﻿192 
  H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  

  

  actually 
  intersect, 
  and 
  the 
  comet 
  in 
  its 
  unperturbed 
  orbit 
  

   reaches 
  the 
  point 
  of 
  intersection 
  later 
  than 
  the 
  planet 
  and 
  

   when 
  the 
  planet 
  is 
  distant 
  therefrom 
  0'15174. 
  The 
  semi-trans- 
  

   verse 
  axis 
  of 
  the 
  resulting 
  hyperbolic 
  orbit 
  about 
  the 
  sun 
  will 
  

   be 
  36-90. 
  

  

  20. 
  Resulting 
  orbits 
  of 
  maximum 
  perturbation. 
  — 
  The 
  posi- 
  

   tion 
  of 
  the 
  relative 
  orbit 
  about 
  Jupiter 
  in 
  these 
  cases 
  of 
  maxi- 
  

   mum 
  perturbation 
  for 
  given 
  values 
  of 
  co 
  is 
  easily 
  determined. 
  

   From 
  the 
  equations 
  (7), 
  (6) 
  and 
  (15) 
  

  

  tan 
  a=B/A=A 
  sin0/A=sin 
  0/(cos 
  0±1). 
  

  

  The 
  positive 
  sign 
  gives 
  2a 
  = 
  0, 
  and 
  the 
  negative 
  sign 
  gives 
  

   2a=7r 
  + 
  6. 
  But 
  the 
  angle 
  2« 
  in 
  the 
  first 
  case 
  is 
  the 
  angle 
  of 
  

   the 
  asymptotes 
  enclosing 
  the 
  branch 
  of 
  the 
  hyperbola 
  described 
  

   about 
  Jupiter 
  by 
  the 
  comet. 
  Since 
  the 
  two 
  original 
  orbits 
  

   intersect, 
  the 
  plane 
  of 
  the 
  relative 
  orbit 
  contains 
  the 
  planet's 
  

   path, 
  so 
  that 
  the 
  comet 
  passes 
  directly 
  in 
  front 
  of 
  the 
  planet 
  

   and 
  being 
  turned 
  backward 
  leaves 
  Jupiter 
  exactly 
  in 
  the 
  direc- 
  

   tion 
  of 
  Jupiter's 
  quit.* 
  The 
  place 
  of 
  encounter 
  with 
  Jupiter 
  

   will 
  be 
  near 
  an 
  apse 
  of 
  the 
  comet's 
  resulting 
  orbit 
  about 
  the 
  

   sun. 
  The 
  comet 
  leaves 
  the 
  planet 
  with 
  the 
  relative 
  velocity 
  

   v 
  , 
  so 
  that 
  if 
  s<^l 
  the 
  motion 
  about 
  the 
  sun 
  in 
  the 
  new 
  orbit 
  

   will 
  be 
  direct; 
  if 
  s>l 
  the 
  motion 
  in 
  the 
  new 
  orbit 
  will 
  be 
  

   retrograde. 
  That 
  is, 
  by 
  (4) 
  when 
  co 
  <^^tt 
  the 
  resulting 
  motion 
  

   is 
  direct 
  ; 
  when 
  co 
  > 
  \ir 
  the 
  resulting 
  motion 
  is 
  retrograde. 
  

  

  In 
  the 
  second 
  case 
  the 
  angle 
  2a, 
  being 
  greater 
  than 
  180°, 
  

   stands 
  for 
  the 
  angle 
  between 
  the 
  asymptotes 
  exterior 
  to 
  the 
  

   orbit. 
  Hence 
  the 
  comet 
  passing 
  behind 
  the 
  planet 
  will 
  be 
  

   turned 
  forward 
  and 
  will 
  leave 
  the 
  planet 
  in 
  the 
  direction 
  of 
  

   Jupiter's 
  goal, 
  and 
  have 
  a 
  velocity 
  that 
  will 
  send 
  it 
  perma- 
  

   nently 
  out 
  of 
  the 
  solar 
  system. 
  

  

  21. 
  The 
  results 
  of 
  Art. 
  20 
  assume 
  that 
  co 
  is 
  given. 
  To 
  

   find 
  for 
  what 
  value 
  of 
  co 
  the 
  period 
  of 
  the 
  resulting 
  orbit 
  is 
  

   the 
  shortest 
  possible 
  we 
  may 
  put 
  As*=mr 
  and 
  1 
  — 
  s 
  2 
  =2s 
  cos 
  

   in 
  (15) 
  so 
  that 
  

  

  w 
  1 
  — 
  s- 
  ± 
  2s 
  

  

  To 
  find 
  the 
  minimum 
  for 
  @ 
  place 
  — 
  =0 
  in 
  this 
  equation. 
  

  

  This 
  gives 
  s=±l, 
  in 
  which 
  result 
  since 
  s 
  is 
  inherently 
  positive 
  

   only 
  the 
  positive 
  sign 
  is 
  used. 
  But 
  when 
  *=1, 
  @=-Jr, 
  h=mr 
  

   and 
  (o=\tt. 
  Hence 
  tke 
  greatest 
  effect 
  of 
  perturbation 
  of 
  a 
  

   planet 
  momng 
  in 
  a 
  circular 
  orbit 
  in 
  shortening 
  the 
  periodic 
  

   time 
  of 
  a 
  comet 
  originally 
  momng 
  in 
  a 
  parabola 
  is 
  obtained 
  

   if 
  the 
  comers 
  original 
  orbit 
  actually 
  intersects 
  the 
  planers 
  

   orbit 
  at 
  an 
  angle 
  of 
  45°, 
  and 
  if 
  the 
  comet 
  is 
  due 
  first 
  at 
  the 
  

  

  * 
  The 
  goal 
  and 
  the 
  quit 
  of 
  a 
  moving 
  body 
  are 
  those 
  two 
  points 
  on 
  the 
  celestial 
  

   sphere 
  towards 
  which 
  and 
  from 
  which 
  the 
  body 
  is 
  moving. 
  

  

  