Pluto motion from image 1 to image 13:
image 1: x1 = 346 y1 = 220
image 13: x2 = 352 y2 = 217
dx = 6 pixels * 1.117"/pixel = 6.702"
dy = 3 pixels * 1.074"/pixel = 3.222"
dr = 7.436"
dt = 12 frames = 7200 sec
--> 5.007e-9 rad/sec
From this, determine how far from Sun Pluto orbits.
acceleration = (G*M_Sun)/r^2 = v^2/r
At opposition, the difference between Earth's, Pluto's velocity determines the angular motion:
d(theta)/dt = (v_Earth - v_Pluto)/(r_Pluto - r_Earth)
let r_Pluto = a*r_Earth
d(theta)/dt = (v_Earth - sqrt(G*M_Sun/r_Pluto)) / (r_Earth*(a-1))
= (v_Earth - sqrt(G*M_Sun/(a*r_Earth)) / (r_Earth*(a-1))
= (v_Earth - v_Earth/sqrt(a)) / (r_Earth*(a-1))
= (v_Earth / r_Earth) * (1-1/sqrt(a)) / (a-1)
let b = d(theta)/dt * r_Earth / v_Earth
= d(theta)/dt * r_Earth / (2*pi*r_Earth / T_Earth)
= d(theta)/dt * T_Earth / (2*pi) (Eq. 1)
Measured value: b = 0.02513
(1-1/sqrt(a)) / (a-1) = b (Eq. 2)
(sqrt(a) - 1) / (sqrt(a) (sqrt(a)+1) (sqrt(a)-1)) = b
drop a=1 solution,
a + sqrt(a) - 1/b = 0
keeping only sqrt(a)>0 solution (assuming b>0),
sqrt(a) = -1/2 + (1/2)sqrt(1 + 4/b)
a = (1/4) (sqrt(1 + 4/b) - 1)^2
---> a = 33.96 au
Actual value for Pluto:
a = 31.85 in 2010 (actually it varies a lot since the orbit is
elliptical, but this is the current distance-- Observer's
Handbook 2010)
:)