ASTRONOMICAL LECTURES
BY PROF. ORSON PRATT.
LECTURE FOURTH.
In our last Lecture we demonstrated upon strict mechanical principles, that the gravitating force which preserves the moon in its orbit, would be entirely insufficient to preserve a body at the distance of the sun in an orbit that should be described in one year.
But should any one feel disposed to doubt that the force that binds the moon in her orbit was extended to the sun, and suppose that the sun revolves around the earth under an entire different force from that exerted upon the moon, then we should be under the necessity of searching for some other phenomena to decide the question whether the sun’s annual motion were apparent or real.
Fortunately the great discovery of the aberration of light furnishes us with the most incontestible evidence of the earth’s annual motion around the sun. Roemer, a Danish astronomer, in the year 1667 from a comparison of the observed times of the eclipses of Jupiter’s satelites with their computed times, discovered the progressive motion of light. He found its velocity to be about 192,000 miles per second..- A little over a half century afterwards, Dr. Bradley, an English astronomer of great eminence, commenced a series of observations of great accuracy upon the fixed stars to ascertain, if possible, whether they were subject to any minute apparent changes in their relative positions in consequence of the annual revolution of the earth in its orbit.–He soon found that there were apparent changes constantly taking place; but not such as should result from the different positions of the earth in its orbit. Each star in the heavens seemed to revolve, once a year, in a very small elliptical orbit whose greatest diameter never exceeded 41s of a degree. Those stars situated near the poles of the ecliptic, or at right angles to the plane in which the sun performs its apparent annual motion, appeared to revolve in small orbits very nearly approaching circles. As you proceed from these polar points towards the ecliptic, the eccentricity of these small elliptical orbits seemed to increase. The major axis in all these ellipses was observed to remain constant, being equal to 40.72 sec., while the minor axis seemed to vary in proportion to the latitude of the stars, decreasing as the latitude decreased. A star situated in the ecliptic seemed to oscillate in a straight line, the minor axis being reduced to nothing. These strange phenomena were such as could not be accounted for upon the principle of the annual parallax of the stars, or upon any other principle then known.
Dr. Bradley, after many trials to reduce these phenomena to a general law, at last happily succeeded in discovering the true causes of these curious appearances. He demonstrated that they were the results of the combination of the motion of light with the motion of the earth in its annual revolution around the sun.
If the motion of light were instantaneous, that is, if it required no appreciable time to come from a distant luminous body to the eye, then all these displacements of the stars in the form of little elliptical orbits would entirely cease, and the stars would be seen in their true places, directly in the centre of those ellipses, whether the earth were at rest or in motion; but the velocity of light is an appreciable quantity, and when combined with the velocity of the earth, it appears to come from a different direction from what it would if the earth were at rest.
This may be illustrated in the following manner. Suppose that the wind should blow directly from the north with a velocity of 30 miles per hour; a weather cock or vane on the top of a railroad car at rest would point out the true direction of the wind.– Suppose now that the rail car should be put in motion due east with a velocity of 30 miles per hour, the direction of the wind would no longer appear to be north, but it would seem to come from the northeast. Next let the car go directly to the west with the same velocity; the vane upon the top will now point to the north-west, which will be the apparent direction of the wind. The less the velocity of the car, the less will be the angle of the apparent displacement of the wind from its true direction.–Suppose then that the velocity of the car be about 10,000 times less than that of the wind; the apparent displacement of the wind would then be about 20s of a deg. from its true place.
Now let us apply this principle to the combination of the motion of light with the motion of the earth. Let us suppose the light to come from a star in the pole of the ecliptic with a velocity of 192,000 miles per second; if the earth were at rest, we should see the star in its true place. Suppose next that the earth should be put in motion in the plane of the ecliptic to the eastward with a velocity of about 19 miles per second; we should no longer see the star in its true place, but the rays of light would have an apparent displacement of about 20s of a degree to the to the eastward of its true place. If the velocity of the earth were increased to 192,000 miles per second, the star would apparently be displaced 45 deg., being equal to the apparent displacement of the wind when the velocity of the car was equal to the velocity of the wind. As the velocity of the earth decreases so will the angle of apparent displacement decrease. If the earth should go to the westward in the ecliptic at the rate of 19 miles per second, the apparent position of the star would be 20s of a degree to the westward of its true place.– In whatever direction the earth may be moving in the plane of the ecliptic, the stars will appear to be displaced in a direction parallel to that motion, towards a point in the heavens which the earth, for the moment, seems to be approaching.
Hence, as the motion of the earth is not in a straight line, but nearly in a circle, it is evident that a star situated in the pole of the ecliptic, perpendicular to that motion, must constantly alter its apparent direction as the earth in its orbit alters its direction. And, therefore, it must necessarily have an apparent annual revolution in a very small orbit around its true place, which will be exactly in the centre.
We will suppose next that the wind was blowing from the north-east, while the car was going with the same velocity east, the vane would now point east-north-east; hence the displacement of the apparent direction of the wind would be only one-half what it would have been, had the motion of the car been at right angles to the motion of the wind.– Were the true direction of the wind to the north-west, while the direction of the car was east, the vane would point, if the velocities were equal, to the north-north-west. In this case, the apparent direction would be more northerly than the true direction; while in the other case, it would be more southerly; and in both cases, the wind, by the motion of the car, would seem to shift 22 1.2 degrees towards the point to which the car was moving.
Thus it will be seen that the nearer the true direction of the wind is to the line of the motion of the car, the less will be its apparent displacement. When it blows in the direction that the car is moving, or in the opposite direction, its displacement will be nothing, and the vane will point out its true direction.
Now if the motion of the earth be taken for that of the car, and the motion of light for that of the wind, phenomena precisely of the same kind will happen in regard to the apparent direction of a star situated in the ecliptic; for instance, if, about the time of sunset on the 21st of December we observe a star situated in that portion of the ecliptic which is on our meridian, we will see it in its true place, because the earth will be going in its orbit directly from the star; if the star be observed every day for three months to come, it will be seen to move apparently to the westward, arriving at its maximum distance about the 20th of March next, when the direction of the motion of the earth would be at right angles to the motion of light from the star; it would then gradually begin to recede back again towards its true position; and at the end of three months more, or about the 21st of June, it would be seen in its true position, as the motion of the earth would then be directly towards the star; during the next three months its apparent motion would be to the eastward in the ecliptic, obtaining its maximum distance about the 21st of September, when the motion of the earth would be again at right angles to the motion of the rays of light from the star; and for the next three months it would apparently recede back again to its true place, at the end of which it would be seen in the same position as it occupied one year before. Thus each year it would appear to oscillate in a straight line in the ecliptic, deviating from its true place on each side about 20 seconds.
These phenomena will perhaps be more clearly perceived by supposing a railroad car to be drawn round the circumference of a circle once a year, while the wind blows constantly from one direction, say from the north; let the car start from a point on the east side of the circle towards the south with a velocity 10,000 times less than the wind, which is about the proportion of the earth’s velocity to that of light; when the car starts it will be going directly from the wind, hence the vane will point the true direction of the wind. While the ear gradually describes the first quarter of the circle, it will gradually deviate from a southern to a western direction, during which the weather vane will deviate more and more from the north towards the west, and at the end of three months the direction of the car will be due west, or at right angles to the true direction of the wind, the deviation of the vane will now be at its maximum, pointing about 20s of a degree west of north. As the car describes the next quarter of the circle, the weather vane will gradually recede back again, and at the end of the second quarter the direction of the car would be due north, and the vane would point out the true direction of the wind. In the third quarter, while the car is going from the west to the north point of its circumference, the weather vane would again deviate by degrees to the east of north, and at the end of the third quarter the direction of the car would be due east or at right angles to the direction of the wind; and the deviation of the weather vane would again be at its maximum value, namely 20s east of north. During the last quarter the wind would apparently recede again towards its true position; and having performed one entire revolution the direction of the car would again be due south, and the vane would again point the true direction of the wind. Thus a true north wind would, by the motion of a car in a circle, appear to oscillate each side of its true position, precisely in the same manner that a star in the ecliptic appears to oscillate as the earth moves in its circular orbit.
As all the stars of heaven are affected by the combination of the earth’s motion with that of light, it is evident that their true places cannot be known only as they are deduced from their apparent places. Tables of aberration have been calculated by which these deductions can be conveniently made.
The phenomena of the aberration of light are evidences which can never be controverted in proof of the annual motion of the earth round the sun; for if the sun revolved around the earth while the earth remained at rest, there would be no appearances of aberration.
As we have demonstrated that the earth has an annual revolution around the sun, let us next enquire, what the form of the orbit is? It was supposed for many centuries, during the dark ages, that all the heavenly bodies revolved in exact circles; but modern astronomy has overthrown this conjecture, and has proved that the planetary orbits deviate from the circular form. We shall now point out the process by which this is ascertained.
If the sun be observed at different seasons of the year, he will be seen to vary in his apparent angular diameter. This can be easily determined by measuring with some accurate instrument the apparent breadth of his disc. It will be found that about the first of January the sun will subtend an angle of 32 min. 34.6 sec.; on the first of April, his apparent diameter will be 32m. 1.6s, having decreased 33 sec. in three months: on the first of July his diameter will appear smaller than at any other time of year, being only 31m 30.2s; on the first of October his apparent diameter will be the same as on the first of April. Thus it will be seen that from the first of January to the first of July, the sun decreases in its apparent size;– and from the first of July to the first of January he increases in size. The difference between the greatest and least apparent diameters, is 1m. 04.4s.
Now it cannot for one moment be supposed that the real magnitude of the sun undergoes a periodical change; therefore the difference in his apparent size MUST result from a change of distance. One-half the sum of the greatest and least diameters is equal to 32m. 02.4s, which is the mean diameter or the diameter which the sun gives on the 31st of March and 3d of October in the year 1852. This mean diameter must correspond to the sun’s mean distance from the earth; while the greatest and least diameters correspond to the least and greatest distances of the sun from the earth. If we call the sun’s mean distance 1, then the greatest and least distances may be found by the following proportions: As the sun’s mean apparent diam’r (=32m 02.4s) is to the sun’s greatest apparent diameter (=32s.34.6s.) as the sun’s mean distance (=1) is to the sun’s greatest distance (=1.01675). Also the mean diameter is to the least diameter, as the mean distance is to the least distance (=0.98325). Thus it is ascertained that the greatest, the mean, and the least distances of the sun from the earth are in the respective proportions of the numbers 1.01675, 1.00000 and 0.98325. These numbers are very nearly in the proportion of 1 1-60, 1, and 59-60.
Now if the earth revolved around the sun in a circular orbit with the sun in the centre, his apparent diameter and distance would be precisely the same the year round. But from the above numbers, it will be clearly perceived, that the situation of the sun within the earth’s orbit is ECCENTRIC, the ECCENTRICITY amounting to 0.01675, or nearly 1-60 of the mean distance. These observations and calculations do not demonstrate that the orbit of the earth about the sun is not a circle, but they merely demonstrate that the sun is placed nearly 1-60th of his mean distance from the centre of the orbit.
In order to obtain the true form of the earth’s orbit, let the sun’s apparent diameter be taken when he is at the beginning of each of the twelve signs in the ecliptic, or in other words, observe his apparent diameter for every 30 deg. of longitude; from these observations, calculate the proportional distances corresponding to the apparent diameters, assuming the mean distance equal to 1.00000. With these data the form of the orbit can be delineated upon paper in the following manner: let any point upon the paper be chosen representing the place of the sun; from this point lay off the proportional distances, making an angle with each other of 30 deg., connect the extremities of these distances by continuous curves; it is evident that this will be a correct representation of the orbit of the earth about the sun. The curve thus constructed will be perceived to deviate from a circular figure, being longer than it is broad, that is of an elliptical form. – The point representing the position of the sun, will not be in the centre of the ellipse, but will be in one of the foci at a distance from the centre equal to about 1-60th part of the mean distance of the sun from the earth.
This representation will be still more accurate if the sun’s longitude and apparent diameters be observed a greater number of times during the year; as for instance, every day, and his proportional distances be calculated from the observed diameters according to the above rule; for then, if each of these 365 proportional distances be drawn from any point on a sheet of paper, making angles with each other equal to the observed daily differences of longitude, the extremities of these lines will determine a greater number of points in the continuous curve connecting them, and consequently the form of the curve will be more accurately represented.
The form of the curve may be exactly determined by referring to the properties of the ellipse. If an ellipse be described whose eccentricity is equal to about 1-60 of its semi-major axis, any point in this ellipse may be expressed in terms of its angular distance in respect to the major axis and one of the foci. Now let different points in this ellipse be chosen, corresponding to the observed longitudes of the sun, or to the angle which they make with the earth’s major axis; let the distances of these points from the focus be calculated, and they will be found to coincide most perfectly with those derived from the calculations founded on the measurement of the sun’s apparent diameters.
In this way the elliptical form of the earth’s orbit has been demonstrated, and the amount of its eccentricity determined to a very great degree of exactness. It will be very difficult for those who are unacquainted with the geometrical properties of the ellipse, to fully comprehend these demonstrations; therefore such will be under the necessity of relying upon the testimony of mathematicians until they shall qualify themselves to understand the nature of such demonstrations.
We will now more fully define some terms, that will be of frequent use in our future investigations.
The Mean Distance of a planet from the sun, or of a satellite from a planet, is equal to the semi-major axis of its orbit, or half of the longest diameter; or in other words, one half the sum of its greatest and least distances. The distance from either focus of an ellipse to either extremity of its shortest diameter is equal to the mean distance.
The Major and Minor Axes of an elliptic orbit are respectively the longest and shortest diameters.
The Foci of an elliptic orbit are two points situated in the major axis at equal distances from the centre and at the mean distance from the extremities of the minor axis.
The Eccentricity of an elliptic orbit is the distance from its centre to either focus, – expressed in fractional parts of its semi-major axis.
That point of the elliptic orbit of a planet, which is the nearest to the sun, is called the Perihelion, and the most distant point of the orbit from the sun, is called the Aphelion.
The nearest and most distant points of the moon’s orbit, or of the sun’s apparent orbit about the earth, are called, respectively, the Perigee and the Apogee.
These same points are also called Apsides; the former is called the Lower Apsis, and the latter the Higher Apsis. The line joining these points, or the Major Axis, is termed the Line of Apsides.
The Equator is a great circle of the heavens, equally distant from the two poles, the plane of which is at right angles to the earth’s axis.
The Ecliptic is a great circle of the heavens, the plane of which contains the elliptic orbit of the earth as also the apparent orbit of the sun.
The Obliquity or the Ecliptic is the inclination of its plane to that of the equator, which is equal to 23 deg. 27m. 31s.
The Poles of the Ecliptic are two points in the heavens 90 deg. distant from the ecliptic, the line joining the poles is at right angles to the plane of the ecliptic, and is inclined to the earth’s axis at an angle equal to the obliquity of the ecliptic.
The Vernal Equinox is that point in the ecliptic intersected by the equator through which the sun apparently passes from the south to the north side of the equator.
The Autumnal Equinox is that point in the ecliptic through which the sun apparently passes from the north to the south side of the equator.
The Right Ascension of a heavenly body is reckoned on the equator, and is its angular distance east of the vernal equinox.
The Declination of a heavenly body is its angular distance north or south of the equator.
The Longitude of a heavenly body is its angular distance reckoned eastward from the vernal equinox on the Ecliptic.
The Latitude of a heavenly body is its angular distance north or south of the ecliptic, reckoned in a direction at right angles to the ecliptic.
The Tropics are two smaller circles, situated on each side of the equator at an angular distance of 23 deg 27m. 31s., and whose planes are parallel to the plane of the equator. The northern tropic is called the Tropic of Cancer, the southern, the Tropic of Capricorn.
The Solstitial Points are two points in the ecliptic which touch the Tropics, and are 90 deg. distant from the vernal and autumnal equinox.
The Summer Solstice lies north of the equator; the Winter Solstice lies south.
The Polar Circles are two circles parallel to the Tropics; their angular distance from the poles is equal to the obliquity of the ecliptic. The one to the north is called the Arctic Circle; the one to the south, the Antarctic Circle.
The Poles of the ecliptic are contained in the polar circles, being 23 deg. 27m. 31s. from the centres of the polar circles; these centres are the Poles of the earth prolonged to the heavens.
Having proved that the earth has an annual motion around the sun in an elliptic orbit, and that the sun is not situated in the centre of the ellipse, but in one of the foci: and that the eccentricity of the orbit, or the distance of the sun from the centre of the ellipse is equal to nearly 1-60th of his mean distance from the earth, we shall next proceed to investigate the law of the angular velocity of the earth around the sun; this will evidently be the same as the apparent angular velocity of the sun around the earth, supposing the earth and sun to exchange places.
Now let us suppose that the real velocity of the earth, in its elliptic orbit was uniform, it is evident that its angular velocity around the focus of the ellipse would be different at different distances; that is, the greater the distance, the less the angular velocity. A body moving at right angles to the line of vision at twice the distance with a uniform motion, would have one-half the angular velocity; at three times the distance, one-third the angular velocity, and so on. Now when we observe the apparent angular velocity of the sun, or, which is the same thing, his daily change of longitude in different parts of his apparent orbit, we find that about the 31st of December, when the sun is nearest to the earth, his apparent angular velocity is the greatest, amounting to 1 deg. 01m. 09.7s in 24 mean solar hours; and about the 1st of July, when he is the most distant from the earth, his apparent angular velocity is the least, being only 57m. 10.2s. in a mean solar day. The average change of longitude in a day is found by dividing 360 deg. by 365.24224, which is the number of mean solar days in a tropical year; the quotient amounts to 59m.08.33s. Thus it will be perceived that from the perigee to the apogee, the sun’s apparent angular velocity decreases, and from the apogee to the perigee it increases. Does this variation depend wholly upon a change of distance, or is there actually a change of real velocity in different parts of the orbit?
This question may be determined by comparing the rate of variation in the angular velocity with the rate of variation in the distance. If the mean distance, and also the mean angular velocity be each assumed equal to unity or 1.00000, then the extremes of distance will be 1.01675, 0.98325, and the extremes of angular velocity will be 1.03420, 0.96671.
By a comparison of these numbers, it will be seen that the deviation of the angular velocity from the mean is much greater than the deviation of the distance from the mean. Therefore the rate of variation of the angular velocity must be much greater than what would result from a mere change of distance alone; hence the excess must be dependent upon a real change of velocity.
If the extremes of distance be compared with the extremes of angular velocity, the latter will be found to be nearly equal to the inverse squares of the former; they would be quite equal were the observations from which they were deduced perfect. And if we compare the angular velocities at any other points of the earth’s orbit they will be found to vary exactly as the inverse squares of their respective distances from the sun. The real motion of the earth, therefore, in its orbit, cannot be uniform; its actual velocity decreases from the perihelion to the aphelion, and increases from the aphelion to the perihelion. At corresponding points on each side of the major axis, its velocity is equal.
The law of the angular velocity having been determined to be as the inverse square of the distances, we will next investigate the law of its actual velocity.
If we suppose a line drawn from the earth to the sun, it is evident that it will sweep over the whole surface or area of the elliptic orbit in one year. — This line is called the Radius Vector. Now it has been determined by observation that the radius vector moves over equal areas of the ellipse in equal times. If the velocity of the earth were uniform, this could not take place; for the radius vector as it increased in length would, with equal velocities, describe an increased area; therefore as the radius vector increases in length, the velocity of the earth must decrease in such a proportion as to have the areas, swept over in equal times, exactly equal. – Consequently the areas described must be proportional to the times. This is the law of the actual velocity of the earth in its orbit.
[Transcribed by Shannon Devenport, Marlene Peine, and Mauri Pratt; May 2012]
Orson Pratt, Astronomical Lectures: Lecture Fourth. Unknown newspaper, 21 February 1852.