34 PUBLICATIONS of the CELESTIAL CARTOGRAPHY By

34 PUBLICATIONS OF THE

By Roy S. Farmer

Abstract.—The purpose of this discussion of celestial cartography is to increase the appreciation of the proper application of maps to the demonstration or interpretation of problems relating to the distribution of celestial objects. The divisions of celestial mapping considered in the paragraphs that follow may be classified into five groups: (1) The principles of the projection of maps on to the developable surfaces of planes, cylinders, cones, and mathematical figures ; (2) the relevant features of selected map projections ; (3) the uses of celestial maps ; (4) obtaining the desired co-ordinate system of maps of the celestial sphere ; (5) a few suggestions concerning the construction of maps; (6) bibliography.

PRINCIPLES OF MAP PROJECTIONS

Since the continuously curving surface of the imaginary celestial sphere cannot be developed graphically or mathemati- cally into a continuous plane surface (map) without distortion of one or more of several involved irreconcilables, it is apparent that any sphere-to-plane projection must be a compromise between the elements of the projection. Certain types of maps may show correctly the scale or area or azimuth of plotted objects. But a single map cannot present, in a wholly undistorted form, all three elements of the ideal map. For this reason, the choice of a projection to be used for any one map is arbitrary and may be determined only when a desirable balance is found between the type of data to be shown, the area to be covered in one map, and the most relevant feature (scale, area, or azimuth) to be demonstrated. The most simple types of projection, which are adapted only to the presentation of small angular inclusions of a few degrees in declination and right ascension, are those of a plane tangent to, and receiving the projection from, the celestial sphere. Fig- ure 1 represents four projections fulfilling such a set of conditions. In each of the examples shown, the plane tangent to the sphere receives the projection in a manner that is determined by the location of the point of origin of the projection. The Gnomic projection, which has its point of origin at the center of the sphere, projects the true distances from the sphere to

the plane in a distorted manner so they become too large when measured on the plane, as will be seen by the difference in length between the tangent solid line (representing a tangent plane)

METHODS OF PROJECTION

\SX/

/a/ // \ ' u 1 \ >' 1

V/ /

GNOMIC STEREOGRAPHIC

\ V// ι ! Κ \\ i / U \ i / / / \ L'

CLARKE'S ORTHOGRAPHIC Fig. I.—For simplifying the conditions under which tangent plane projections may be demonstrated, the tangent solid lines in this figure represent two-dimensional planes in contact with a pole of the spheres. In practice, the sphere is imaginary and the pole of projection (point of tangency) may be placed at any point on the imaginary sphere. The out- lined bar above each method of projection indicates the true length of the projected arc when measured on the surface of the sphere. The Gnomic projection distorts co-ordinates most, while Clarke's projection is one of minimum error.

and the hollow line just above it which represents the true linear arc distance when measured on the surface of the sphere of the projection. The Stereographic projection places the point of origin at the sphere's surface opposite to the pole of projection between sphere and plane. Distances projected by the Stereographic method are still too great, but the error of dis- crepancy is decreasing, and in Clarke's projection is found the minimum distortion of scale. The fourth type of projection shown in Figure 1 is the Orthographic, which is based on the use of parallel lines to carry the projection from sphere to plane. In this method of construction the projected distances are smaller than their true lengths on the sphere. From a consideration of the trigonometric functions of a plane tangent to a sphere, it is obvious that the errors of scale in such a projection relationship are dependent upon the distance of the boundary of the projection from the point of tangency (pole of projection). It is for this reason that the use of a plane tangent to a sphere is limited to the mapping of small areas of say fifty square degrees placed concentrically about the pole of projection. A simple expedient to increase the area of a map that may be included within the limits of an allowable distortion boundary is the forming of a tangent plane into a cylinder rolled about the sphere, usually with its line of tangency congruent with the equator of the sphere, as shown in the left half of Figure 2. The use of the tangent cylinder extends the length of practicable projection through 360° of right ascension, although it does not increase the range of allowable error of projection in declination. A cylindrical projection will permit an area of 7° in declination by 360° in right ascension to be mapped with a scale error in declination of only one-half of one per cent. With a more generous error allowance, an area of 3600 square degrees (10° X 360°) to 18,000 square degrees (50° X 360°) may be shown on a single map. Should the scale errors of a tangent cylinder projection exceed two or three per cent, the cylinder may advantageously be placed secant to the sphere as .is shown in the right half of Figure 2. This method of placing the cylinder creates two

standard parallels along which projected scale errors are zero and between which the scale is too small and outside of which the scale is too large. When a cylindrical projection is carried into high latitudes the projection becomes so distorted that, to restore conformality, an adaption of the cylindrical surface becomes desirable and the cylinder is replaced by a cone or by several cones with radii that are successively shorter as higher and higher latitudes are mapped. In the left half of Figure 3 is shown a single cone tangent to the sphere. In the right half of the same figure the cone is placed secant to the sphere. The use of several adjoining cones instead of the single one shown in the figure permits the surfaces which receive the projection to conform more closely

CYLINDRICAL PROJECTION CONICAL PROJECTION Fig. 2 Fig. 3

Fig. 2.—The cylinder which is to receive a projection from a sphere may be placed tangent (left) or secant (right) to the imaginary celestial sphere. The secant projection creates two standard parallels along which scale errors in right ascension are zero. Mercator's projection is based on the use of a tangent cylinder and the Gnomic method (Fig. 1) of carry- ing co-ordinates from the undevelopable surface of the sphere to the developable surface of the cylinder. Fig. 3.—The use of a cone instead of a cylinder for projections of areas including high latitudes is a particular purpose adaption of the cylinder and may be considered as such regarding the placing of the cone, or cones, in relation to the imaginary sphere which they will enclose.

MOLLWEIDE

. SINUSOIDAL Fig. 4.—The Mollweide and Sinusoidal (sometimes called Sanson- Flamsteed) projections of the entire sphere. The parallels are straight, equidistant lines and the meridians are located by dividing each parallel into the same number of parts equally spaced between , the V axis of the projection and the bounding ellipse. The Sinusoidal projection has straight, equidistant parallels crossed by sine curves of differing amplitudes which represent the meridians of right ascension.

to the surface of the sphere. This arrangement produces some of the most accurate projections possible between an undevelopable surface (sphere) and a developable one (cone). Since two adjoining cones of differing radii will be in complete edge contact only when they are formed as cones, it is necessary, upon their being flattened into planes (maps), to adjust the scale of the projection on each cone until the adjacent edges of succes- sive flattened cones are in contact throughout their length.

AITOFF EQUAL-AREA

Fig. 5.—AitoiFs Equal-Area projection preserves the equality of area between the spherical co-ordinate quadrangles and the quadrangles of the projected representation of the sphere. The co-ordinate network of this projection is constructed from formulae which are solved and assembled in usable form in Special Publication 68 of the Coast and Geodetic Sur- vey. The network in printed form may also be purchased from the Survey.

To the astronomer interested in stellar distribution in areas of limited size, not greater than a hemisphere, the methods of the plane, cylinder, cone, or polyconical map construction will furnish co-ordinate networks fulfilling his requirements. Very often, however, an analysis of distribution demands one continuous unserrated representation of the entire celestial sphere ex- tending 180° in declination and 360° in right ascension, a total of approximately 41,250 square degrees. To construct a map that

COMPARISON

OF SELECTED QUADRANGLES

NORMAL

AITOFF EQUAL-AREA

SINUSOIDAL

Fig. 6.—This romparison of configurations of quadrangles from the sphere and from three projections of the sphere demonstrates the shape of selected quadrangles from several points along the galactic equator when it is projection on to the equatorial co-ordinates of the Aitoff, Moll- weide, and Sinusoidal projections. It is apparent that the Aitoff projection at least partially preserves the right-angled relationship between the parallels and meridians of a sphere.

will cover as large an area as this, it becomes necessary to use a mathematical means of plotting the X axis and V axis loci of the co-ordinate system. Figure 4 shows two mathematical co-ordinate structures : Mollweide's, based on the figure presented by an ellipse having a ratio of 2:1 between its major and minor axes; and the Sinusoidal projection, a figure consisting of sine curves of several amplitudes sheared 180° along their X axis. AitoiTs Equal-Area projection (Figure 5) is the co-ordinate system of greatest use to astronomers because of its partial retention of the shape of co-ordinate quadrangles upon their being transferred from sphere to projection figure. Figure 6, the Mollweide, Sinusoidal, and Aitoiï projections, will be considered individually in the paragraphs under the following sub- title.

RELEVANT FEATURES OF PROJECTIONS ; USES OF CELESTIAL MAPS

From the preceding summary of the basic principles of map projections, the astronomer merely wishing to plot data of distribution for interpretation or demonstration may logically conclude that celestial cartography is a complex balance of com- promises. Table I will serve as an aid in clarifying the status of ten types of projections and their characteristics. Of the ten projections, two are for use in mapping small areas and are based on the tangent plane method of construction; five are for areas of medium size or various extents in declination and right ascension and are constructed on the cylindrical or conic prin- ciple ; and three are for showing areas of great angular extent in declination and right ascension and are more or less satisfactory mathematical constructions of the co-ordinate system of an entire sphere. Strangely, the scope of most celestial maps tends to include only a very small or a very large area; the middle range of included area of, say, a hemisphere, is seldom used by astronomers. Considering first the maps covering small fields of 25 square degrees or less, there are two types of projections that will be of use in mapping stellar phenomena of distribution. The azimuthal equidistant projection is of value because of its

'Ö 'Ö'Td T-jtrtw (V (υ<υ h h'-· NN Ν Ν ,Ν ξ u u '? '? "τ "? 'τ Λ .Ä ^ O ^ artaoj ^oJ art^jrt ; ^ .^(D.ria;^(L> .¿(U^íU Vh l·-!ω uα 2ω gçti gΛ (L>rta->nj <υπ} OViOH'riα>α}<υα3 ^ ·^ΓΙ ·"gΓΗ ^WOT WWW

cd a o ci 1) '€ Ö ^ o Λ '.2■Μ ^ cd O 1)Ü ο <υ O O H On >% 13ω Ü2 G -Mrt Tj I—IO .S 3 H Λ g U § I t-l T, w 4) <ί en «Λ a <$ t—> m <υ ω o o <υ O o o o >Η >Η PH o c/) I—IU u. ^ H α; Ο 0 0 0 s I 0 0 0 0 to u-j _i, ^ ^ ^ 3 5 £ J? S ^ ^ >η ^ !z¡ w en 1 I H υ ο .2 cr rt αϊ <υ Ph C <υ .Β U ' 71 ^ ai α Λ a ω —» « ä 4-> > 'û .S a ai 3 1 π 1 ε ι Η _ CT er <υ ^ Ν >> (U y ^ as o 13 s ^ cS ο » 3 'rt Α ι (Λ cr (U cd Ο U Ul <υ JO T3 -5 τΰ υ -4-·Ο t: 0 'o π s ^ rt (U ^ s c C υ #o ^ u .ÍH ^ 'S Ο Vh Q

preservation of scale in declination and azimuth relations between any plotted point and the pole of projection of the map. The parallels of declination in this projection are spaced so they maintain the exact scale of the parallels of declination of the sphere from which the projection is made. The constant spread of the meridians of right ascension prevents the map from showing the true scale of the sphere in the direction of longitude. A typical use of the azimuthal equidistant projection is in the mapping of the members of a globular cluster where the feature to be demonstrated is the shape of the cluster about its zero point (center). The configuration of a line connecting points of equal density of distribution will be a direct function of the distance and angle of the members as referred to the center of the cluster. The azimuthal equal-area projection sacrifices the preservation of exact scale but it does maintain the correct azimuthal relations and proportionality of area between sphere and figure of projection.. The co-ordinate systems of either the azimuthal equidistant or azimuthal equal-area projections are very simple to construct when their pole of projection is congruent with a pole of the galactic or equatorial celestial sphere {polar azimuthal equidistant or polar azimuthal equal-area maps) but if the pole of projection is centered elsewhere on the sphere, the co-ordinate form becomes more complicated and is then most easily constructed by the use of computed tables. (See the Bibliography at the end of this article.) Disregarding the seldom-used projections of areas of medium size, at the macroscopic end of the astronomer's carto- graphic interests are the mathematical constructions allowing the 64,800 square degrees of the entire celestial sphere to be shown in one continuous unserrated figure of convenient form. Of the two types of mathematical constructions shown in Fig- ure 4, and the Aitoff Equal-Area projection shown in Figure 5, two, the Mollweide and the Sinusoidal projections, have none of the three attributes of an ideal map which, as mentioned above, are preservation of the relations between scale, area, and azimuth. The Aitofii Equal-Area projection does preserve the single property of presenting co-ordinate quadrangles equal in

area, but not in shape, to the quadrangles of the sphere from which the projection is taken. But the Aitoff projection does not maintain scale in declination or right ascension, nor does it maintain the proper azimuth of plotted objects. The meridians and parallels of the Aitoiï projection cross more nearly at right angles (as on the sphere) than do the co-ordinates of the Moll- weide and Sinusoidal projections. It is this desirable property that imparts to the Aitoff projection greater value than the other two mathematical co-ordinate structures of the celestial sphere (Fig. 6). It appears, then, that no projection is entirely satisfactory for showing the whole celestial sphere in one map. But it is to be noted that the distribution of stellar objects throughout the universe is such that many of them appear to be included within a space of lenticular shape centered in a general way about the galactic plane. Figures 4 and 5 demonstrate the shapes acquired by the galactic equator when it is projected on to several methods of representing the equatorial co-ordinate system. In'projecting the galactic co-ordinates upon the equatorial system, the galactic equator is shaped into a regular curve reaching the rather high, and consequently unsatisfactory, latitude of 62° north and south of the equatorial system equator. The galactic plane may be brought to the less-distorted central region of a map by the simple process of redesignating the values of the co-ordinates of declination and right ascension of the equatorial system to new values on the galactic system and plotting the data of distribution on the galactic rather than the equatorial co-ordinate network. Perhaps the most concentrated use of maps of the celestial sphere to be found is in Handbuch der Astrophysik, Band VI, Erster Teil. Following is a list of several of the types of projections used in that volume. Under each projection is listed a number of applications of the projection to the actual charting of data of distribution. 1. Azimuthal equidistant projection a) Galactic distribution of large spiral nebulae and globular clusters b) Star density (in Messier 19)

2. Mollweide's projection a) Distribution of spiral nebulae h) The Milky Way

3. AitoiTs Equal-Area projection a) Intensity of calcium lines in various parts of the sky b) Distribution of clusters and galaxies c) Galactic distribution of spiral nebulae based on radial- velocity determinations d) Distribution of planetary, dark, and diffuse nebulae e) Distribution of globular clusters (in galactic co-ordinates) /) Iso-photometry of the Milky Way

4. Equally spaced meridians and parallels, cylindrical projection a) Distribution of helium stars h) Structure of the Milky Way c) Iso-photometry of the Milky Way d) Distribution of globular clusters in the F-Z plane e) Distribution of globular clusters in the X-Z plane ./) Distribution of globular clusters in the plane of the galaxy

OBTAINING A CO-ORDINATE SYSTEM

Tpie first problem to confront the celestial cartographer after his choice of projection has been determined by a consideration of the data to be presented, the area to be covered in a single map, and the most relevant feature to be retained, is the method of obtaining the chosen co-ordinate system upon which he will plot his material. The most basic method of constructing a co- ordinate network is to project it graphically from a sphere, but the inaccuracies that would result from such an operation would make it of little value. A second method of construction is by plotting the numerical values of computed tables, which may be of purely mathematical origin, or by the graphic plotting of X axis and Y axis loci derived from formulae. Tables computed from formulae will be found in the Publications of the United States Coast and Geodetic Survey listed in the Bibli- ography at the end of this article. The tables published by the Survey are based on the use of a spheroidal figure of the Earth

rather than on the true sphere, but the error introduced by the use of these tables will be so small as to be entirely negligible in the plotting of the celestial sphere. A third method of obtaining co-ordinate systems is to adapt them from maps published by commercial map makers. Commercial maps are printed in a wide variety of sizes, prices,.types of projections, and location of the poles of projection. Small maps may be obtained in a convenient 834 X 11-inch size from several companies specializ- ing in maps for school use. The cost is usually less than five cents each. Since the co-ordinate system is the only feature of a commercial map desired by the celestial cartographer, the network of meridians and parallels may be traced or scaled from the printed map, omitting the continental outlines. If the published map is centered correctly in latitude but not in the desired longitude, the values of the meridians of longitude (right ascension) may be redesignated to new values, bringing the desired longitude into the scope of the map. The values of parallels of latitude (declination) cannot be changed without chang- ing the form of the parallels. The fourth and most simple way to obtain a co-ordinate system without the continental outlines of a terrestrial projection is to purchase printed copies of the network. Unfortunately, the only type of projection that is obtainable in printed form, without continental outlines, is AitoiFs Equal-Area projection, which may be purchased from the Coast and Geodetic Survey, Washington, D.C. The size of the printed co-ordinates is approximately 18χ25 inches, and the cost of the map, about 15 cents, is very reasonable considering the long and tedious job of hand drafting it from computed tables.

SUGGESTIONS CONCERNING THE CONSTRUCTION OF MAPS

A few suggestions in outline form will serve to recapitulate the practical ¾sρects of celestial cartography: 1. For maps of small fields use: a) Azimuthal equidistant or l·) Azimuthal equal-area projections 2. For maps of medium-sized fields use : a) Cylindrical projections for long narrow maps of equatorial regions

h) Conic projections for long narrow maps in high latitudes c) Polyconic projections for long wide maps in high latitudes 3. For maps of the entire celestial sphere use : a) AitoiFs Equal-Area projection or h) Mollweide's projection 4. Use galactic co-ordinates for data based on galactic distribution 5. Co-ordinate systems may be: a) Drawn from loci given in computed tables h) Adapted from published maps c) Purchased (AitoiTs Equal-Area projection only) 6. Use strong lines for co-ordinates, neat, good lettering for notes, and clear, simple symbols for indicators if the map is hand drafted. A well-made map will maintain confidence in correctly plotted data; a poor or careless construction de- stroys confidence in a map without regard for the value of the data presented. In concluding this discussion of celestial cartography, I wish to express my appreciation to Mr. V. Krat for several conversations concerning the mathematical basis of map projections.

BIBLIOGRAPHY

All of the publications of the Coast and Geodetic Survey listed below are obtainable from the Superintendent of Docu- ments, Washington, D.C. The prices listed are subject to revi- sion. Special Publication 5. Tables for Polyconic Projections, 30 cents Special Publication 38, Serial 47. Elements of Chart Making, 40 cents Special Publication 47, Serial 77. Lambert Conformai Conic Projection, 25 cents Special Publication 49, Serial 82. Lambert Projection Tables, 25 cents Special Publication 57, Serial 110. General Theory of Polyconic Projections, 25 cents

Special Publication 68. Elements of Map Projection, 75 cents Special Publication 130, Serial 378. Tables for Albers Projec- tion, 5 cents Of the seven Publications listed, Special Publication 68 is probably the most valuable title for astronomers and celestial cartographers.

Los Angeles, California December, 1937