But it is desirable to know what sort of influence the Sun would exert even at distances beyond the limits of his direct control. Let us set ourselves to form adequate conceptions of the Sun's energy of gravitation. The measure of all gravitation is that force which the Earth exerts as we know to draw bodies to her surface. We must therefore first ascertain what proportion the energy of the Sun's attraction bears to that familiar attraction exerted by the Earth. For this purpose we may proceed either directly by comparing the amount of velocity which the Earth's attraction communicates to falling bodies, with the actual motion towards himself which the Sun causes in the case of any planet; or indirectly by comparing the motion of any planet round the Sun with the Moon's motion round the Earth. I select the second method as being the simplest.* I adopt also a way of apply wholly away from his dominion. It follows that, setting aside the abovementioned influences, a comet which travels in an unclosed orbit round one star can never travel in a closed orbit round any star, but will continue to flit from star to star through all time. *It is worthy of notice, however, that the other method, though seldom applied, is in effect quite as available as the method in which reference is made to the Moon's motion. Thus we know that in one second a body falls about 161 feet towards the Earth, acquiring a velocity of twice this number of feet, or 32.2 feet per second. Now the Earth circuits the Sun at a rate of about eighteen miles per second, and supposing e é (fig. 19) to represent this distance on the orbit e é E, then eL, obtained by drawing a perpendicular from e' on the diameter e s E, represents the amount by which in effect the Earth has been drawn towards s, and a velocity of twice e L per second measures the Sun's gravity at the distance se. But by Euclid vi. 8, ee' is a mean proportional (ee) between e L and e E; in other words, e L is equal to e E and therefore ing it by which certain difficulties of conception are removed. The Earth at a distance from the Moon of about 238,800 miles has power to change the direction of the Moon's motion through four right angles in 27-322 days, the Moon moving in her orbit round the Earth the Sun's gravity at e is measured by a velocity of or in feet (18)2 x 1760 × 3 91,500,000 (e e') es per second, Hence at a distance equal to the Earth's radius, the Sun (supposing all his mass collected at one point) would exert a force represented by a velocity of about (18) × 1760 × 3 × 91,500,000 (4,000)2 feet per second, which reduces to 9,783,180 feet (or about 1,853 miles per second). Comparing this with the measure of the force of terres FIG. 19. trial gravity, or 32-2 feet, we see that the Sun's mass must be, according to this rough process, fully 300,000 times as great as the Earth's. As a matter of fact it is estimated at 315,000 times the Earth's, a result we should have closely approximated to had we taken the Earth's radius at 3,960 instead of 4,000. The following general theorem is often useful. Take r the Earth's radius, E her mass and g terrestrial gravity, R the radius of a circular orbit described by a body of mass m, about a larger body of mass M with velocity v (in feet per second). Then the attraction between the bodies, or the gravity of m towards м, is represented by the expression v2R and the mass м+m is equal to E. If m is relatively very small r2g v2 Ꭱ 238.800 27.322 with a velocity which we may represent by Now the Sun at a distance from the Earth of about 91,500,000 miles has power to change the direction of her motion through four right angles in 365-256 days, the Earth moving in her orbit with a velocity which we may represent by Now clearly, since gravity varies inversely as the square of the distance, the Sun would require (were other things equal) to have an attractive power greater than the Earth's in the ratio 91,500,000 238,800 91,500,000 365 256⚫ (100) to produce the same effect on her that she produces on the Moon; and secondly, since to change the direction of a body's motion through any angle is a work which will be done at a rate proportioned to the force which operates, it is clear that the Sun's attractive power would have (were other things equal) to be less than the Earth's, in the ratio, to accomplish in one sidereal year what the Earth accomplishes in one synodical month; while lastly, since the faster a body moves the greater the force necessary to deflect its course through a given angle, it is obvious that the Sun's attractive power would have (were other things equal) to be greater than the Earth's in the proportion of 91,500,000 to that is, in the ratio 91.500.000×27-322 -to 365-256 238.800 238.800x365*256 (as in the case of the Earth compared with the Sun), and the radius of the larger mass is p, then gravity at the surface of the larger mass is equal to v2.R *We need not trouble ourselves to determine the velocity in miles per second, or minute, or hour; because relative and not absolute velocities are in question. Hence we can represent the Moon's velocity by the radius of her orbit divided by the period, provided we represent the Earth's velocity in like manner. produce a given change on the quickly moving Earth in the same time that the Earth produces such a change on the less swiftly moving Moon. Now we have only to combine these three proportions, which take into account every circumstance in which the Sun's action on the Earth differs from the Earth's action on the Moon, in order to deduce the relation between the Sun's real attractive energy, and the Earth's (at equal distances from the centre of either). This gives the proportion which reduces to 314,798, in (91,500,000)3 (238.800)3 (27-322)2 which proportion the Sun's mass exceeds the Earth's. We may take 315,000 (the value given in tables of the elements) as representing in round numbers the true proportion, which as it depends on the Sun's distance, cannot be determined so accurately that the last three figures of the number can be regarded as significant. At equal distances, then, the Sun exerts 315,000 times as much force on any body as the Earth. So that if the Earth's mass were as great as the Sun's, her dimensions remaining unchanged, a mass which now weighs one pound, would weigh more than 143 tons. A man now of average weight would be crushed down by a weight of more than 20,000 tons. A body, if raised but a single inch and let fall, would strike the ground with a velocity three times as great as that of the swiftest express train. But now that we have thus ascertained the proportion which the Sun's attractive energy bears to that exerted by the Earth, and so are able to measure the Sun's might as ruler over his system by direct comparison with the familiar force of terrestrial gravity, we must endeavour to form an estimate of the extent of force exerted by the Sun at different distances. We are to inquire what are the limits of the Sun's effective reign, not as regards distance alone, but as regards also the activity of matter,—that is, the velocity with which it is travelling. We may begin with the Earth. We know that the Earth is completely subject to the Sun's attraction. Notwithstanding the inconceivable velocity with which she moves, and therefore the inconceivable energy of the tendency she has to travel onwards in a right line and so to free herself from the Sun's control, she is compelled to travel in a nearly circular course around him. At one time her velocity has reached its maximum and she has power for awhile to increase her distance from the Sun. But she has derived that very power from him. Anon her speed is reduced to its minimum, and then she is compelled slowly to approach the ruling centre. But throughout her course there is one constant relation from which there is no escape. The Earth's velocity and distance are the two quantities which measure the extent of the Earth's partial freedom. When one is reduced, the other is increased, and vice versâ, in such sort that there is absolutely no change in her condition regarded as depending on these two combined relations. The one law from which there is no escape, let distance and velocity change as they may during the Earth's circuit |