10 



INTENSITY OF SUN'S HEAT AND LIGHT. 



SECTION II. 



LAW OF THE SUN'S INTENSITY UPON THE PLANETS IN RELATION TO THEIR ORBITS. 



The preceding Section represents the Sun's action upon a distant planet at a 

 given distance, or at rest. It is here proposed to examine the effect when the 

 distance is variable ; that is, supposing the planet to commence its motion from a 

 state of rest, in an elliptical orbit, to determine the intensity received during its 

 passage through any part, or the whole of its orbit. 



In the annexed figure, let S denote the Sun situated in one focus ; P the Planet's 



position at a given time ; A, the perihelion 

 or point in the orbit nearest the sun, and 

 B, the aphelion or point farthest from the 

 sun. 



Let SP or p denote the radius-vector; 

 A S P or 0, the true anomaly ; e, the ratio 

 of eccentricity; and a+nt, the mean ano- 

 maly ; n, being the mean motion in the unit 

 of time. 



If A denote the semi-transverse axis, it is 

 well known that A 2 n </ 1 — e 2 will express 



the whole area of the ellipse, and J \ p 2 dd, 



the area of the elliptic sector corresponding to 0, where 7t denotes 3.14 1592, or a 

 semi-circumference. Hence by Kepler's law, that equal areas are described by the 

 radius-vector in equal times, 



A 2 it V 1 — e 2 : J i p 2 d d : : 2n : a + nt. 



Reducing to an equation and differentiating, 



1 " (80 



p" A 2 ndt -v/ 1 — e 2 

 Since heat and light vary inversely as the square of the distance p, the second 

 member evidently measures their intensity at any instant. Then, as pointed 

 out in the Calculus, we may regard the second member as the ordinate, and 

 the time t as the abscissa of a curve. Multiplying the equation by dt, there- 

 fore, and integrating between the limits of any two anomalies, and 6', we obtain 

 for the sum of the intensities, 



(8.) 



Si 



dt = 



P" A 2 n V 1 — e 2 ' 



In interpreting this result, we know that the orbitual motion of a planet is not 

 uniform, being accelerated in perihelion and retarded in aphelion. Hence, in the 

 annual variations of radius- vector, the Earth does not receive equal increments of 

 heat and light in equal times ; but the amount received in any given interval, is 

 exactly proportional to the true anomaly or true Longitude described in that interval. 



