, be taken into accountin the derivation of secular variation from field-observations. The practical computationof this activity-measureis doneas follows: From the ordinary daily meansof H or of X, whichevermay be published in the observatory'syear-book, consecutivedifferences from day to day are formed. We assignto eachday the differenceof its meanfrom that of the precedingday; in this way the greaterdifferences which occurin the

OCTOBER NOV[UB._'R,, ' I,q2• •4 m •8 •0 2z z4 z• Iz • •0 • 3 5 • 0 • •3 •5 •

1•405 I II ,,,SEDD,IN

18365 I / xl - • L t -

183•5

• • • I HUANCAYO• • k/

• •545

.... I• / V WATHERO0

24800 I/

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,

0 Fro. 1--The world-widenature of post-perturbationindicated by similarityof three curvesshowing the daily means magneti• north component at Seddin(Germany) and horizontalintensity at Huancayo(Peru) and at Watheroo(Western Australia),October 1,• to November19, 1928(Plotted means for consecu- tive 24-hourperiods, 6 hoursapart; those for 24 hourscentered at (;reenwichmidnight coincide with verticallines) transitionsfrom a quietto a disturbedday are ascribedto the latter. Monthlymeans of thesedifferences, taken regardless of sign, are formed to the nearesttenth of a gamma(1•/--0.00001 c. g.s.). Thusthe mean forJuly, for instance, is theaverage of 31differences, the first being the changeof ti fromJune 30 to Julyl•that is,the mean value for July 1 minusthe meanvalue for June30--and the last beingthe changefrom July 30 to July 31. A singulardifficulty arises in all numericalmeasures of magnetic activitydue to occasionalfailure to obtaincomplete photographic records;such losses occur mostly during great disturbances, when the recordinglight-spot either leaves the photographicpaper, or movesso 6 .L BA RTEL,S' [Vo•.. 37, No. •1 fast as to leave no trace. ()nly a few ot•servatories(amr)ng those whose records are used in this I)aI')er, l•otsdam, I3omb•ty, •rntl B•ttavia) have effective safeguardsto insure continuity of their records, l"')t•taf!er the stations for the International !:•olar Vear of 1932-33 will have been equippedwith special insensitivemagnetographs there is hope for a change in lhe phi!o.sophicalattitude of those stations which so fre- quently lose recordsin the most interestingintervals of the history of terrestrial magnetism. In •tny case, it is not r)ossil)lesimply t(• ignore these lossesin deducing an average measure of activity, since this would lead to a grossunderestimate. For caseswhen recordswere lost.at an obserwttory because(•f magnetic storms, no attempt was made in this discussionto interpolate and no monthly mean was deduced f(•r that observatoryfor that particular month; when, h(.•wever,the lossof trace occurredduring quiet times, int'erpol;•ttir•nwith the hell) (•f or:herstations wasoften possitfieand wasai•plied, t)r•vide(l the monthly mean difference could therel)y be ol)(aine(I within an accuracy of ()he t)r two per cent. The next ste!• is t,l•e c()ml)in;tt'i(>x•of the results fr()m a numl)er of observat.ories in ()rder io {)l•tain an ex!>re,qsir•n()f the magnetic activity for t.he Earth as a whole, that is, ;t universal value f()r eacl• month. To a fair degree()f approximaiit)x•,(.he pr.•st-t)eriurl):'ttioncan be regarded•ø as a uniform magneticfield, s•[y P, varying in time, l•ut. always parallel to the Earth's magnetic axis (f()r which we lake the direct"ionfrom the Earth's center 't:(•a point i• 78ø.5i•t)rt..h, 291 ø east {)f (.;teenwith). Sup- pose, for a certain ()[')serva((>ry,ihe interdiurnal w,triabilit'y, U, of t! or of X, has ]>eendetermined as descril>ed;t. he angle, in sI)ace, bet.ween the directionsof II or of X and of the magx•eti('axis is easily calculated, and may l•e called [•. If then the w'u'iltl)iliiy of If r)r (')f X were due only to changesin P, it woul(l register thesechanges ()nly ii• the ratit> cos •:1. Thereft)re, t.he interdiurn•fi wtriat)ilit.y r•f P !.)ase(1Ul)t>i• U determined from an ()bservat:ory'srecr)rds w()uld !•e (1) •t = U/cos ;• If, for the mrm•ent, we designate the magnetic equator •ts the great circle peri'>endiculart..o the m;tg;•eticaxis, we may, slightly idealizing, call •t the inlerdiur•talvari, bi!ily of thehorizonlet! componen. t at lhe mag•t.elic equalor. t;rrr•mt he value !; me;tsuringt t•e interdiurna! wtri;'fi•ili t.y (>f t.l•enorth coml)r•net'•t.¾ at. Sertdin(•e;tr l'(•ls(lam), f(•' instance,we (fl•!-.,tin,there- fore, with I5;: 55ø.0 1.745U U as well as tt will be expressedin units of !()-• (=().0()()1 (:. g. s. unit), in order to make u c)fthe order of magnitude 1, and thereforect)mparable with the character-figures. This relation (l a) was used in orelet to cal- culate, from the Seddin recordsalr>ne, a preliminary value of z•,,say, u,s. This is the only application made of fr)rmula (1); the values U obtained from the records of the other observatories were reduced to the standard of •tsin the followingway. There is now a total of 114 months for which we have data from both Seddin and \Vatheroo, from which U6, and Uw may be determined. The sumof t he 1! 4 monthly meansof the unrevised measureu = 1.745 U.s,expressed in the unit 10•, is 104.07, while the sum of the 114 means of U)). for the same months is 79.81 in the same unit. RELATIONS MAGNETIC ACTIVITY TO SOLAR PHENOMENA 7

If, therefore,each monthly mean for U,r is multipliedby the conversion- factor k= 104.07/79.81= 1.30, we get from the Watheroo data a second seriesof preliminarymonthly meansfor u, say uw. Accordingto the methodof calculation,uw is connectedwith us only insofaras, for all monthsin whichthe interdiurnalvariability is availablefor both stations Seddin and Watheroo,the mean averageof u• is equal to that of us. This standardizing does not bias then fluctuations of uw and u• from month to month, in which we are alone interested,serving only to guaranteethe homogeneityof the seriesof u. U dependsto someextent on the choiceof the 24-hourly interval (for instance,0 h to 0 h, or 1 • to 1h, etc.) for which the daily meansare formed, and therefore the conversion- factorsk must be determinedseparately for eachgroup of yearsin which the observatory adhered to the same method of forming daily means; a new discussionof the physical significance*of this influence of the beginning of the day will, however, be postponedfor the present, since the daily variation of activity will not be discussedin this paper. Recordsof the followingobservatories were used (L denotesthe limit of the 24-hourly intervals used for the daily means in Greenwich mean time)' Seddin (52ø.3 N, 13ø0 E), 1905-28; X with L=0•.0; values of U for 1905-23 are published* and those for 1924-28 were computed from the daily means of X as given in the year-books. t•otsdam (52ø.4 N, 13ø.1 E), 1891-1904; H with L=23h.6; values of U are published*. Greenwich(51ø.5 N, 0ø.0), !872-90; H, astronomicalreckoning is used through 1884with L = 12h.0, and civil reckoningwith L =0h.0 thereafter. The Greenwich serieswas used to provide an extra-tropical station for the years before the establishmentof the Potsdam Observatory. The unusual form in which the early Greenwich observations are published caused some inconvenience. Daily means for the disturbed days are not published,but had to be computed,which was done by selecting and averagingabout 96 values nearestto each quarter of an hour. Up to 1882, no correctionhad been applied for temperature; useof the uncor- rected values would be expected to yield somewhat higher values for U than the correctedvalues, but several tests showedthat the systematic differenceis negligible---.lessthan one per cent. The publisheddaily meansare expressedin parts of the horizontalintensity tt; the reduction of the Greenwichseries to the Bombay standard (which is derived, to a high degreeof accuracy,from the Seddinstandard) does not require knowledgeof t-[ for Greenwlch,but in orderto obtaina comparable expressionfor the reduction-factork the mean/-/was taken as 18040•,. On the whole, the resulting value of uo from the old Greenwichseries appearsto be lessreliable than the Bombayseries, probably because of shifts in the base-line, which are comparatively more dangerous in higher latitudes where the interdiurnalvariability itself is smaller. Therefore,Greenwich values were only given half weightin the formation of the final means for u. Bombay(18ø.6 N, 72ø.9E), 1872-1927;from 1872-1920a homo- geneousseries for U hasbeen published by N. A. F. Moosin hisfamous discussionsof the Bombay observationsn and has been used in the • Magneticobservations made at the Governm.entObservatory, Bombay (Colaba magnetic data), 18•,6to 1905, Part 2, p. 456 (1910); continuedin the later volumesfor I906-10, 1910-15,and 19!6-20. 8 .I. B,,1R7'I,2L,"; [voL,37, N•. 11 f•)r•nert)al':)ers 7. t'•I{)wever, sin('e it wasf•und that a fewmonthly means f•',' [,;•tt B(m!lmywere affected ]'•y It•ss {ff rec.{•rds, the relative weight thesetmrti•'uli•r monl..hly •nea•s was made(me-half. Through....1920, //-v;;tluesare give• with L ....':5}'.1; itfter 1921, frequent changes occurred, L lining19h.{/, !9}'.1, ;tn(l (lb.() [(>r 1921-24, 1925-26, a•d 1927,respect- ively. Balavht-.Bu[lenzor•(6ø.6 N, 106ø.8E), 1884-99,!902-26; II from 188•- 99 with L=17h.4; X from •91)2-19wi(h L=17h.4, and from 1920-26 with L = 17 •.(). Ih)nolulu(21•.3 N, 2()1ø.9E), 19{)2-3();//hourlyvalues from 1902-!• wiI h L = 11 h.(), and fromm1915-3() witl• L = 11 •'.0. PortoRico (18ø,2 N, 294•'.6E), 10()2-16;H fr{)m19{)2-14 with L a n d f rr)n• 1915 - 16 w i th L = 4 )'.(). Y'ucson(32•.2 N, 249ø.2tS), 1917-3();!• wit]• L=7h.(). (The daily m(,•tnvalue s ()f • f()rl.t,)•{)lult• amt Tttcsox•. thr()t•gh 193() w'ere kindly ft.•r•i.•l•e(li)ri()r l()l)•l)ti('•tti{')n, hy the !•nite•l SixtiesC()ast ant! (;e()detic 5t,rv:ey, Wasl•ingt()n, I). [<[Stlhert)t•(3(?.3 S, 115ø.2 !';•, 1919-30; !I with i• 'I'itl)Ie 1 tim ('•)nversi()n-fa('tr•rsk •tre given as computedfor the three ttsual se;ts()n;tlgr()t•l)S {)f m•)•ths, it• ()rder t() show ht)w the ratios ()f tim i•tertliur•;•l v;tri;tl)ililies wtry with seas{)n. For the actual ]•t•W(•ver,oIlly (]le valtles c)f k (lerive(1 from all months }•v(.t !)ee• use.•l. 'l'l•e stal)ility •)f t l•esev•tI•es ft)r k fr)r eachh()mr)geneous

'I'^I•1[,I,;1...... Vttlt•e,• qf • tmversion.f•trtork

) N•)v.] Mar. May X•'e•tr.•}I)ec. l Al)r, ]jt•ne All

1•15 I•128'1 7451 7,151 745 [1"884...... 1 ]Jan. ]Sel). [•t•. months I8')1 lo{).l'l 67 I,•)7 1872 18821,10 1,37 1,2t 11o02...... •9'2011.0• 1•.06 1•,20 • o9 ]1021.... 1926/I,06 11,00 11.16 1 10 11902 10i611.16 [1,12 I 1.16 I 15 •1•17 102811.17 [1,11 It.•s I(•()()i ()1()•(') .3 () (,)3 ()I 11020 103(1[..... l, 1 14 i•iI 1()2()'•(')"•')t) () 02 ]1903.101611.28 / I ! 22 11o17...... 1O2811,44 11 ! 44 [1929, !030/ ..... I, 1, 44 I025!2)26•)')7 1)'•8 05 •191o....,1928/1.26 I

. seriesis remarkable...... thus, for example,as long as Bombay did not alter the manner (>f deriving daily means,k was always 0.96, so that it was safe enoughto take this value alsofor the periodbefore 189!, where no direct comparisonwith Potsdamor Seddin is possible. Similar con- siderationslead t(.)extrapolated values of k for Batavia(1884-90) and for H()x•½)lulu,Tucson, and Watheroo (1929-30). The low value of k for (3reenwich,determined by comparisonwith Bombay,indicates spurious variati(•nsin Greenwichand is the main argumentfor assigningone-half weightto the (;reenwichvalues, as mentionedabove. The preliminary RELATIONS MAGNETIC ACTIVITY TO SOLAR PHENOMENA 9 10 J. BAR7VœLS [VOL. 37, No. 11 values of the magnetic activity u for all other observatories,however, were averagedwith equal weight when deriving the final value of u given for each month in Table 2. The values for Sitka, derived in a former paper7, are not included since the discussionhad shown that the inter- diurnal variability of [-I at polar stations is not so exclusivelygoverned by the world-wide post-perturbation vector as at the non-polar stations. (The discussionof magnetic activity in polar regionswill be postponed till the results of the International Polar Year 1932-33 will be available.) The number of available observatories was 2 from 1872 to 1883, 3 from 1884 to 1899, 2 from 1900 to !901 (this interval of the reorganization of the Batavia Observatory fell fortunately in a period of low magnetic activity,' so that the average of Potsdam and Bombay is sufficient), 4 for 1902, 5 from 1.903 to 1918, 6 from 1919 to 1926, 5 for 1927, 4: for 1928, and 3 for 1929 and 1930. The seriesis based on more than 80,000 singlechanges fr•m'• day to day.

4...... -Annual meansof activity since I834 More for illustration than for actual use, the series was extended backward to 1835. Before !872, no satisfactory data for the calculation of interdiurnal variabilities are available. It was, therefore, necessary for these years to use measuresderived from the diurnal variation. The chief drawback7 of such measures is the great change of the diurnal magnetic variation, and all its characteristics,with season;indeed, one of the chief advantagesof the u-measureis its comparative freedom from this seasonal influence which is not easily eliminated. Therefore, it was nt•t attempted to derive monthly means prior to !872, but only annual means,centered at the beginningand in the middle of each year; thus the mean entered f(•r 1836.0 is the average •f the twelve months July 1835 to June 1836, that entered for !836.5 is for Jan•iary to !)ecem- her 1836. Early recordsof magneti.c {)t•servatoriesbefore 1872 are available, but are lacking in uniformity r•f procedure;many stationsdid not observe on Sundays. Only two fairly homogeneousseries of observationst>efore 1872 were found. One of them, the "Einheitliche l)eklinati(•ns-Vari- at:ione•" of R. Wolf and A. Wolfer •v, which we call E, is fairly homo- geneouslmck to 1835.0;it is chiefly deducedfrom the daily range {•f the declination as observed in (h'eenwich at certain fixed hours. The other seriesavailable since 1847.5, for the "summed ranges" s, is clueto Moos•:•; s is derived from the mean diurnal variation of IJ' at Bombay for each single month, expressedin departures from the average, and is the sum of these departures,summed without regard to sign. The standardiza- tion was done as follows: For the annual means !872-1901, the values of u and s have the high tort'elation-coefficient+0.94 and stand in the linear relation (derived by least-squareadjustment) (2) u = o.0040 (s- 82) Preliminary annual values of u were calculated from this formula for 1847-72; comparisonwith E, for the same years, gave correlation-co- efficient +0.83, and the linear relation (3) u =0.164 (E-- 3.54) •, Astr. Mitt, Nr. 61, 11 if., contains a table of E for the years 1781 to 1880. •' Colaba magnetic data 1846 to 1905, Part 2, pp. 294 and 691 (1910). RELATIONS MAGNETIC ACTIVITY TO SOLAR PHENOMENA 11

This formula was used to calculate a secondpreliminary series for u from 1835 to 1872. The'two series, deduced from s and from E, were then combined to give final values for u; starting with 1847..5where the seriesfrom s begins, the values deducedfrom s were given double weight relative to those deduced from E. The values in Table 3 and Figure 2 may be regarded as homogeneous,but it is understood that they are least reliable for 1835 to 1847, better for 1847.5 to 1872.0, and satis-

T.•Bt•E 3--Revised annual means,u-measure, 1835-1930

Mean epoch

0.73 0.66 0.59 0 56 0 55 0 48 0 49 0 58 0 72 1835.5 0 87

1836.0 0.94 1836.5 0.90 1837.0 0.92 1837.5 1.14 1838.0 1.21 1838.5 !.15 I839.0 1 16 1839.5 1 23 1840.0 1 29 i840.5 1 !4

1841.0 1 10 1841 5 0 98 1842 0 0 79 1842 5 0 74 1843 0 0 66 1843 5 0 66 1844 0 071 1844 5 0 74 1845 0 0 73 1845 5 0 84

1846 0 111 1846 5 I !8 1847 0 ! 05 !847 5 10l 1848 0 o 89 1848 5 o 99 1849 0 1 14 1849 5 1 05 1850 o o 96 1850 5 1 oo

NoTv.--The value entered for 1835.0 is the mean for the 12 months July 1834 to June 1835, that for 1835.5 is the mean for the 12 months January to December 835, etc. 12 .I. BAR7't•:J.,.$' tyro,.,37, No. 1] factory for !872.5 to 193().5. In Figure 2, u is plotted together with the relative sunspot-numl•erR t• show the l 1-years' cycle in both. Figure 3 showsthe monthly means•f •t.and of R from 190()-30.

5..... Activity and ener,•yof disturbam,e If A•, Y, Z are the rectangularcomponents of the magneticforce, the magneticenergy of the field is equalto the volume-integralof (X•+ +Z•)/8•, taken throughoutthe field. If the field changesfrom (X t', Y0, Z0) to (Xo+•Xt•, Y.+•Yo, Z+•Zo), the magnetic in- creasesby the integral of (4) 2 (X0•X0+ I• Y,+Zo•Zo)/Sr+(•Xo•+ • Yo•+ S. Chapman•4 called the first term "joint-energyintegral" of the dis- turl>ance;since the vari•rtitms(•X, • Y, •Z) •tre usually small com- paredwith theaverage v•dues (X0, V,•,Z0), thesectrod term, called "self- ener,gyintegral," is generallynegligil)Ie in ctm•parisonwith the first. All measures()f magnelic activity wl•ich invt•lve Stluares(•f the ranges of the magnetic:elements, such as the measureI)rOt)osed by F. Bidlin,g- maier, ta!cei•!to account•only the small "self-energy"•m(1 •rt'e therefibre in no way real measuresof the energy(•f disturban(:e. The measureproi)osed l)y A. Crichtt•nMitchell, and a(lot•l'edl•y the international Geophysical[h•ion •a is .X'•Rx+VoRv+Z•Rz, where X•, Y0,Z0 are the meai• valuesof the (:(>mi)onents,marl Rx, Rv, /qz are tl•e al)soluteranges, that is, (liffereIl(:esl•etween lhe highestand lowest valueso[ X, Y, Z, respectively,in the courseof a (;teenwithday. The form of this measurereminds (•ne of t:hat of the joint-energy integral but this similarityis, •>fcourse, der:el)Vive, since the extremew. dues X, with range Rx, o(x:urgei•erally ;ti. (•II•el- times than the extreme values of Y and Z. Theref(•re, whatever tl•e merits ()f these measuresof activily may l)e, il•ey ceri.ainlyCa!1Ilol lm regardedas more significantlh•u• others in a.i•l•ysicaI asI)ect as yiel(ling values propt)rti(}nal to the ener•/yof the magneticdisturban. ce. As to our,u-measure, it was possible 7 to estimateroughly a lowerlimit. for the total purelymagnetic energy involved in the posl-perturt)ati()• undervery simI)lifiedassumptions TM. The •tveragemagnetic u imt)lies supplyof energyat the rate of ().6X10•7ttergs per secon(l.The r•tther highwdue of u= 2.5 correst)(mdstherefore t() twentymillion horse-I)r•wer. The direct solar radia/irmreceived 1)y the Earth on the wholed•tyligl;t hemisphereis about2XI0 "4erg's per se('o•cl,that, is, 1()7 timesthe rate of supplyof purelymagnetic energy, even in hig'hlydisturt)ed tarruths. "While the expenditurer)f energyduri•g a magnelicstorm is very great, it is quite insignificantcompared with the supplycontinually l•eing receivedby the Earth throughthe ordinarysolar radialion" (S. man14). ' 6.... l'he ½t•-measztre While the relations between sunspot-numbersR and magnetic activity u will be discussedlater (• 12), someremarks on the frequency • Mort. Not. R. Astr. Soc.,79, 70-83(19191. The theoreticalhonception of a magneticstorm has sincebeen greatly modified---see S. Chapman and V. C. A. Ferraro,Terr. Mag.,36, 77-97,171-186 (193I). report• Unionin Bull.Geod.No. 7, Geophys. 57 if. (1929).Internat., See Sectionalso G. Mag. vanDijk, f,;lectr. Meded.Terr., enBull. Verh., No. 8,205l/rtrecht, if.(193•)½No. also(!922),previous L. A. Bauer,Terr. Mag.,27, 31-34(1922); Ad. Schmidt, Ergebnisse der erdmagnetisehen Beobachtungen in Pots- dam und Seddinim Jahre1915, 29-36, Berlin, Ver6ff. Met. Inst., No. 293 (1917);C. R. Duvall,Terr. Mag., 36, 311-314 (1931). RELATIONS MAGNETIC ACTIVITY TO SOLAR PHENOMENA !3 14 .f. BART'I'JLS [w., 37, No. •!

•)f l•)w •rl hig]• m•l•tl•ly nm•tns •tre {lt•ite necessaryhere. The exist- e•!'e •',f •t [;tirlv {tefi•ite l•wer limit is a fea'ture common to broth R and u; tllis limit is (1 f•r R, an{l •tl•'•t•t ().4 for it. Values in the neighl•orhood •f these !•wer lirails ;tre (tulle frerluent; judging from t.he 708 monthly means•f the years 1872-193(},10 I>ercent of all monthshave valuesof R trotween(I •t•rl 3, an(! as many have values of u equal to or less than I).52. No strohcomlmraiive!y sharp limits exist for the high valuesof R and •t. But while the frerlue•cy-(tistrit)utionof the monthly means of R is, at the t;pperlimit, m•re or lesssimilar t(• the familiarexponential decrease •f frequency,which is tylfical for frertuency-curvesof the types called after (;auss •r l"•{fiss(m•*;, Ihe (listritmtion of •t is c•aracterized by the •'c•tsio•;tl (•cct•rre•ce •f is•l•tted extremely high values. This is, of crmrst:.,;• •;ttur;tl c(•nSe(ltleI•'e(•f the well-kn(•wn fact, that some few magnet..i•'st•,rms :t,'e •f titlite (mtstai•tli•gintensity as comparedwith the rest. The three l•ighestmo•tl•ly valt•es(ff 'tt,si'tme 19()1), are due the vi•l(?•l st,,rms (,f At•gust 1917 lit:= 2.37, ('(,{,•c{(l{ngwith the highest m•,nthly,nea,, (154} (,f R si,,('elg72], M•rch •92() (zt=2.$4),and May 1(}21 (u =:•:•:2.7(•). Every measure(,f magneticaclivity that is (leftnedas an averageof n•ges, (,r (•f s(it•ares(•f ravages,(•,' in a similarway, will showthe same feature ;is •t, th;tl is, s{•meextremely high mr•n!hlymeans. The monthly means•f the i•tcrnati(mal ('haract.er-figures,however, have a different freqt•ency41istrii•t•ti•n,l•ecause the highestdaily figures,C= 1.9 or 2.0, are attri}mled witchout ft•rlher discrimination l• all days with great stortns that exceed a cert;tin range. 'l'akc, for i•stance, March 192(I (with tt=..•2.54),•t•t(t ('(msi(lerthe i•flttence (•f the str•rm (m March 22 ;,•(..123 (,,• the averagei•l.er{liurnld wtri;tlfility 1; (•f the nr>rthc()mponcni X at •e(l(ii•. The •tver:tgeU f(•r lhe 27 (!iffcrencesfr(m• Fel•rt•ary29 •, M•t,'('l•21, •t•{l [r(m• M•trch 25 t(, Ma,'d• 31, is 7.8•; the averageU f(,r tl•(' wl•,!e •n(•nth,t,y incl•(li•g !he f(mr l•ig (liffercncesMarch •,•, M;•r{'l• 25, {• r;t.ise(! t(, •3.6•. !:(,' c(•mlmris(m,the averageinter- •tt•(•tl {.l•ar•t('ter.fi,guref(,r the 27 daysMarch 1 to 21 and March 26 t•, 31 is l' ...... (i.65, •t•(l [(,r lhe wh•le mr:,•th,only slightlyhigher, C=0.79. 'i'ttefew sl,,r•-d•tys theref(,re nearly (1ouble the valiseof U (andccrty t•) thrt•ttles()mewhitl the influenceof the exceptionally gre•tt{tisturt)imces. Since these great storms, as will t')e discussed later (t 17),r!• notf•llr•w s.•me •f thei)eriodicities exhibited l)y thelesser stt•rms,!he investigationof theseperiodicities would be hamperedby theuse r•f ;• measurethat emi)hasizes them. Themost general expression w(•uldcall for the intr{)(lut'tirmof a suitableweight-function • of the rl!fft,,.iC,I1•.t:... 6 from (11•Vto day, with p decreasingwhen • is increasing (at,/aa<()),and the'f(•rmltti(.)n()f m(mthly average such as "l'l•ecr)mptlting lal)()r [{)r sucha reduction,however, would be pro- • R, v•mM•ses, V•)rlesungen al!8 •tt•HI (;ebiete der angewandten Mathemaiik 1, Wahrscheinlich- koitqrechnung und ihrv Anwendung in der Statistik und lheoretischen Physik tLciI)zig 1931). R.ELATIONS M, 4GNœTIC ACTIVITY TO SOLAR PttENOMENA 15

t . 16 3'. BARTELS IX'or,.37, No. 1] hibitive. Therefore, the monthly meansof the modified measure,zt•, was simply defined as a function of the monthly mean of zt; the function was chosenso that the monthly means of u• had a frequency-distrit>ution similar to that of the sunspot-numbers R, at least for the high values. After some trials, the combination of linear and quadratic functions indicated in (5) was found suitable.

For u.=<0.6, u• = 100u-30 0.6:5 u '51.6, ut=30+100 (u-0.6)-30 (u-0.6):' u•= 100+40 (u-1.6.)-10 (u- 1.6)• 3.6, •= 140

u• as a functionof u is represented in Figure 4. The monthly and annual means of • are given in TaMes 4 and 5. It may be menti(med that the variable scale chosenfor u in the diagram / /i [ i •:^Lœ of the monthly means (Fig. 3) is e(tuivalent to the introduction I:•(;. 4•Thc ,•-measttre of magnetic act:ivity as derived from the monthly means of u of 'ltt, and has served to make that diagram clearer. The ad- vantage of •t• over u will also appear in õõ 10 and 16.

T^m.E 5--.Annual means, .u,-measure,1872-1930

Mean M ea n

19112 29}• 1922.() I 48 881 i ]• 191)119,)2 t) •2.5 I 36I1922..51921.5 •I 4257 iA55:'1873.5 882883 , i • 19,)3 2{ 3 13.()13.5I i923 5 36 1874.01874.5 884 : r •i 19,)4 •{ 3 1924 () 40 g/[19,)419115 192451925 () 42 i'9•95 I 19255 5l 19•36 1926 0 71 187650 3430 886 19•36 1926 5 75 19•)7 1927 0 66 187750 3136 887 i• •'• 3 i9•7[ :i } 16'51•7.5117.0/ 1927 5 64 18780 31 888 i]i • ]i 18.0/ 191938 38 1928 50 5461 18790 30 889 i ]• 1939• .• 19.0 / 19290 72 t878528888 i • 19:39[ • tS.Sl1929 5 67 1910 1930 • 62 188003089018801879 5 4630 889890 i• 1910< .• 20.519.5 I 1930 5 63 NoTg•The value entered for 1 t72.5 is the mean for the 12 months January to December !872, that for 1873.0 is the mean for the 12 months July 1872 to June 1873, etc. RELATIONS MAGNETIC ACTIVITY TO SOLAR PttENOMEN:i 17

7--The annual variation of magneticactivity The 59 years, 1872-1930, were classifiedinto three groups of high, medium, and low activity, which comprised 20, 19, and 20 years, re- spectively,and were selectedaccording to the annual meansof u• (u• >- 58, 57 >-u•>-43, and 42 >__u•). The averagemonthly meansof u• in each group and for all 59 years are shownin Table 6; in preparingFigure 5, the monthly means were slightly smoothed, according to the formula b'=(a+2b-½c)/4, where a, b, and c are three consecutivemonthly means in Table 6. The well-known maxima near the two (March and September)and the minima near the solstices(June and December) are clearly shown in all curves. The standard deviations, •, of the monthly means in each of the 4 lines of Table 6 (formed in the usual way using the departures of the monthly means from the respectiveannual means and computing the squareroot of the averagesquare of the departures)are' High, 6.6; medium, 5.8; low, 3.2; all, 4.7 On the other hand, if we form for eachsingle year the departures of the 12 monthly means from their respective mean of the year, we get as standard deviation v of these departures High, 19.5; medium, 17.9; low, 12.0; all, 16.7 If we consider• to be causedby two more or lessindependent influences, a regularannual variation (with standard deviationv•), and variations of other kinds (with standarddeviation v'), we may assumethe relation (6) •-•= • d- Since v and v• are known, we get for v' High, 18.3; medium, 16.9; low, 11.5; all, 16.0

TABLE 6•Average monthlymeans u•-measure of activity, 1872-1930 YearsofNumber Jan.IFeb.lMar.!Apr. IMay[JunelJuly[Aug. lsep. IOct. INov.[Dec. IMean High..... /-- 20...... 15-•t•-•77 7-•.9' 7--•. 66-•-'-'-• •-•7-•6-• 7-• 7-•.48-•.2 7--• •-• .[ 19 [40.8t47.4154.350.9153.1144.6143.8150.8159.4/55.2/60.5149.4[ 50.8 LowMedium' ..... I 20[28.1130.4135. 6131.8134.6130.0126.5128.8135.5136. 6134.0129.0131.7 All...... / s9 142.9149.5155.6151.1150.4/44.5145.1150.4156.4157.3]55.2/46.6150.4 The ratio v•/a' is High, 0.36; medium, 0.34; low, 0.28; all, 0.29 The regularannual variation is thereforenot only absolutely.more pro- nouncedin yearswith highactivity, but also relatively as comparedwittz the fluctuationsof thesingle monthly means. In a roughaverage, the magni- tude of the annual variations, expressedby v•, is about one-third that of the non-seasonalfluctuations, expressedby v'. If it were thought necessaryto prove the physicalreality of the annual variation even more definitely, one could use the standard deviations in the following way: Supposethat v does not contain a 18 J. BARTELS [Vot.. •, No. 11

systematic part such as the annual variation. According to the law of propagationof accidentalerrors, we should then expect that the average monthly means, taken for a number of years N, should have the standard deviation•/x/.2V--• the ratio •' (•/x/•) couldtherefore be regardedas an index for the reality of the annual variations. This ratio, easily calculated from the values given above [for instance, for high activity, 6.6-(19.5/x/•-)], is High, 1.5; medium, 1.4; low, 1.2; all, 2.2 That this ratio is always greater than unity, especiallyin the mean of all years, indicates the reality of the annual variation t7.

JAN FEB MAR APR MAY JUN JUL AUG.$œP OCT NOV DEC 80 I ! I I I I I I I I

JAN FEB MARAPR MAY JUN JUl,, AUGSEI a OCT NOV D•C ?0.6

,o- • ....• ••,,.•

20 ...... I .I i ,, i i i, i ! I t !

FIG,$ .! I •. I . I I I I I I I , t:t{;. 5--Annual variation of the magnetic activity u in years of high, medium, and low activity F•(•. 6•Annual variation of sunspot-numbersin years with many, average, and few sunspots

8•The annual variation of sunspot-nurnbers("Earth-effect?") It has been suggested•a that an "Earth-effect" can be traced in the sunspots,in the form of a small, but significant, annual variation of the sunspot-numbersR. This can be tested in the way just indicated for annual variation of the magnetic activity. From the 59 years 1872-1930, three groupsof years with many, average, and few sunspotswere selected according to annual means of R (R?=54, 54> R>=20, and 20>R). The average monthly means of R in each group are given in Table 7, and smoothed graphs accordingto b' = (aq- 2bq-c)/4 are given in Figure 6. In the order of many, average, few, all sunspots,the standard deviations, defined as above, are •' 3.18, 2.34, 1.08, and 1.55; and •' 16.4, !4.7, 6.1, and 13.1. The crucialratios, •' (•/x/•)), are' 0.9, 0.7, 0.8, and •* This test for reality of an average variation, which does not refer to sine-waves, was first applied by the author in Ann. Hydrogr., 51, 153-160 (1923), and amplified in •. J. R. Met. Soc., 51, 173-176 (1926). It resembles the periodogram-analysis in the form given by E. T. Whittaker and G. Robinson in The calculus of observations, 345 if., London (!924). •" L. A. Bauer, Terr. Mag., 26, 114 (1921); L. A. Bauer and C. R. Duvall, Terr. Mag., 30, 200-201 (1925); H. H. Clayton, World records, Smithsonian Misc. Collect., 79, page V (1927). The series for the of since 1919, published by C. G. Abbot, Smithsonian Misc. Collect., 85, No. 1 (1931), has been discussed,as to an annual period, by Fr. Baur, Met. Zs., 49, 15-18 (1932). The low corrdation-coefficients between monthly means of the solar constant and sunspot-numbers, as compared with the high correlations between sunspot-numbers and magentic activity (õõ 12, 15), did not encourage a discussion of relations between the solar constant and magnetic activity. RELATIONS M.4GNETIC ACTIVITY TO SOLAR PHENOMENA 19

0.9. Since these ratios are all smaller than unity, no physically real an- nual period is indicated, so that the curvesin Figure 6 are all accidental in nature. In selectinggroups of years with many or few sunspotsand calculating the annual variation for these groups separately, the possibility of a systematic spurious "curvature-effect"•9 must be considered. In the caseof many sunspots,fbr instance,this selectionis equivalentto the superposition of 12 monthly intervals cut out from the crests of the 11-year cycles. On the crests,however, the curves have a predominant convexcurvature (negativesecond derivative); this systematiccurvature is not eliminated or smoothed by the superposition. The order of magnitude of this effect can be judged as follows: Idealize the 11.-year cycle of R as consistingof perfect sine-waves,each with a total amplitude of 100. If 3 intervals, of 12 months each, are cut from every crest, the average annual variation would show a value of R for the month in the middle of the 12-monthlyinterval that would be 1.8 units higher than the average of the two months at the ends of the interval. Intervals cut from January 1 to December 31 would thereforeshow higher values of R for June and July than for Jan.uary and December; but this could never be interpreted as an "Earth-effect," for the simple reason that

T.•,BLE 7--Average monthly meanssunspot-numbers, 1872-I930

7 Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec. Mean SunspotsNumberYearsof Ijan ' •an--'•...... 2'0• i6-• 7-• 75-5.9 757;7-77.7 6-•.'.'•--•70.8 •verage.. 19 /34.637.6 38.0 33.6 35.2 38.7 42.2 38.1 •:2.0 37.7 32.7 36.3 37.2 7.8 11.4 8.4 8.3 8.8 8.6 7.0 8.7 8.9 8.7 7.6 8.4 AllFew...... ?I 5920 J37.27.2 41.0 39.O 38.0 38.2 40.2 41.8 39.6 39.4 38.1 37.0 36.6 38.8 intervalscut from July 1 to June30 wouldshow the spuriouseffect in the otherdirection, that is, comparativelyhigh values in Januaryand December,low valuesin Juneand July. Yearsof minimumsunspots would show concave curves for the annual variation. The natureof thiscurvature-effect implies that it is not eliminatedby the useof longseries of observations,but, on the contrary,is evenmore pronouncedin relationto accidentalvariations. From the 100 years, 1826 to 1925, two setsof years (monthsJanuary to December)were selected,namely, 5 aroundeach of the sunspot-maxima,and 5'around the sunspot-minima.Half-yearly meanswere derived, for northern summer(April to September)and northernwinter (January to March, Octoberto December).The resultingaverage sunspot-numbers during 1826 to 1925 are as shown- Summer Northern Northern minus Years summer win ter win ter 45 maximum 77 73 45 minimum 17 19 - 2 All 100, 1826-1925 46 46 0 •J. Barrels,Wien-Harms, Handbuch der Experimentalphysik,25, I, I66 (1928);see also Beitr. Physik. frei. Atmos., !1, 51-60 (1923). 20 .f. BARTEL3 Wo,-. 37, No. l]

Since the "summer" half-year forms the central part of the selectedinter- vals of years, the last column shows the marked curvature-effect. Fortunately, however, the curvature-effectis in the interval 1872 to 1930, with which we deal, so small as compared with the accidental variations, both for R and u•, that the values in Tables 6 and 7 are not noticeably affected by it. It was even not found necessaryto apply in these tables any correctionsfor non-cyclic changes,that is, for pro- gressivequasi-linear changes throughout the year. Harmonic analysisof the annual variation of R will be discussedlate:r (õ 10) and will confirm the results already obtained.

9--Tests for explanationsof the equinoctialmaxima of activity The Sun passesthe celestialequator on about March 21 and Sep- tember 23, the time of the equinoxes. The Sun's axis of rotation is inclined 7ø.2 to the ecliptic; the greatest inclination of the north pole of the Sun, that is, the time when we seemost of the northern hemisphere of the Sun, occurs about September 7, while about March 5 the Sun shows us most of its southern hemisphere. Otherwise expressed,the radius from the Sun's center towards the Earth attains 7ø.2 northern heliographiclatitude aboutSeptember 7, and 7ø.2southern hellographic latitude about March 5. For convenience,let us refer by the symbols 'Eq and Ax to the times of the astrononomicalphenomena, just men- tioned; they follow each other within only 16 days. Argumentshave beenbrought forward which assign the causefor the equinoctialmaxima of activity to eitherof Ax or Eq, that is, to particular values of either the Earth's heliocentric coordinates or the Sun's geo- centtic coordinates. The first explanation is'2ø based on the fact that sunspotsoccur most frequently in he!iographiclatitudes 10ø-15 ø, while the equatorialbelt of the Sun showscomparatively few spots. If, then, the corpuscularsolar streamsstart from the samebelts in which the sunspotsoccur, and leave the Sun's surfacein a radial direction,they shouldbe morelikely to sweepacross the Earth in September,if they are emitted from the northern spotted belts, and in March, if they are emitted from the southern belt. S. Chapman" simplifies lhe question in the followingway' Supposethe streamsto be shapedlike spherical cones,with the apex in the Sun and with angularbreadths a (angle formedby an axial cross-section);the axesof the conesmay be inclined 12ø.5north and south of the Sun's equator. Near the equinoxes,then, of all streamswith !1ø•a540 ø, only those from one zone will sweep across the Earth, while near the solsticesstreams of both zones•will traversethe Earth, but only if a } 25ø. The Ax-exp!anationof the two maxima in the annual variation of activity would therefore imply that narrow streams with a < ll ø are more than twice as numerous as those with a > 25 ø. The-last statement can at once be tested in the following way' A streamof angula1'breadth e doesnot take more than aX(27/360)= 0.075a days for traversingthe Earth. The values11 ø and 25ø for a would thereforecorrespond to 0.8 and !.9 days, respectively,and the Ax-explanationwould imply that disturbedperiods of duration0.8 day •0A. L. Cortie,Mon. Not. R. Astr. $oc.,73, 52-60(1913) and 76, 15-18(1916); L. Rod6s,Terr. Mag., 127-131 (1927). •xS: Chapman, Mon. Not. R. Astr. Sot., 89, 465 (1929). RELATIO•.¾S gfAG.¾ETIC ACTIVITY TO SOLAR PHENOMENA 21 were more than twice as numerous as those ot• two-days' duration. This inference is not at all supported by the character-figures,which only seldom show isolated disturbed days, but, as a rule, two or more disturbed days in succession(see Fig. 18). However, two hypotheses offer themselvesfor saving the Ax-explanation- First, it is quite likely that the streams are not spherically symmetrical, but have elliptical form, with the longer diameter parallel to the Sun's equator; or, what is equivalent, the disturbedregions on the Sun are more extendedalong the Sun's circles of latitude than along the Sun's meridians (This would hold for sunspots,which "invariably stream out in longitude," but not for faculae,which "frequently appear in streaksroughly at right-angles to the direction of the Sun's rotation", see õ 18); and, second, the magnetic disturbance may last for some time after the corpuscular stream has traversed the Earth. More fuhdamental are three other, independent,tests of the Ax- explanation. The two hemispheresof the Sun vary, to a certain degree, inde- pendently in the l 1-year cycle. For instance,for years in succession the sQuthernhemisphere may show more sunspotsthan the northern hemisphere(thus in the years 1883-89, 1895-1900, 1907-12). In such years the Ax-explanation would lead us to expect higher magnetic activity in February, March, and April than in August, September, and October. An attempt to apply this test meetsthe formal difficulty that the average sunspot-areas,separated for northern and southern hemispheres,are publishedonly for intervals correspondingto solar rotations,and not to months, for which ,t• is available. Fortunately, E. W. Maunder22 has published a suitabletable for the meandaily areas of faculae,corrected for foreshorteningand expressedin millionthsof the Sun's visible disc, for each month from 1886-1915. This table was used in the followingway- First only the monthsFebruary, March, April, and August, September,October were considered,in which the Sun's axisis mostinclined towards the Earth and of thesemonths 'only those in which the total area was more than 300, and in which the area of faculaeon the northern(or southern)hemisphere exceeded the area on the otherhemisphere at leastin the ratio 3:2. Then two groupswere formedfor' (1) the "favorable"group for whichall thosesingle months February, March, and April were selectedin which the southernhemis- pherehad more faculae than the northern,and all thosesingle months August,September, and October in whichthe northern hemisphere had morefaculae than the southern; (2)the "unfavorable"group containing thosein whichthe otherhemisphere was preferred. Means of the areas and of the magneticactivity u• were formed as shownin Table 8. The totalarea of faculaeis practicallythe same for bothgroups, and sois the magnetic activity ux, in spiteof the fact that the solar hemisphere turnedtowards the Earthcontains for the "favorable"grouping more than doubleand for the "unfavorable"grouping only half as many facu!aeas the otherhemisphere. This is certainlya seriousargument againstthe Ax-explanation.For, althoughthe active areas (the M-regions,see õ !8) on the Sun'ssurface, which are responsiblefor terrestrial-magneticactivity, cannot always be identified with sunspots or faculae,it is fairlysafe to assumethat the M-regionswill, on the • Mon. Not. R. Astr. So½.,80, 724-738(1920). 22 J. BARTELS [VOL.3?, NO. 1]

8--Comparisonof areas of facu!ae in millionthsof Sun's disc and of magnetic activity, u•, for favorableand unfavorablegroupings

Item "Favorable" "Uhfavorab!e" grouping grouping

Number of months in group ...... 29 33

Mean areas of faculae on notbern or southern hemisphere' On solar hemisphereturned towards the Earth ..... 835 380 On solar hemisphereturned away from the Earth... 345 778 On whole disc ...... 1180 1158

Mean magnetic activity u• ...... 50.2 52.3

average,also be more frequent on that hemispherewhere' faculae are abundant.

10.-Tests using harmonicdial and probable-errorcircle The secondtest consistsin the following determination of the exact time of occurrence of the two maxima in the annual variation. The 12 monthly meansof ut for eachsingle year, 1872-1930,were submittedto harmonic analysis='•, in the usual form

(7) a• cos t-kb• sin t+% cos 2t-½b.,.sin 2t=c•. sin (t-½•)q-c.2 sin (2t+•) where t is the time, increasingfrom the beginningof January to the end of December from 0ø to 360 ø. The harmonic coefficientswere corrected for non-cyclicchange and for the effectof smoothing(because of the use of monthly means). The harmonicanalysis of the mean valuesfor the threegroups of years(as given in Table 6) is givenin Table 9. (The monthlymeans were regarded as representing equidistant values centered at the middle of each monthly interval used for calculating •t, so that t=15 ø for the day January 16.0, etc. Some calculationsshowed that the dates given for the maxima would not be affectedby more than about one day, if the different lengthsof the monthswere taken into account.) To estimate the reliability of the harmonicanalyses, the harmonic constantsfor eachsingle year were subjectedto the period-testas intro- ducedby the author=4. It will be explainedfor the caseof the second harmonic(6-month!y period). The coefficientsa= and b=for each year are used as rectangularcoordinates (a•. positive upward, b•. positive towardsthe right) in a "harmonicdial" (Fig. 7), so that eachyear is representedby a point; the distanceof this point from the origin is equal to the amplitudeco. on a certainscale, and the directionof the vector drawnfrom the originto the point givesthe phase,indicated on the dial by the times (two in a year) when the maximaoccur. The 59 pointsare well scattered,but the maximashow a definite preferencefor the two sectorsof equinoctialmonths, March and April or • For the method used see J. Barrels, Beitr. Geophysik, 28, 1-10 (1930). • Berlin,Ver6ff. Met. Inst., Nr. 346 (1927);Zs. Geoph¾sik,3, 389-397(19271. An expositionof these methods, in their relation to others, is in l•reparation. RELATIONS MAGNETIC ACTIVITY TO SOLAR t>ttENOMF•NA 23

TABLE 9--Results harmonic analysis magneticactivity u•, 1872-1930 Yearsof 12-monthl.ypar..•t 6-monthl•y par.•t.IMaximum Maxima activity Ampli. I Phase Ampli. I Phase!12-monthly 6-monthly

Medium ...... 3.24 • 166.8 6.85 • 245.4 I Oct. 14 Apr. 13, Oct. 12 •o•...... 0.s8i x33.0•.30 i 24s.•i No•, X7 Apr.13, Oct. 12 All...... 2.15[ 160.96.34 I 261.0I Oct. 20 Apr.5, Oct. 4

Septemberand October. The centerof gravity of all points corresponds to the harmonic coefficientsof the 6-monthly wave as computed from all years. The distances&, du, .... d•0of all single points from this center

SCALEOF AkiPLII'UDES

UNITSOFU t FzG.7--Harmonic dial for the 6-monthly waves in themagnetic activity u• foreach of 59 years, !872 to 1930 of gravity were determined. The standarddeviation M2 for the second harmonicof eachsingle year fromthat for the meanfor all yearsis determined by (8) Thestandard deviation m2 for themean second harmonic of 59years is then 24 J. BARTELS [VOL. 37, No. 1]

(9) rn2= and the probableerror for the mean is (10) p•.=0.833 (Equation (10) gives,if the distribution is normal, p•.in the usual meaning that as many deviations are greater as are smaller than p•.. If the normality of the distribution is not tested, it is understoodthat p=means nothing but a constant multiple of the standard deviation m•..) Standard deviations were also computedfor each of the three groups,by comparison of the harmonicconstants for the singleyears with thosefor the respective group-means. The same calculationswere made for the first harmonic, that is, the 12-monthly sine-wave, as given by the coefficientsa•, b•. The advantage of the method used here, as compared with the periodogram-methodin its usual form, is the comparisonof the average amplitude of a period with a standard deviation which is gained by analysesrestricted to periodsof the samelen,gths. It will be readily recog-

Fie.. 8--Harmonic dial for the 12-monthly waves of magnetic activity ut in the average for groupsof years with high (//), medium (M), and low (L) activity, and for all years 1872-1930•circles have radii equal to the respectiveprobable errors p, Fro. 9•Harmonic dial for the 6-monthly waves in magneticactivity u,, drawn for the groups of Figure 8•probable-error circles with po as radii

nized that the method of the probable-errorcircle is nothing t0ut a devel- opment of the method used in õ 7, as specializedfor pure sine-waves. Table 10 contains the standard deviations and probable errors. The last two columnsgive the values of the ratios c•/p• and c=/p.o,that is, of the averageamplitudes (as given in Table 9) to their respectiveprobable errors. Harmonic dials for the average 12-month!y and 6-monthly waves and their probableerrors have beendrawn in Figures8 and 9. The meaning of the probable-errorcircle may be explained for the mean 6-monthly wave for all years: The center of the circle is the average 6-monthly wave for the 59 years 1872-1930. Suppose the series of observations could be extended as to include, say, 2V intervals, of 59 years each and supposefurther all theseintervals would show the samegeneral statistical aspect for • as the interval 1872-1930. The average6-monthly waves, as computed from each of these • intervals, would not be the same as thai computedfrom the interval 1872-1930;the points marking each wave, would thereforenot coincidewith the point marked "all" in Figure 9, but would be scattered on the harmonic dial. Though we do not know' RELATIONS MAGNETIC ACTIVITY TO SOLAR PtfENOMENA 25 where every single point for future intervals of 59 years will be located on the dial, we know the standard deviation m•. that governs approxi- mately the distribution of the points. The fundamental theorems of the theory of probabilityø-5 assure that we are justified in assuming that the distributionwill be normal, sinceevery point representsa mean of 59 single years. That would imply that approximately N/2 points

TABLE 10--Standarddeviations for singleyears and probableerrors for meansof groups, first and secondharmonics in annualvariation of magneticactivity u•, and ratios averageamplitudes to probableerrors, 1872-1930 (The units for M•, Mo, p•, poare those of u•, seeTables 5 and 6)

Stan. Prob. Stan. Prob. Years of activity dev'n error dev'n error Ratio Ratio M• P• c•/p• c.o/p.o

High ...... 11.9 2.22 14.2 2.64 1.2 3.2 Medium ...... 9.9 1.90 11.4 2.18 1.7 3.1 Lmv ...... 5.5 1.03 7.9 1.48 0.6 2.9

All ...... 9.6 1.04 1I .9 1.29 2.1 4.9 would be located inside, and 3[/2 points outside the probable-error circlearound the final average. The relative frequency(or probability) that a point liesoutside a circlewith radiusr/m,. = (r//0.833) p2 is simply e-•; for example,for the probable-errorcircle, ,/=0.833 and e-ø.Saa = 1/2. A circle, with its center in the point marked "all" in Figure 9, and with its peripherypassing through the origin,would have (accordingto Table 10) a value ,/=0.833X4.9, and e-• =e -ø-Saa/•.92= (1,/2) st or about 10-7 . It would be thereforeextremely improbable (with the odds ! :10,000,000)that a pointwould fall outsidethis circle. The correspond- ing calculationfor the first harmonic,with r/=0.833X2.1, givesonly = = /20. This considerationshows the superiorphysical significance of the 6-monthlywave as corn.pared with the12-monthly wave. The 12-monthly wave that enhancesthe values of u• in Septemberand October over thosein March, fairly conspicuousin Figure 5, seemstherefore to be a more or lessaccidental feature for the interval 1872-1930,which is not very likely to return in the future. This is satisfactoryinsofar as the reduction-factorsk (Table 1) showedsome seasonal variation, which might havenbut fortunatelyhas not---introduceda slight spurious annual variation in u and u•. The Earth'ssurface receives at thetime of theperihelion (beginning of January) about 6 per cent more radiation from the Sun than at the timeof theaphelion (beginning of July). A similar12-monthly period in magneticactivity might have been expected,but cannot be traced in the observations. The 6-monthlywave is shown to bethe significantphysical character- isticof the annualvariation. It hasa fairly constantphase, but its amplitudein disturbedyears is morethan •wice as greatas in quiet '• R. vonMiss, Wahrscheinlichkeitsre[•hnung, õ8 (193I). 4 26 J. BARTELS [vm.. 37, No. 11 years. If the origin of the time-variablet is chosenat the beginningof a sunspot-maximumyear, and if it is assumedthat the amplitudeof the 6-monthly wave changesregularly with the l l-year cycle, this wave can be roughly expressedas (11) [6.5+2.6 cos (t/11)] sin (2t+261 ø) In the well-known manner, similar to that in tidal theory when the luni- solar terms K• and O, are derived from the lunar tidal potential2i•, we can transform our expressionso that it is a sum of three terms with constant amplitudes as in (12) 6.5 sin (2t+261ø)+1.3 sin [2t+(t/11)+261ø]+l.3 sin [2t- (t/11) +261 ø] The frequencies(per year) of the threeterms are 2, 23/11, and 21/11, with periods 6.00, 5.74, and 6.29 months, respectively. Ordinary periodogram-analysiswould therefore yield, besidesthe main 6-monthly wave, two wavesof about 1/5 of its amplitude,and periodsof 5.74 and 6.29 months,respectively; but this result differsonly in form, not in physicalcontent, from the statementof a 6-monthlywave of constant phasebut variable amplitude,as expressedin Table 9. The harmonicdial (Fig. 9) makesit evident that the maxima of the 6-monthlywave in activity occurnearer to the equinoxesthan to those times when the Sun's axis is most inclined towards the Earth. This agreement.in favor of the Eq-explanationof the annual variation, as againstthe Ax-explanation,can be put in a quantitativeform' We con- siderthe average6-monthly wave for all years. Table 10 givesthe param- eters of the hypotheticalbell-shaped frequency-distribution of points representingthe averagesecond harmonic in a set of intervalsof 59 years each. From thesewe can calculatethe probabilitiesthat the maxima occur in one of three intervals of ten days each, centered on April 5 (thetime of maximumfound from the years 1872-1930), March 21 (equinoxes),and March 5 (Sun'saxis most inclined),and the corre- spondingten-day intervals half a year later. Theseprobabilities cal- culated by integrationover 10-day sectorsin the harmonicdial, are i:).686,0.096, and 0.000022,respectively. in other words,even though the maximaof the 6-monthlywave in activity u•, as computedfrom the interval 1872-1930,occur about 14 days after the equinoxes,the chances arefairly high (0.096:0.686 = 1:7)that thisis merelyaccidental, due to the shortnessof the series,so that the maxima, as computedfrom a much longerseries of years,might as well occur at the equinoxes.On the other hand, the chancesin favor of the earlier dates A x are very small (0.000022:0.686- 1:31,000). It maybe mentioned that thissharp distinction between the twodates was madeonly possibleby the introductionof the measureu•. For the averageof all years1872-1930, the originalu-measures gives, by a curious coincidence,the sametime for the maxima of the 6-monthly wave as u•, but the scatteringof the valuesfor the individualyears is muchhigher for u than for u•, sothat the ratio c•/p• for the averageof all yearsis only 2.9 for u, insteadof 4.9 as for u• (Table 10). Therefore,the original valuesu wouldhave permitted no sharpestimate of the relativeproba- bilities;even the chancesin favor of the earlierAx-dates would be rather • Handbuchder Geophysik,herausgegeben yon B. Gutenberg,1,331 (1932). RELATIONS MAGNETIC ACTIVITY TO SOLAR PHENOMENA 27

high and indecisive, namely, 1:36, instead of the decisive low value 1:31,000 which was obtained by using The series 1906-30 of the international character-figuresC was also analyzed for its annual variation. The averagemonthly means,shown in Table 11, give the following harmonic coefficients,corresponding to equation (7) 0.63!-30.011 sin (t+49ø)+0.065 sin (2t+294ø.6) Analysis of the single years gives the standard deviations M•=0.062, M2=0.061, the probable errors (of the 25-yearly average harmonic amplitudes)p•=0.0104, p•.=0.0102, and the chara•:teristicratios c•/p• = 1.1, c2/p,.=6.4. Again the 12-monthlywave is not significant,while the 6-monthly wave is very pronounced,with maxima on March 16 and September18. The relativechances for the Eg-datesand Ax-dates TABLE11--Average monthly means to showannual variation international magnetic character-figures,1906-30

Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec. Mean 0.6120.6820.7120.6210.6390.5760.5780.619 0.696 0.696 0.576 0.568 0.631

.

.....

TAW.E12•Results harmonic a•alysis annual variation relative sunspot-numbers 1872. 1930, same groups as Table 7

First harmonic Second harmonic N u tuber sunspots Ampli. Phasel dev'n I error I Ratio dev'n[ error Ratio [Stan. [Prob. I Ampli.IPhase Stan.[ Prob. ½..,/p,• ......

o

Many...... 2.16 1.2 Average .... 2.37 308237 [11.3[] 14.4 12.65/ 0.8 2.36 42810.6 I 1.98 1.4 Few ...... 0.42 342[ 4.2 }0.78[2.17 [ 0.51.1 0.a22.28 /[ 2s6 8.3[3.9[0.721.59 1.1 All ...... 1.27 1.5 are againin favorof the former,though the oddsare only 7:l---which is not unexpectedsince the availableseries of observationsfor C is less than half aslong as that for u•. The third test of the Ax-explanation will be givenin õ 14 and is likewise unfavorable to it. Somephysical considerations regarding the Eq-explanationhave beengiven previously '-'•,in connectionwith the influence ofthe beginning of the day on u, andwill not be repeatedhere. For completeness,the annualvariation of the relativesunspot- numbersR was testedby harmonicanalysis in the samemanner as describedfor the magneticactivity. The resultscollected in Table 12 confirmthe conclusion that neither a 12-monthlynor a 6-monthlywave of physicalsignificance can be detectedin the series1872-1930. The accidentalvariations veil also the curvature-effect (see {} 8), whichwould be expectedto produceopposite phases (•=270 ø and90 ø) of the 12- monthlywavein the years with manyand few sunspots; the curvature- effectis also not noticeable in the magnetic activity (Table 9). 28 J. BARTELS [vm.. 37, No. II

11--The questionof annual recurrences(influence of cometsand meteors) Simultaneous appearancesof meteors and aurorae have been noticed in some casesmore than 100 years ago, but no particular weight has been attached to them sincethe coincidencesseemed to be explicable by pure chance. H. Fritz •'s in 1881 summarized his opinion as follows, after mentioning observationsof Wrangel in !821 to 1823:

"Likewise, no particular weight can be attributed to the simultaneous appearanceof northern with meteor-showers,as they occurredin the years 1833 and 1838 from November 12 to 13, since the strongeraurora (1838) coincided with the weaker meteor-shower, since, furthermore, many impor- tant meteor-showersoccurred without intense , and since, finally, the time about November 12 is as little marked for numerouq aurorae as the time about August l 1 or as other days of the year which are conspicuousfor meteors or meteorites."

While most magneticiansadopted the same opinion, H. B. Maris '2ø has recently revived the •tuestionby assertingthe discoveryof three groups of annual recurrencesof magnetic storms. While the first two groupsfall before 1875,the third groupis comparativelyrecent and can easily be tested. It is saidto have started May 16, !913, and to be almost unbroken sincethen. The observational data seem, in the present writer's opinion, to be at variancewith the existenceof this group, becausethe days May 15, 16, and 17, 1913, have very low international magnetic character-figures,namely, 0.5, 0.1, and 0.2, respectively,and therefore. can hardly be claimed as starting a seriesof annually recurringmagnetic storms. In fact, the extension of Maunder's list of magnetic storms which H. B. Maris usesin his paper•, and also in a former joint paper with E. O. Hulburt :•ø,appears not quite satisfactory. For instance,the total number of magnetic storms, for the 78 years from 1848 to 1926, is given as 1,550. Somedates of "storms"given aø for yearsafter 1915 have, however, internationalcharacter-figures C as low as 0.5 and 0.6 and should hardly be rated as disturbed; judging from L. W. Pollak's frequency- list for C, there have been 1,728 days with C.•I.0 in the interval 1906 to 1925 alone, that is, more than the whole list of storms for the 78 years contains. This indicates a shift in the rules for selecting storms, as compared with those used by Maunder, and also throws some doubt on the basis of the statistical considerations :• with which the authors attempt to substantiate their interesting hypothesisof a correlation between magnetic and comet activity. An independentsearch for annual recurrenceswas made with the help of the international character-figuresC. In the years 1906 to 1930 values C_>-1.6have been assignedto a total of 392 days--on the average, about 16 per year. The monthly frequency-distributionis given in Table 13. In order to show the exact occurrence of each of these days, a chart (Fig. 10) was drawn to scaleshowing for eachday of each of the 25 years (in leap years, the dates before March 1 are shifted one day toward the left) by distinctivesymbols those days with C=1.6, 1.7,

• See* and Wien-Harms, Handbuch der Experimentalphysik, 25, I, 663-664 (1928). • Das Polarlicht, p. 289 (1881). '• Phys. Rev., 37, 1680-1681 (1931); 39, 504-514 (1932). a0.Phys. Rev., 33, 1046-1060 (1929). a• 1. c. p. 1057. RELATIONS MAGNETIC ACTIVITY TO SOLAR PI-IENOMENA 29

T•,m,E 13--Nu.mber of days with international character-figuresC betweenvalues 1.6 and 2.0, 1906-30 CI Jan. Feb. Apr•.May Nov.[Dec.Total 2 2 2 I 2 I 6 I 6 I 8 8 5 -i- 61![41519 2 3 61 1.8 .-5 3 8 1 I 4 ] 4 ] 14 [ 14 5 0 72 1.7 7 8 6 I 3 110 [ 13 I 9 7 8 95 6 10 116 Total ;- 35 29 / '24 392

1.8, 1.9, or 2.0, respectively. The total area coveredby the symbolsis shown for each year at the right, and for each month at the bottom. The right-handcolumn exhibits the ll-year cycles,and annual variation appearsat the bottom. Apart from a generalbut not exclusiveprefer- ence for the times about the equinoxes,no vertical string of symbols can be discernedthat would indicate a particular date preferredby magneticstorms year after year. This holdsin particularfor the date May 16, alsoMay 11 and November12 or adjacentdays.

YEARLY TOTAL ß i

MONTHLY' -.----TOTAL THEDAILY INTERNATIONAL CHARACTER-NUMBERS AREINDICATED ASFOLLOWS' CHARACTER-NUI•BER1.6 1.7 1.8 1.9 2.0 SYMBOL ' , I • [ F•(;. !0•Days of greatmagnetic disturbance, 1906-30 30 J. BARTELS Wo•.. 37, No. l]

12--.Relalionsbetween a•nztal meansof solar activity ct•zdterrestrial- magneticactivity In the next paragraphs the relations between two variables x and y will be expressedin the following usual way3ø-:The given values may be x• and yn; n=l, 2, ...... N. The standard deviations of the x• and y• are called rr• and vv, respectively. A linear relation between x and y is establishedaccording to the methods of least squares; that is, coefficientsa and b are determined, so that the computed values (13) yJ=ax,•+b differ from the observedvalues 3'• as little as possible,or, more accurately, so that the sum of the squaresof the residuals (!4) Zxy•=y,,-y,• • is a minimum. If xt,, Yoare the average values of x•, yn, the correlation- coefficient r is defined as

(15) Z(x,•-Xo)(y,•-yo)/No'•o' v and the coefficientsin (13) are (16) a=rtrv/cr•, b=yo-axo For normalized variables, which are expressedin such units that vv= 1, a is simply equal to r. In the general case,the standard deviation of y•' is rcrv,and that of the residuals /xy• is vx/1-r '2. in other words, y can be conceivedas the sum of two parts, of which one, y', is a strictly linear function of x, and the other, /Xy, is not correlated with x, or, in the suitable notation used by H. B. Heywoodas, /Xy is orthogonal to x. The relative magnitude of both parts y' and /xy can be estimated from the ratio r:w/1-r u of their respective standard deviations; this ratio remains the same when x and y change their role in (13). The reliability of r increasesof coursewith the number N of available obser- vations,and is expressedby its standarderror, (1 - r?)/ The average values of sunspotsR, magneticactiwity tt and •, during 1872-1930,for N = 59, are 38.9, 0.854, 50.4, respectively,and the standard deviations 27.5, 0.22l, 16.9. The very high correlation-coefficients, +0.869 between R and u and +0.884 between R and ut, indicate how closethe relationsare. The ratios r' x/l•' r•.are 1.76 and !.89. As could be expected, ze• furnishesslightly better correlation with R than the original measure u; it is therefore preferred in the discussionswhich follow. The average values of s2tnspotsR, international magneticcharacter- figures C, and magneticactivity u•, 1906 to 1930 for N= 25 are 42.3, 0.630, 53.4, respectively,and the standard deviations 27.4, 0.087, 15.9. The correlation-coefficients are +0.570 between R and C, +0.820 between R and u•, and +0.719 between C and z•, and the values of ratios r:x/1-r'" 0.69, 1.43, and 1.03. The inferiority of the annual mean values of the

• Ad. Schmidt, Met. Zs., 43, 329-334 (1926). a• Proc. R. Soc., A, 134, 486-501 (1931). RELATIONS MAGNETIC ACTIVITY TO SOLAR PHENO•'-I/IENA 31 character-figures C, as compared with those of u or u•, is revealed in these correlation-coefficients, because they gix,e the paradoxical result that one good measure of terrestrial-magnetic activity, namely is better correlated with the sunspot-numbersR than with another, but unsatisfactory measure of terrestrial-magnetic activity, namely C (see also õ 13). We have from Greenwich the areas of sunspotsS and of faculae F during 1882-1930 for N=49 for comparisonwith relative sunspot-numbers R and magnetic activity u•. Since 1882 the Greenwich mean areas of sunspotsand faculae34 are based on photographs taken on practically every day. The projected areas, uncorrected for foreshortening, ex- pressedin millionths of the Sun's visible disc are used here as they are more likely to be of geophysical significance than the corrected values. Average values of S, F. R, and u• are 849, 1181, 39.7, and 50.9, re- spectively, and the standard deviations 616, 785, 26.9, and 16.7. The correlation-coefficients of u• are +0.854 with S, +0.795 with F, +0.858 with R, while the correlation-coefficientof the areasof spots $ with the areas of faculae F is +0.898 and with the relative sunspot-numbersR is -3-0.981. Sunspot-areasS are thereforenot superior to the relativesun- spot-numbersR, since both give the same high correlation-coefficients with terrestrial-magneticactivity ut, while the areas of faculae F yield a definitelyless correlation-factor with ut than both S and R. The high correlation between the series S and R emphasizesthe satisfactory qualities of both; with r=0.981, the ratio r: x/'l-r '• is as high as 5.31 (see õ 13).

13.--Testsof homogeneityfor measuresof solarand terrestrialactivity The correlationsfound in õ 12 are close enough to warrant tests of the followingkind: The determinationof the relative sunspot-numbers R aswell asof areasof sunspotsS and,especially, faculae F aS,is naturally uncertainand difficult enoughto make specialsafeguards necessary in order to keep the series homogeneous. The close connection of the terrestrial-magneticactivity ut with the measuresof solar activity makes an independenttest of homogeneitypossible, by actual calcula- tions of the linear formulae(equation 13) and the residuals(equation 14). Inhomogeneityin any of the variableswould be revealedby a conspicuoustrend in the residuals;which one of the variables is taken as y and whichone as x (equation13) is not important,as longas it is not forgottenthat the regressionof y on x is not the sameas that of x on y. Startingfrom the annualmeans 1882 to 1930,the variablesF, R, andu• wereexpressed as linear functions of the sunspot-areasS, namely (17) F= 1.146S+208 (18) R = 0.0428S+3.3 (19) ut = 0.229S-3-31.4 Figure 11 showsthe observedvalues of S, F, R, and u•, and the residuals /XF, /XR, and /Xu• (observedminus computedvalues). The ratios of a•Mon. Not. R. Astr.Soc., 49• 381 (1889);63, 465 (1903);76, 402 (1916);84, 742 (1924);91, 1005 (1931). a• Mon. Not. R. Astr. Sot., 8•t, 96-99 (!92•). 32 Y'. .BA R TELS the standard deviations of F, R, and u•, to the standard deviations of the residuals AF, zSR, and /Xm' are r:M'i-P', forTwhich the correlation- coefficientsin õ 12 give 2.04, 5.31, and 1.64. The scalesfor the observed values have been approximately normalized, that is, chosenso that the amplitudes of the various curves are not too different. The scalesfor

1882 1890 Iõ00 1010 1920 1030 '1i i i i i i i i i I'"'"F"i i i i i'1 i i i i i I i i I i i i i Ji i...... J"l"'li • ...... /•' ..'."-OBSE •v•...... VALUES

/' . ooo

--

, k 1' ...... ß ,o

...... • +i• .... • •SI[,UALS...... '-};;) '......

.... - ...... --

,

• •_ -IO • *•o...... • ......

_• ...... l 1--Tests oœhomogeneity for sunspot-areasS, faculae-areasF (both projected, uncorrectedfor foreshortening,and in millionthsof the Sun'svisible disc), relative sunspot-numbersR, and terrestrial-magneticactivity u•, 1882-1930 (Upper curves give observed values F, S, R, and u•, while lower curves give residuals &F, AR, and &u•, that is, algebraic excessesof observed va1 u es over t hoseco mputed fro m S by 1east-sq uare li near-formu!ae)

F and /x F are equal, also those for u•.and/Xu•, but the scaleof AR was magnified four times that of R in order to show the details. •x F doesnot changeits sign irregularly from year to year, but shows definite persistence. The large value for 1892 is puzzling, also the long interval of positive signsfrom 1903 to 1911, and thi• negative signsfrom 1917 to 1924. /xR is satisfactorily small; it is an interesting feature that 'for th.ree years in succession,1916 to 1918,the observedrelative sunspot-numberR has been more than '[1 units higher than the simul- RELATIONS 3IAGNETIC A CTIIqTY TO SOLAR .PHENOMENA 33

taneoussunspot-areas S would have suggested. So, either R is too high or S is too low for these years; the former alternative seemslikely from the comparisonof R and ul (õ 14). Figure 12 indicates that the larger residuals /xF and /xR occur, as would be expected,in the years of sunspot-maximaand that they cannot be explainedby non-linearregression, that is, by systematic deviations of the general relations from straight lines. A slight improvement would be possiblewere the regressionline drawn through the origin. Further commenton the observationalor physicalnature of the residuals /xF and AR which we have shown in Figures 11 and !2 is a matter mainly of astrophysicaI interest and is therefore not discussed. The linear relation between the 25 annual ,ooot . ] means(20) ofCandC= Ul 0.00392u•+0.421is The residuals /xC between the observed values of the international magnetic charac- ter-figures C and those computed from it• are considerable, the standard deviations of /XC and C being nearly equal. This is ,•oo verifiedver by Figure 13. The outstanding residualres5 for the year 1930 indicates that C • '••eeee wasWa• estimated, on the average, 0.16 unit higher in that year than the value for zt• wouldWO1 suggest. The suspicionof a spurious shift in the character-estimation C is con- ' SUNSPOT-AREAS rS)-- firmedlift2 by the fact that, in the whole series of 25 years, 1930 has the highest annual and April 1930 the highest monthlym •tl mean (1.04), while both intervals are not at all outstandingif judged by the 01Zobjective•C' measures u or u• or other numerical measuresm, LS1 of activity whichare basedon daily. ranges(C. R. Duvallt*). It is not unlikely •o• •. thrathat ,•e this signifiesa breakin the homogeneity of the series of international character- figures,which may bedue to the newpractice, adopted by someobservatories, of determin- ing the character of actual measurement of ranges, etc., contrary to the original con- Fro. 12--Linear adjustments ception (õ 2). The increasedfrequency of of relations areaõ of faculae F the higher values for C is noticeable in our to sunspot-areas$ (upper) and Figures 10, 18, and 19, but does not vitiate of sunspot-numbersR to sun- the conclusionsdrawn from these diagrams spot-areas S (lower), each dot and has, especially, no influence on the represen.ting a pair of annual means •n the 49-year series, identificationof the 27-day sequencesin 1882-1930 Figures !8 and 19. 14•The lag of the annual meansof terrestrial-magneticactivity behind thoseof solar activity The last curvein Figure ll showssuch definite persistence in the signsfor /Xu• for severalsuccessive years that it was thoughtworth 5 34 J. BARTELS [VOL.37, NO.]1

Fzc,. 13mObservedannual meansmagnetic activity u•, international magneticchar- acter-figuresC, and residuals/xC=C-C' betweenthe observedvalues C and values C'=0.00392,•+0.42!, computed by linear least-squareadjustment

while to studythis phenomenonby usingall the annualmeans for the series1872 to 1930 as given in Table 5. Figure 14 showsthat the fit of the linear formula (21) ul = 0.543Rq-29.3 can be improvedby takinginto accountthe characteristicsteeper ascent for the lowervalues; accordingly, the curvedregression-line L shownin Figure14 waschosen. From R=0 to R=20, u• increases,on L, by 20 .units insteadof only 11 units on the straight line; this characteristic feature(that thesame increase in sunspot-numbersR affects the magnetic activity u, generallynearer sunspot-minimum than maximum)is well broughtout by the distributionof the dots in Figure 14. The sensitivityof the magneticactivity for smallchanges of R near sunspot-minimumis noticeable in 1923(Fig. 2), whenu• did not reach the low values of 1901 and !913, because"1923 did not have the pro- nouncedcharacter of a sunspot-minimumyear as,for example,1901 and 1913" (A. Wolfera•). The regression-lineL defines ,u•' as a functionof R. The graph representingu•' for the series1872-1930 (Fig. 15) is thereforenothing but a transformationof the graphrepresenting R in Figure2, consisting in a deformationof scale, accordingto L; if the R-graph in Figure 2 were drawn on some elastic material, the u•'-graph in Figure 15 could. be obtainedfrom it by stretchingor compressingthe verticaldistances between consecutivehorizontal lines, till the non-uniform scale of R on the right-handside of Figure15 wouldbe obtained.The residuals /Xu•=u•-•' are plottedin the secondrow of Figure15; superposition

•Astr. Mitt., Nr. 114, 117 (1926). RELATIONS MAGNETIC ACTIVITY TO SOLAR PHENOMENA 35

FIG. 14--Annual meansof relative sunspot-numbersR and terrestrial-magneticactivity u•, each dot marking a pair of values, for each of 117 annual means (for intervals January to Decemberand July to June), 1872-1930;straight line by least- square adjustment and a curved regression-lineL are shown of thetwo graphs for u•' and for /Xu•would give the graph o• theob- served values u•. On the whole, no progressivetrend can be detected in /Xu•. The result of our test indicates therefore that the sunspot-numbersR as well as the measureu, of magneticactivity are fairly homogeneousthroughout the interval 1872-1930. As to the discrepancy between sunspot-areas S and sunspot-numbersR in the years 1916-18, mentioned above, it would seem now that it is due to high values of R, since the values u•', as

187;:) 1880 1890 1900 19I0 I920 1930

• i I I -- '• 80 •! ..Jk• I I! / •I ß ...... ! '"-* •/ A " ----60--100•

u 40 --20 o

• ...... --0

,

-• • • ...... , ...... ,,, ,, '• ..•

Fro. 15--AnnuaImeans of magneticactivity u•' as computedfrom sunspot-figures by regression-lineL in Figure1•, residualsAu•=(u•-uF) and average distance(latitude) I of sunspotsfrom Sun'sequator 36 J. BARTELS [VOL.37, No. II computed from R, are also too high as shown by the negative in Figure 15. . The characteristicpersistence of the sign of/xu• for severalsuccessive years,which was noticeablein Figure 11, reappearsin Figure 15. The phenomenonis pronouncedin the interval 1900-25,when negative signs of /ku• prevail during the ascentto a sunspot-maximumand positivesigns prevail duringthe descentto a sunspot-minimum.This amountsto a lag of the annualvalues of u• behindR, and substantiates previousresults obtained from shorterseriesa7. ' Before 1900, the lag wasonly noticeablefrom 1886-89. The lag (whenit occurs)can also be expressedin the statementthat a certainvalue of the magneticactivity u• in the descendingphase of the sunspot-cycleis connectedwith a relative sunspot-numberR, that is about 20 units lower than that number R which would be connected with the same u• in the ascending phase. This phenomenonhas beenexplained by the well-knownfact that the sunspotsof a newcycle appear in highlatitudes of the Sun and graduallyapproach the Sun'sequator. The averagesolar latitudes of sunspots,that is, their averagedistance from the solarequator, have beenplotted in Figure15 accordingto the Greenwichobservations for eachyear. Radialstreams of solarcorpuscles are thereforemore likely to traverse the Earth's orbit in the descendingphase of the cycle,when sunspot-latitudesare low. This explanationfollows qualitatively the samelines as the Ax- explanationfor the annualvariation of magneticactivity (õ 9), but a quantitativetest is againunfavorable to the latter. Accordingto Table 9 (meanfor all years),u• is near..theequinoxes 2c== 12.7 units higher than nearthe solstices;therefore, judging from the linearequation (21), relative sunspot-numbersR producing the sameactivity u, must be roughly12.7/0.534=23 units highernear the solsticesthan near the equinoxes.If this largeeffect were produced by the comparatively small changesin the Earth's heliographiclatitude within the year, as the Ax-exp!anationasserts, then the great changesin the sunspot- latitudesoccurring in the solarcycle should result in a greaterand more systematiclag-effect than is actuallyfound in Figure15.

15---Relationsbetween monthly means of solar activityand terrestrial- magnetic activity Beyonda generallysimilar trend, Figure 3 showssome coincident featuresin the monthlymeans of sunspot-numbersR and activity u, for instance,in November 1905, February 1907, September1908, August1917, March 1920,March 1922,December 1929. Othersimi- laritiesare the drop in activityduring October 1906, in the midstof the sunspot-maximum;the revivalof activity,September to December 1909; the fact that August1917, the monthwith the highestsunspot- number,was also magnetically very active. The lag of u behindR in the l 1-yearcycle is alsopronounced in the monthlymeans. Notable discrepancidsare, for instance,the highmagnetic activity of October to December1903, at thebeginning of a newsunspot-cycle, and the small a•See * and.,0; also L. A. Bauer,Terr. Mag., 23, 66 (1918);H. W. Fisk,Terr. tVIag., 34, 147-150 (1929_). High diurnalranges of declinationat Greenwichare alsomore frequent in the descendingphase of the solarcycle than in theascending phase, as can be seen from S. Chapman'sfrequency-curves based on 63 years of observations,Phil. Trans. R. Sot., A, 225, 59 (I925). RELA T!ONS MAGNETIC A CT!VITY TO SOLAR PHENOMENA 37

value for R duringMay 1921,the monthwith the highestu of the whole series as. The periodicityof somewhatover 8 months,which is indicatedin the monthly sunspot-numbers 1905-08, and which is characteristic for the weaksunspot-maxima 39,is not soeasily recognized in magneticactivity. Diagrams correspondingto Figure 14 and showingthe relations betweenthe monthly means of zqand R, weredrawn for the threegroups of yearswith high, medium,and low activity (õ 7), and for all years (Fig. 16). Monthswith R=0 are sonumerous that theyhad to be indi- catedby dotsto the left of the lineR = 0; theseare considerably scattered

,,

LOW •_

-

-

-

80- 80-

=•-

40-

~

I t I I I I I I I I I I I I I I I . I I I I I o o --

.

MEDIUM ALL

ß

.

_

,,,

80-

.

o

40-

FIc. 16•Monthlymeans of relativesunspot-numbers R andterrestrial-magnetic activity- u•, 1872-I930: "High" for 20 years with annual means u•_- > 58- "low" for20 yearswith annual means u•-<42; "medium" for 19years with intermediateannual means; "all" for the 59 years--708months • Nevertheless,thegreat storms ofMay 1921 are undoubtedly ofsolar origin, since an extremely largenaked-eye spot-group, covering 1/700 of the Sun's disc, crossed thecentral meridian onMay 141 it wasthe largest group ever photographed atGreenwich onthe solar equator. The relative sunspot-number aR case forMaywhere 14 the was, sunspot-areahowever, onlyS was 63, more because significant even the than largest R. spot contributes only10 units to R. Thisis • A. Wolfer,Astr. Nachs., 100, 336 (1909). 38 J. BARTELS [VOL.37, No. II in the vertical, indicating that low solar activity can be accompanied by fairly high magnetic activity. The distribution of dots in the dia- grams "medium" and "high" resemblesthe elliptical type of "normal correlation," while in the diagram "low" the dots crowd toward the line R = 0. That the monthly means of u• and R are more loosely connected with each other than th6 annual means, is clearly seen from a com- parison of Figures 14 and 16. The following reasons,in the supposed order of magnitude, are offered- (1) Since the relation between R and is statistical, not functional, it must appear clearer the more casesof nearly equal character are combined. This holds for the combination of monthly means into annual means (õ 16). (2) Since magnetic activity lags behind solar activity by some days, use of simultaneousintervals for R and u• introduces a certain boundary-effect which is, of course, relatively greater in monthly means than in annual means. (3) Because of the annual variation of u•, the like values of u• are, in the equinoctial months, coordinated to simultaneous values of R that are about 20 units lower than the simultaneous values of R in the solstitial months

, 16.--General remarks on correlationin serieswith after-effect(monthly and annual means) The series of monthly means for u• as well as for R show definite "after-effects" insofar as means for successive months differ much less than would be expected were they independent. The statistical treat- ment of such series has been developed mainly for series of a single variable, in the Lexis theory, or in the theory of Brownian movement and other fluctuations •;. The correlation between zt• nad R offers a good example of the relations between two series of this type and the following general remarks may be useful for similar cases,for instance, in meteorological correlations. N pairs of annual means A•, B• (n=l, 2, .... , N) may be given with the average values A0, B0. They are derived from NM monthly means an•, b• [n=l, 2, .... , N and m=!, 2, .... , M; for monthly means M = 12]. The departuresof the monthly means from the respective annual mean are called e•, •, thus (22) a• =a•.•--A• and •, =b•-B• The standard deviations of the annual means, monthly means, and monthly departures, respectively, are called •x, •,; •, *•; •=, *•; the correlation-coe•cients are called ra•, r•o, r,•. By straightforward appli- cation of the well-known formula of the type

(23) Z•=•• A • (A• Ao)•+NAo•=N (•x•+Ao •) (24) N N +AoB0) we obtain the symmetrical relations ß (25) • = •.4•+• • and RELATIONS M'AGNETIC .4CTIt"ITY TO SOLAR PHENOMENA 39

The latter formula, with (27) k•=•r•/.•t and k•=•r•/• gives the useful relation (28) ra, = (rA•q-k•k•r•)/x/(1 +k• 'ø)(1 +k• •) The correlation rabbetween the monthly means will therefore be nearly the same as that of the annual means, if the monthly departures are small in comparisonto,the scattering of the annual means (k• and ka small). The value of r•b may become numerically greater than r•, if r•a and k• and k• are great; when k• and ka tend to infinity, r• becomes equal to r•. If, however, r•a is numerically smaller than r•., r•, will also be numerically smaller than r•z. Assigningthe letters A and 23 to R and u•, respectively,the data for the three groupsof years and for all years, 1872-1930,have been collected in Table 14. Since r• is small in all cases,the correlations r• between the monthly means are always smaller than those r•. between the annual means. That the coefficients r• are smaller for the three groups (chosenas in õ 7) than for all years was to be expected,since the correspondingdistribution of dots in the groupswas simply obtained

TABLE 14•Standard deviationsand correlation-coefficientsfor.sunspot-numbers R and magnetic activity u•, 1872 to 1930

Sunspot-numbers R Magnetic activity Item Dev'n[HighIMedium Low All StandardMeanvalues deviations:...... -Ti;.--J 1-- --'•.-7• -4071i • } 9.0l 4.8 7.2 15.9 Annualmeans ...... •r.4 18.3[ 15.1I 8.712 24.4•21.5 [ 07 ½b } 21.5 18.6 14.0 23.8 MonthIvMonthly departures..means....:. •Va 16.1[ 14.9]160:14131 3 [19.sl17.911.9 16.7 Ratiosk...... k• 0.8810.99 10.7010. k• 2.17 3.73 1.65 1.05

Correlation-coefficients

I tern High Medium Low All

Number of years ...... 2O 19 20 59

Correlation for annual means,rAB .... +0.530 +0.601 +0.569 +0.884

Correlationfor monthly means,tab.. +0.301 +0.365 +0.377 +0. 654

Correlationfor monthly departures, +0.234 +0.378 +0.264 +0.292 by dividingthe diagramof Figure14 by two horizontallines at u•=57.5 andu• =42.5 into three horizontal strips. This reduces rr. in thegroups, and explainsthe high ratios k•=•r•/**. It is also characteristicthat the ratiosk• fox'R areall smallerthan the ratiosk• for u•,indicating 40 3 •. BARTELS [VOL. 37, No. 1] a smoother run, or higher after-effect in the monthly means of sunspot- numbers as compared with magnetic activity; this is already noticeable in Figure 3. The transition from the correlation-coefficient rx, for the annual means to that for the monthly means,ra•, which has just been discussed arithmetically, is equivalent 'to the transition from Figure 14 to Figure 16 and can therefore be described graphically, as indicated for three years (of high, medium, and low activity) in Figure 17. The large dot, representing the annual means for 1918 in the left-hand graph of Figure 17, is the mass-centerof the 12 dots representingthe months; or, other- wise expressed,the dots for the monthly means are formed by an "ex- plosion" of the dot for the annual means, resulting for the three years in the three "" in Figure 17. In the left-hand graph of Figure 17 the distribution of the annual means determines ,.{, •, and rAB, and

+40

+20 '

-2O

OSCaleof'R -40 5•ileofR i 40i• , i 6,,i ...... i 8,0 i I00i ..... i i, _-20 i, 1 0i i +20I , Fie,. 17--Relative sunspot-numbersl{ and terrestrial-magnetic activity u• •n year of high activity (1918), medium activity (1880), and low activity (190!), demon- strating relations between correlation-coefficientsfor annual means, monthly means, and monthly departures; monthly and annual means at left and monthly departures from annual means at right the distribution of the monthly means determines •a, •,, and ra•; superpositionof the three "stars," with the annual means as center, gives the distribution in the right-hand graph of Figure 17, which determines •, •, and r•. The correlation-coefficientr correspondsto a given distribution of dots in the following manner: Supposethe scalesof the two variables (say x and y) being normalized,that is, chosenso that their standard deviations are equal; this is graphically realized by extending or com- pressingthe graph (like Fig. 17) in the direction of one axis. If then, according to the methods of least squares, ellipsesof equal frequency are constructed, with their axes along the 45 ø lines x=y and x=-y, the ratio of the axis alongthe diagonalx =y to the axisalong the diagonal x=-y is, for each ellipse, equal to (29) x/(l +r)/(1-r) for instance, equal to 2 for r=0.6 and equal to 1 (circle) for r=0. The ellipticity of the distribution is thereforea measurefor the correlation. These indicationswill be sufficienthelp for visualizingthe contentof the RELATIONS MAGNETIC ACTIVITt" TO SOLAR PHENOMENA 41

analyticald.iscussion given above and of Table 14; Figure17 makesit clearhow important are the ratiosk• andk• betweenthe amountsof the "explosion"of the annualmeans and the scatteringof the annualmeans for the combinationof r.4, and r• into ra•. And the reasonsfor the introductionof the u•-measure({}6) may now be simply summarized in the statement, that the greater scattering of the monthly means of the originalu-measure would have impliedgreater values of k and therefore smaller correlations for the monthly means. • is, of course,the samequantity that was designatedby • in equation(6), õ 7. Wemight have obtained slightly higher correlations betweenu• and R if we had usedmonthly means of u•, whichwere freed from the annual variation, and which had thereforethe standarddevia- tion v' as definedin (6). However, the numericaldifferences between • and a', as givenin õ 7, are too small to make this eliminationnecessary. In conclusion,it must be mentionedthat in certainselected groups of years,at the end of a sunspot-maximum,the correlationsr•, between the monthlymeans of R and u• are muchpoorer than Table 14 indicates for the whole series. The correlation-coefficient r• for the 36 monthly means1918-20 is only +0.06 and for 1928-30is +0.01. This remarkable obliteration of the otherwise close relations between sunspots and magneticactivity in certainintervals is not removedby the useof other measuresof solaractivity (õ 18) and thereforeimposes caution in drawing conclusions from observations in a limited number of years. 17•The individual 27-day recurrences,1906-31, and their relation to ., sunspots The main results of the extensive work of C. Chree and J. M. Stagg4ø - were that disturbed and quiet magnetic conditionstend to recur after intervals of 27 days, and that no systematiclengthening or shortening of this interval could be detected when groups of years with many or few sunspotsor with high or low sunspot-latitudeswere formed. Con- trary to opinionsheld by H. Deslandres,no trace was found of the existenceof any disturbance-interval which is an exact submultiple of 27 days, that is, of any symmetrical pattern of the active regionsaround the Sun's circumference: According to W. M. H. Greaves and H. W. Newton 4• the recurrence-characteristic is mainly a property of the storms of smaller range, while the intense storms (with an average mean range in the three componentsover 1803/as recordedat Greenwich) are generally followed neither by another storm nor even by a subsidiary disturbance. While the investigationsjust mentioned deal mainly with averagesfor many cases,it seemedto be of interest to investigate the 27-day phe- nomenon individually. Therefore, a day-by-day record of magnetic activity, as measuredby the international magnetic character-figuresC, was prepared indicating C by suitable symbols. The record reads like a book. The date of the first day in each row is indicated on the left, each successiverow beginning 27 days later; at the end of each row the first nine days of the next row have been repeatedin order to emphasize the continuity of the series. For reference,the days in each row are numbered 1 to 27. The symbols were chosen so that there are in all •0 C. Ct•ree and J. M. Stagg, Pl•il. Trans. R. Soc., A, 227, 21-62 (1927); J. M. Stagg, Met. Office, London, Geophys. Mem., 4, No. 40, 8 pp. (1927). For a review of these papers and others sect. • Nlon. Not. R. Astr. Sot., 89, 641-646 (1929). 6 42 J. BARTELS [VOIo.37, No. 1] about as many black as red symbols on the chart. The values for C to September193! have been provisionallyderived from the quarterly lists. The years 1906-31 have been arranged in two vertical rows, so that intervals which are !1 years apart appear side by side. The information which is contained in this arrangement of more than 9,000 symbolswill be useful insofar as most days can now easily be assignedto a 27-day sequenceof quiet or disturbed days; by sequences we mean one of the many vertical columns of symbols of the same or nearly the samekind, which are so conspicuouson the diagram and which demonstrate the 27-day recurrence-phenomenonmost convincingly4'-'. Somecharacteristic features may be pointedout 43 [inset (Fig. 18), opposite]: (a) Years of sunspot-minimum are marked by prevalence of red symbols, and years of sunspot-maximumby frequent black symbols. However, there is not a singlerow of 27 days which containsnothing but black or nothingbut red symbols;on the contrary, pronouncedsequences of quiet days persist:near a sunspot-maximum, for example, in 1917, 1918, and since 1926, and sequencesof disturbed days persist near a sunspot-minimum(1911 and 1923). (b) The synodicalrotation-period of the Sun's surface,as determined from recurrent sunspots,is 26.9 days on the solar equator, 27.1 days in 10ø latitude, 27.5 days in 20ø, 28.3 days in 30ø, 28.8 days in 35ø latitude. Higher levels of the Sun's rotate faster and therefore face the Earth after shorter synodicrotation-periods, as shownin Table 1544. A recurrence-interval shorter or longer than 27 days would appear in Figure 18 as a gradual shift of the sequencetoward the left or right, respectively; from the systematic change in sunspot-latitude during the 11-year cycle (Fig. !5), longer intervals would be expectedat the begin- ning and shorter intervals at the end of a sunspot-maximum. The slight systematic movementsin longitude of the spots4'•, which depend on their age, are not great enough as to bring about essential differencesfrom the average rotation-period which is proper for the latitude. A 28-day interval seems to be indicated in the minor disturbances during the secondhalf of 1913 (after day 13), but otherwisethe diagram does not show a regular effect of' that kind. In fact, 27-day intervals seem to prevail and in long sequencessystematic shifts of more than half a day within a are rare. 'I',xmm 15•Synodic rotation-periodsfor differentlevels of the Sun

Heliographic Reversing Calcium- latitude Sunspots layer line 4227 (near Sun's limb)

¸ days days days days 0 26.9 26.4 25.9 25.7

15 27.3 27.0 26.0 25.9

3o 28.3 28.4 27.0 26.4

• The original diagram was prepared for the Annual Exhibition of the Carnegie Institution of Wash- ington in December 1931. • The solar data in this discussion are taken mainly from the Greenwich photoheliographic results and their annual summaries in Mort. Not. R. Astr. Soc., and from the Astr. Mitt. by A. Wolfer and A. Brunner. • Based on the table of angular velocities in Handbuch der Astrophysik, 4, 167 (1929). • Greenwich photoheliographic results for 1024, pp. D79-D85.

RELATIONS 3(AGNETIC A CTIVITt' TO SOLAR PItENOMENA 43

(c) Great storms appear often isolated, that is, within an otherwise quiet sequence. Outstanding cases (with C->1.7) are the following' September 11-12 and 29-30, November 17, 1908; August 6, 1912; July 1, 1916; August 15-16, 1918 (seemsto recur after two rotations); Sep- tember 22, December 26, 1920; September 14, 1922; September 26-27, 1923; March 5, 1926; October 24-25, 1928; July 10 and October 16, 1929; December 3-4, 1930. Other great storms, however, appear in, or lead, pronounced sequences. G. E. Hale 46has recently discussedthe casesin x•hich solar eruptions could be connectedwith the following outbreaks of terrestrial-magnetic storms. In the interval for which Figure 18 is constructed, several of thesestorms occurred in definite sequences,namely, September 11, 1908; September 2•,, 1909; January 26 and February 23-25, 1926 (this pair is part of the same sequence). Someof the'stormsappeared at the end of disturbed sequencesand were followed by quiet days, namely, May 14, 1909; June 15, 1915; October 14, 1926. The solar eruption of November 10, 1916, was perhaps not (as Dr. Hale assumes) without a terrestrial effect, because November 12 had a small but widespreadmagnetic disturbance(C= 1.8). This is interesting becausethis disturbancehas in Figure 18 quite the appearanceof the "isolated storms" enumerated above, and for some of which Dr. Maris ?• suggestsnon-solar origin; about the same statement holds for the three storms listed at the end of the last paragraph. It is, therefore, quite likely that more isolatedstorms will becomelinked to strong but short- lived solar eruptions when Dr. Hale's program of continuous solar observationswill be realized, so that the solar origin, which for the minor disturbances is guaranteed by their 27-day recurrence, will also be assured for the isolated storms. (d) Especiallylong sequences seem to occurat the end of the sunspot- maximum, when the spotsare near the equator, while the first disturb- ancesof the new cycleare more irregular, giving those parts of the dia- gram a spotty appearance. The transition from the old to the new cycle is well markedin 1914and especiallyin 1923. The fine sequencein the first half of 1923 (day 18), with its sharp recurrencesafter 27 days, belongsobviously to the old cycle; it is remarkablebecause it persists throughtimes (in Februaryand May) when,f br severalweeks in succession, •tot a single sunspotwas visible/ There were no spotson the Sun from April 8 to luly 8 (92 days), 1913,and after a small'group had lastedfrom July 9' to 14 therewere again no spotsfrom July 15 to August 22 (39 days). "No year since !810 hasbeen so barren of sunspotsas 1913. For sevenmonths (March to September)the Sunwas nearlyalways free from spots."4' However, sequencesof minor magneticdisturbances persist through this interval; the 28~dayinterval already mentionedis probably due to disturbed regionson the Sun, whichwere in the samehigh latitudes(28 ø ) as the few spotsseen in June and July. In 1917,according to A. Wolfer,rhythmic recurrences of spotsafter onesolar rotation are almost entirely absent and spots never lived longer

• Astroph.J., 73, 379-412(1931); reviewedby W. Grotrian, Naturw., 20, 55-56 (1932). Methods of recordingobservations with the spectrohelioscopeare discussed by Dr. Hale in Astroph.J., ?4, 214,-222 (1931). •* Mort. Not. R. Astr. Sot., 75, 19 (1915). 44 J. BARTELS [Vo•.. 37, No. II than through two solar rotations; Figure 18 shows, however, some well- developed disturbed sequences. The remarkable sequencesof 1930 and 1931 will be discussedin õ 18. (e) Though often two or more sequencesrun simultaneously,they do not divide the 27-day interval into regular subdivisions. Even the spotted regions on nearly opposite sides of the Sun, which A. Wolfer points out for the year 1911, are not reflected in magnetic conditions. "During the year 1922, two-thirds of the spot-groupswere confinedto one-half of the Sun, between heliographic longitudes 330ø-0ø-130 øo The region between longitudes 130ø and 330ø contained the remainder, but, with two exceptions,these groups were all very small and usually short-lived."4s Magnetic conditions again do not correspondto these solar conditions, since two well-separated sequencesoccurred in 1922. (f) The greatestdisturbance on the chart is that of May 13-16, 1921, which was accompaniedby the passageof a large naked-eye spot-group on the solar equator through the Sun's central meridian on May 14as. (g) The determination of the lengths of sequencesis somewhat uncertain, but the existence of very long sequencesis obvious. Out- standing examples of disturbed sequences(C=>_0.8) are day 9, July 24 1910, to June !3, 1911, 13 rotations4'a;day 9, December 11, 1921, to November 27, 1922, !4 rotations; day 13, December 9, 1929, to March 13, 1931, 17 rotations. Examples of quiet sequences(C<-0.5) are No- vember 7, 1912, to December 16, 1913, 16 rotations; May 16, 1923, to June 28, !924, 15 rotations; August 28, 1926, to August 14, 1927, 14 rotations. These and other long sequencespersist regardlessof season. This forms an unexpected contrast to the direct solar observations. In her catalogueof recurrent groupsof sunspots,!874-1906, Annie S. D. Maunder •ø says: "Of the 624 instancesof recurring groups catalogued, 468 were seen only in two rotations, 118 appeared in three rotations, 25 in four rotations, 12 in five rotations, and in one instance only, and that somewhat doubtfully, did a spot-group survive to be seen in a sixth apparition. Five months is the longest continuous life-history recorded for any group in the whole of the 33 years covered by the catalogue. It is evident, therefore, that a group of solar spots is essentiallya short- lived phenomenon." Of course, the longer duration of disturbed sequencesin magnetic activity may in some few cases be caused by a more or less accidental continuation of a spot-group in the northern solar hemisphereby one in the same longitude in the southern hemisphere, t0ut it is more satis- factory to mention, in this respect, the longer life of faculae (see õ 18). (h) If the time T of passagefrom the Sun to the Earth would be constant for all corpuscular streams, then our diagram could be con- ceived as a chart of the Sun, indicating the hellographiclongitude of the active regions on the Sun--which we shall call here M-regions. Several investigators'• have shown that T may be as high as 3 or 4 days for moderate disturbances.,while it may be as low as one day for the great magnetic'storms. This latter value is also suggestedby the discussion

•s Mon. Not. R. Astr. Soc., 8•t, 31 (1924). •0 The two long disturbed sequences in 1911 have been discussed by G. Angenheister, Terr. Mag., 27, 69-71 (1922). •0 Greenwich photoheliograI>hic results for 1007, Appendix, p. 6. • Ch. Maurain, Ann. Inst. Phys. Globe, Paris, !5, 86-96 (1927); J. M. Stagg, Met. Office, Geophys. Mem., No. 42 (1928); W. M. H. Greaves and H. W. Newton, Mon. Not. R. Astr. Sot., 88, 556-567 (1928). RELATIONS MAGNETIC ACTIVITY TO SOLAR PtlENOMENA 45

of G. E. Hale46. Since our sequencesmostly consistof minor disturb- ances,our chart incidentallysupports the view that the time T of passage for these,whatever it may be, is certainly fairly constant becauseother- wise such sharp "fronts" of sequencesas in 1923 and !930 could not Occur. (i) The "width" (w) of the columns which indicate the disturbed sequencesvaries from comparatively narrow bands (as in 1923) of about twodays to broadcolumns of five or more days (as in 1930).A, width w= 2 days may often be due to a disturbanceof only a few hours dura- tion, overlapping two successiveGreenwich days. If w could be con- ceived as indicating the extension of the M-region in hellographic longitude, one day would correspondto ,360ø/27=13ø.3. (j) It is not proposedto test in this paper the 30-day recurrencein the larger storms, found by Adolf Schmidtr'"'.

18.--The solar indices,!928-30, comparedwith terrestrial-magneticactivity The identification of the M-regions on the Sun's surface, which are the sourcesof the persistentcorpuscular streams causing the sequencesof disturbed days, will now be attempted. From the discussionin the last paragraph it is already clear that the M-regions cannot simply be co- ordinated to spot-groups;the most convincingargument is that spot- groups have been observedcrossing the Sun's central meridian without causingmagnetic storms, while goodsequences of magneticallydisturbed days occurredin times when no sunspotswere visible. The faculae have often been suggestedas likely to have greater significancefor geophysicalphenomena than the spots. In this respect, the concise summarya• of the Greenwich observations from 1874-1917 may be quoted- "The centresof the chiefzones of the faculaehave a well-definedprogression with the solarcycle of about 11 years . . . , in considerableaccordance with the latitudeprogression of the sunspots.... As comparedwith the spot- zones,however, the correspondingfac•lae-zones are on the averageabout 15ø broader. The extensionis mainlypolewards, and . madeup entirely ofsmall areas of faculae, the largest areas being always'cc•nfined to theregion of the spots. "In character,the faculaeunassociated with sunspotsare small, faint areaslasting at the mostfor two months. This characteristicoffers a striking contrastto the faculaeconnected with sunspots,which develop in a few clays into bright and compactmasses often coveringa great area. Theseafter abouttwo monthsbecome faint and morescattered but they can nearly always be discriminated from those which have at no time been seen with sunspots.With largespot-disturbances, the faculaecan be recognizedfre- quently for several months. "Themean percentage area of faculaeunconnected with spots as compared withthe total areaof all facuiaeis about10 percent. This figureincreases to about30 percent during the minimum years and falls to 5 percent when the Sun is active. "Anarea of very bright, compact faculae indicates that either a spot-group hasbeen very recently connected with it or that onewill appearwithin a few hours. "Theduration of faculaeconnected with a spot-groupis on the average at leastthree times the length of theaccompanying spots, but the proportion isa veryvariable one. Generallyspeaking, the faculae of largespot-groups ascompared with those of smallergroups do notlast proportionally aslong. *•Ad.Schmidt, Met. Zs., 26, 511 (1909); 37, 166(1920); 42, 240 (1925). Astr. Nachr., 21•t, 409-414 (1921). G. Angenheister,Terr. Mag., 27, 57-79(1922). L.W. Pollak•. 46 J. BARTELS [VOL.37, No. II

"The faculae frequently act as a connectinglink between successivespot- disturbancesin the same region. Near the maximum of the , some of these centres have been traced without intermission for over six months and a few for nearly a year. "Faculae frequently appear in streaks roughly at right-anglesto the direc- tion of the Sun's rotation, and have a strong tendency to spread from a spot- disturbance for several degreesin latitude. This feature contrasts with the spots themselveswhich invariably stream out in longitude, whilst there is little trend in the direction of latitude. "The faculae change form rapidly, and individual features can seldom be recognizedat successiveappearances at the Sun'slimbs. The groupsof faculae consideredas entities are, however, more stable than groups of spots. . Theregularity ofstatistics ofsunspots islargely disturbed bythe fact ti•a•: we have an imperfect record of the spotsowing to the Sun's rotation. The longer life of the faculae tends to smooth this. "The investigationof the faculaehas brought more stronglythan wasantici- pated the very closeconnection between them and the sunspots. There are no spots without faculae, and no extensiveareas of faculaewithout spots. "There is a zone of polar faculaein both hemispheres . . . about latitude 70ø, . . . small,short-lived, detached flecks. Their appearanceseems to be somewhat erratic and showsno. pronouncedrelationship with the solar cycle . . . ; they are not associatedwith the polar prominences." From this descriptionit seemsdifficult to separatepurely statistically the geophysicalinfluences of sunspotsand faculae; only the greater persistenceof the faculae is significant in connectionwith the long sequencesin magnetically disturbed days. More definiteresults were expectedfrom the useof the daily character- figuresof solar phenomena,to which we shall refer here as solarindices •a. These indices denote the daily area and intensity of calcium-flocculi and the bright and dark -flocculiby numbers 0 to 5, "0" representingabsence or rarity of flocculiand "5" extremeabundance and intensity. Each observatorystates these numbersseparately; we shall use here the daily mean. As geophysicallymore important, we'shall use the indicesfor the central zoneonly, which in 1928 was the sectorbetween meridians situated 30 ø on either side of the central meridian, and in 1929 and 1930, a central circular surface of a semi-diameter of the Sun's disc; no distinction was found necessarybecause of this change of zone. It is well realized that the solar indices represent the solar activity not nearly as completelyas the international magneticcharacter-figures representmagnetic activity. The main reasonsare the smallernumber of cooperatingobservatories (about 6 againstover 40), frequentinterference of atmosphericconditions, and the small number of observationsper day at each observatory(mostly one, instead of the continuousmagnetic records). Some days have passedwithout a single tany. Brownobservatory. •4, are usedhere The andcorrected found solar satisfactory,indices, as as willgiven be bygeenHowell later; of course, the solar indices, which are derived in a similar way as the international magnetic character-figures, therefore have the same limitations as were discussedin õ 2. For all the material available, that is, the three years !928-30, the internationalmagnetic character-figures C, the relative sunspot-numbers • Internat. Astr. Union, Bull. for character-figures of solar phenomena, published quarterly by the Eidgen/SssischeSteruwarte, Ziirich, since 1928. For a discussionof see, for instance, the article of G. Abetti, Handbuch der Astrophysik, 4 (1929). • Terr. Mag., 35, 237-244 (i930); 36, 345-348 (1931). The establishmentof another standard series of solar indfi:es is announced by A. Brunner in the Bull. for character-figures of solar pt•enomena, No. 13, 52-53 (193i); we shall see later that other methodsof reduction are not likely to affect our conclusions. Though these conclusionswill indicate certain limitations in the expected value of the solar indices, they are not. to be misinterpreted as criticisms of the general scheme of the solar character-figures•without which, in fact, some results of this pai•er could not have been derived. RELATIONS MAGNETIC A CTII,•ITY TO SOLAR IP!:tENOMENA 47

R, for the central zone, and the indices for bright //a-lines were first comparedday by day, the latter being chosenas representativeof the conditionsin a high level in the Sun's atmosphere. In order to make this comparisonas clear and as free from bias as possible,it was decided to prepare 27-day recurrence-diagramsin the manner of Figure 18, and to assign six symbols, called group-indices "0" to "5," so that in each of the three diagrams, for C, Re, and Ha, about the same number of symbols of each lcind occurred. The medium group "2" was made the largest, containing about 28 per cent of all days, while the extreme groups "0" and "5" contained only !3 and 12 per cent, and the groups "1,'" "3,' and "4" about !6 per cent each. On the basis of frequency-tables, the groups were chosenas indicated in the following table. Group- Magnetic character- Central Bright hydrogen index figures sunspots lines

0 0.0, 0.1 0 0.0-0.3 1 0.2, 0.3 1-13 0.4-1.0 2 0.4-0.7 14-28 1.1-1.8 3 0.8-!.0 29-42 1.9-2.4 4 1.1-1.3 43-57 2.5-3.1 5 1.4-2.0 >_58 _>3.2

The diagrams are shown in inset oppositep. 48 (Fig. 19); a few days for which no data for R, or Ha are available had to be left blank. Since the charts are drawn in the same manner as Figure 18, the diagram for C in Figure 19 is only a somewhat generalizedrepetition of the corresponding interval in Figure 18 and shows therefore the same 27-day sequences. Such sequencesare also pronouncedin the two diagrams for R, and Ha, which moreover show the general decreasein solar activity throughout the interval. The interesting feature of Figure 19 is the closeresemblance of the diagrams for _R,and Ha---each quiet or disturbed sequencein one diagram can be recognizedin the other. But no suchresembJance appears betweenthe diagram for C and either of the others,even if the possibility of a general lag of several days in taken into account. A statistical calculation confirms this impression; because of the better observational conditions in the months April to October, as indi- cated by fewer omissionsof days in the original tables, only the 548 complete days of these months, 1928-30, were considered. On 65 of these days C belongedto group "0"; this group has therefore the relative frequency p0'=65/548=0.118. The other relative frequencies were obtained in a similar way and are shown by the following table. Group index 0 1 2 3 4 5 C 0.118 0.153 0.285 0.168 0.159 0.1!7 R• 0.132 0.184 0.307 0.162 0.093 0.122 Hc• 0.117 0.170 0.268 0.185 0.146 0.114

The relative frequency of days, say, in group "4" in C.is p4'=0.159, and of days in group "0" in R, is P0" =0.132. if C and R, were entirely inde- pendent, the relative frequency q4,0of days, which belong at the same time to group "4" in C and group "0" in R,, would be expected to be equal to the product p4'p0"=0.159X0.132=0.0210, and the number of 48 .f. BARTELS [Vo•.. 37, No. II such days mi,0=0.0210X548 =11.5. By actual count, ni,0=11 such days were found. The ratio ni,o/m•,o=qi,o/pi'po"is 1.0 in this case,that is, the observed number of days is equal to the number which would be expected if C and R, were independent. Two quadratic matrices for these ratios, one combining Rc and C, the other combining Rc and Ha have been reproduced in Table 16; these are, of course, equivalent to the usual correlation-tables, which were, however, not immediately applicable in their usual form, becausethe nature of the material made groups of unequal size necessary. The contrast between the matrix for the combination R• and C and the matrix for R• and Ha is striking--- in the first case irregular and accidental oscillations around the value 1.0, indicating lack of correlation, and in the secondcase strong corre- lation. The correlation-coefficientbetween the daily group-indices for Txm,g 16--Ratiosobserved nmnber dgys of groupings of sun}pot-numbersRc,of magnetic charactersC, and of bright t•a-iines, to numberscalculated assuming the .Ohen.omena independent, months April to Octoberduring 1928-30

Sunspot- Magnetic character C Bright !!=.lines ...... number 5

0 7 0 ! 8 9 2 1.6[ .4 0 2 0 0 1.4[1.3 2 3 7 1 6 13 !,5 1 7

4 1 5 3 3 1 7 2 1

5 8 0 6 0 9 38

R, and t/• is as high as +0.745, while that between Rc and C is -0.080, practically 0, because the standard error of the correlation-coefficient, which for 548 independentpairs would be 1/x/548 = 0.043, is certainly higher (becausevalues for successivedays are not at all independent), perhaps1/x/!50'=0.08. An attempt to correlatedaily valuesof Ro with values of C on precedingdays, and thereby to get an estimate of the time T of passagefor the solar streams, was abandoned, since the corre- lation-coefficientscould be only small, and the methodsused by Stagg and Maurain •t are more suitable for determining the lag T between R• and C. Our main result is, that, in spite of thepossibly unsatisfactory observa- tional material or its reduction,the sunspot-numbersand the solar indices for bright Ha-lines are so highly correlatedthat the latter can hardly be expectedto give better results in comparisonswith any geophysicalphenomena than the former. In particular, the M-regions of the Sun, whick are so stronglyindicated by magneticactivity, and which in many casescannot be identified with sunspot-groups,also cannot be associatedwith regions showing bright t!t•-lines. That the other solar indices are also closely correlated with the sunspot-numbers is shown in the following Table 17 of correlation-

RELATIONS MAGNETIC ACTIVITY TO SOLAR PIgENOME2¾A 49 coefficientsfor all the 36 monthly means that are available at present. In the designation used in õ 16, they are coefficientsra•. All the corre- lations between the sunspot-numbers, and the indices for bright and dark !Ia-flocculi and calcium-flocculi are high; lesser correlation is obtained with the radiation. The correlation between terrestrial-magnetic activity u• and all solar indices is negligible; con- trary to expectation,the new indicesfail completelyto improvethe poor correlation between • and R which was found (õ 16) to be peculiar to the period 1928-30. This confirmsthe impressionalready obtained from the daily/ta-indices. Terrestrial-magneticactivity revealstherefore

TABLE 17--Correlation-coe.•cientsbetween the 36 monthly means,i928-30 [u•=terrestrial-magneticactivity; C= internationalmagnetic character-figures; Rc relative sunspot-numbersin central zone; bright //•=character-figuresfor bright /-/a-flocculiin central zone; dark Ha =character-figuresfor dark H•-flocculi in central zone;Ca =character-figures for calcium-flocculiin centralzone; [IV=intensity of the ultraviolet solar radiation--ratio ultraviolet (X=0.32/z) to green (X=0.50/z)]

Element c Brig} Dar] UV

36 --0 -0 07 C -0.44 -0 -0 26! --0 -0 05 --0 ' --¾ 44 86 +0 11 Bright 0.00 -0 46 +0.86 +o 73 90 +0 24 .... 73 Dark +0.02I -o 26 +0.75 +o +0 35 -0.08I -o 59 +0.86 +o +o +0 33 UI / o5 +o +o 33

NOTE--Standarderrors of thesecoefficients are (1-r•)/6, that is, 0.17 for r=0 and 0.05 for r=0.85. solar influences.--recognizedas such by the 27-day recurrences--which cannotbe tracedin the direct astrophysicalobservations. G. E. Hale4•, in his recentdiscussion of the work of the spectro- helioscope,concludes that astrophysicalobservational means have been very inadequatein the past,and advocates that _r/a-photographsbe taken at half-houror shorterintervals throughout the day. The present writer is convincedthat sucha moreor lesscontinuous spectrohelioscopic watchwill bringthe astrophysica!observations more in line with the well-organizedrecords of terrestrial-magneticobservations; byregistering a greaternumber of the sudden solar flares •, it willbe possible tocollect, in thecourse of someyears, more cases of individuallycoordinated solar and magnetic phenomena. Thesesolar observations will alsohelp to decidewhether the solar streamsare nearly continuous or'whether they consist of moreor less separatedclouds of particleswhich the active solar regions emit inter- mittently;in thelatter case, 27-day recurrence asexpressed in Figure 18 •See,for instance, Provisional solar and magnetic character-figures of Mount Wilson Observatory, publishedquarterly inthis JOURNAL since 35; or the observations ofsolar fiocculi made with the spectro- helioscopeat Greenwich, Mort. Not. 1%.Astr., 91, 593-600(1931). 7 50 J. BARTEL,$' Wo•..37, No. II

suggeststhat the intervalsbetween successive eruptions must be of the order of one day or less. As to the daily solar indices,however, they are already, even in their presentdeficient form, so closelycorrelated to sunspot-numbersthat there is not much hopethat even more complete observationswill gi;½ethem much additional value for recognizingthe solar M-regionswhich the minor magneticactivity (C betweensay 0.8 and 1.6) so strongly suggests. The small correlation between •, and C, and the negative correlations between C and the solar indices (Table 17), are prol>at)lydue to a shift ' in the character-estimation C, as indicated in õ 13. The negative coefficientswere then simply due to the spurious increase of C during an interval of declining solar activity, and should be expected to come nearer to zero by usi•g correlation r• between m(•nthly departures (õ 16), sincethese •hould be lessvitiated l)y the shift (•f the scale. In fact, the coefficientsr•a are +0.42 between C and u•, -0.()4 between C and R,, -0.29 between C and ('a; that is, all are algel)raically higher than the correspondingcoefficients r,,• in 'i'alfie 17. A reservationseems appropriate since the discussi(m(•f this t)aragraph was based only on three years (•f ot)servati(ms. However, it will take some time before more material will t)e availalfie, and the chief result cannot be disputed, that namely, in an interval in which the sunspot- numbers failed conspicuouslyto show relations .t(• magnetic activity, the other measuresof solar activity did not dr')better.

SlIMMARY (a) The conceptionof terrestrial-magnetic activity is discussed,with special regard to the daily international magnetic character-figures(•'. As an objective measurefor the average activity of longer intervals, such as monthsand years, the day-to-day change (or interdiurnal varial)i!ity) of the daily means of horizontal intensity is pr•t•osed l•ecause of its uniformity in the non-polar regions, and the u-measure of activity is based on it. [u is the average change from day to day, regardlessof sign, of the horizontal intensit.y at the magnetic equator, expressedin the unit 10-r=0.000! c. g.s.] Mtmth!y means of the u-measure derived by combining the resultsof several obserwttoriesare given for the series 1872 to !930, and, in annual means, extended backward to 1835. (b) The relations between the changes()f energy of the magnetic field and some measuresof magnetic activity are t)riefly discussed. (c) For usein certainstatistical investigations, a new 'tq-measureof activity is derivedfrom the original u-measure. For •q, a function of the monthly means of u is chosen, the frequency-distribution of which re- semblesthat of the sunspot-numbers.The advantage of u• over u is that it repressessomewhat the irregular influenceof exceptionally violent magnetic storms. (d) The annual variation of terrestrial-magneticactivity and of sunspot-numbersis discussed;certain tests for periodsof generalform or of the form of sine-waves(harmonic dial, probable-errorcircle) are describedand applied. Only the semi-annualwave of magneticactivity, with maxima near the equinoxes,appears to be physicallysignificant. RELATIONS MAGNETIC ACTIVITY TO SOLAR .PHENOMENA 51

In different ways it is shownto be improbablethat this semi-annualwave is related to the inclination of the Sun's axis towards the ecliptic. (e) Apart from a general,but not exclusivepreference for the equi- noctial months, magnetic storms have not returned year after year on specificdates. (f) The correlationsbetween the annualmeans of variousmeasures of terrestrial-magneticactivity (u•-measure,international character-figures) and of solar activity (relative sunspot-numbers,areas of sunspotsand of faculae) are discussedand the ensuingmerits of the various measuresare estimated. Linear relations are used to test the homogeneity of the series. Among the noticeable discrepanciesare the abnormally high areasof faculaein 1892, and of the relative sunspot-numbersin the years 1916-18, which seem to be about 10 units higher than the simultaneous sunspot-ar.eas suggest. (•) Due to the higher heliographiclatitude of the sunspotsin the beginningof an l 1-year cycle,they affectterrestrial-magnetic activity less at those times, that is, lessin the ascendingphase than in the de- scendingphase of the cycle. This is inferredfrom an apparentlag of the ' annual means of magnetic activity behind those of sunspot-numbers, which appearsin somecycles. (/•) The statisticalaspect of correlatingmonthly and annualmeans is discussedin general,and for the magneticactivity and sunspot-num- bers in particular. The generalformulae are illustratedby graphs. While the correlation-coefficientbetween the monthly means of the u•-measureand sunspot-numbers for the whole series 1872-1930 is fairly high (4-0.65); there are intervalsfor whichthe correlationvanishes practically.This is the casefor the years1928-30, in whichsolar and terrestrial-magneticactivity appearedto vary independently.Correla- tionsof geophysicalwith solarphenomena may therefore give misleading results,if derivedfrom seriesshorter than oneor two sunspot-cycles. (i) The 27-dayrecurrence phenomenon in the years1906-31 is dis- cussedon the basisof a graphicalday-by-day record, and in relationto solarphenomena. Sequences of recurrences are pronounced, especially in the minordegrees of magneticactivity (international character-figures between,say, 0.8 and 1.6), andare often much longer than observations of recurrentsunspots would suggest. From this discussion we infer the ex- istenceof certainrestricted areas of the Sun'ssurface which are responsible for terrestrial-magneticdisturbances, and whichwe proposeto call M-regions.They appear to bemore long-lived than sunspots. The iden- tificationof the'M-regionswith sunspotsor othersolar phenomena is possiblein somecases only, while in manycases the M-regions lead, so to say, an independentlife. (j) The introductionof thenew character-figures forsolar phenomena (solarindices), available since 1928 for brightand dark hydrogen lines and for calcium-flocculi,does not improve the relations between ter- restrial-magneticand solarphenomena as alreadyobtained by using the relativesunspot-numbers; in particular, the M-regionscannot be coordinatedto any of the solarphenomena which are expressedin the solar indices. Tl:te main reason is found in the strong correlations which exist betweenvarious measures of solar activity, and which de- privethe newsolar indices of theirstatistical independence. 52 J'. BARTELS [vo•..37, No. 11

(k) Observationsof terrestrial-magneticactivity, especiallyof the minor disturbances,reveal persistentsolar influences--recognizedas such by strong27-day recurrences--which cannot at presentbe tracedin the direct astrophysicalobservations of the Sun; in this way, they yield sup- plementaryindependent information about solarconditions. The author is obliged to members of the Department who have given material assistancein the preparationof this paper' Mr. Ennis for his work on the diagrams;Messrs. DuvaI1, Kolar, and Scott for help in the statistical correlationsand in computing, and Mr. Hendrix and Miss Ennis for Figures !8 and 19. I am particularly indebted to Mr. J. A. Flemingfor valuablesuggestions in putting the material and the paper into its final shape. DEPAI•TMENT OF TEI•RESTRIAL MAGNETISM, CARNEGIE INSTITUTION OF WASItlNGTON• Washingtons,D.C.