1912.] 



OF THE GLOBULAR CLUSTERS. 



143 



X. How A Star Entering a Cluster has its Oscillations 

 Damped and is Finally Captured. 



If we recall the familiar equations for an oscillation, as treated in 

 works on physics, 



r) = ae~''^ cos (lit -\- a), or ■,j=zae'''^ sm(nt-\-p), (45) 



where a is the original amplitude of the harmonic oscillation, so that 

 a^"'' ' becomes a coefficient decreasing as t increases, 11 =^ 2Tr/T, T 

 being the period ; we see that as the time f increases the ordinate rj will 

 decrease, though the period T remains constant. The equation 

 (45) thus represents a damped vibration, such as constantly arises 

 where resistance is encountered by vibratory motion. Under these 

 circumstances the harmonic curve rapidly loses amplitude and is of 

 the form : 



Fig. 2. Illustrating damped vibrations. 



The process of damping here brought to light for oscillating 

 particles describing simple harmonic motion has its analogies in the 

 movements of stars in a cluster ; for here too the period of the 

 movement, as we have seen in VIII., is essentially constant, but the 

 amplitude of the oscillation is reduced till it becomes adapted to 

 that of the rest of the system. This is a part of the capture process, 

 because it tends to reduce all the abnormal movements to one dead 

 level. 



Let us now examine the dynamical process by which stars tend 

 to become entrapped in the central region of a cluster. If we con- 

 sider the potential of a spherical shell of stars obeying any law of 



