Re: Wait a moment

[T.L.G.vanderLinden@student.utwente.nl (Timothy van der Linden)]

To Kevin,

>Too late, I already did.  :)  oh well, just ignore my comments

It happened to me several times, but I think some of you comments are still
valid, so I will answer them here.

The minima are:

Vstart  Vexh optimal  Fuel:ship-ratio  Energy per kg of ship (in Joules)
 0.1        0.062          5.36            7.45E14
 0.2        0.121          5.84            3.25E15
 0.3        0.180          6.40            8.11E15
 0.4        0.240          7.06	           1.64E16
 0.5        0.300          7.87            2.97E16
 0.6        0.364          8.91            5.23E16
 0.7        0.433         10.38            9.21E16 
 0.8        0.512         12.72            1.73E17
 0.9        0.615         17.75            4.04E17
 0.99       0.803         52.00            3.12E18
 0.9996     0.906        238.81            2.91E19

Vstart is indeed the velocity that the starship has just before it starts
decelerating.

Energy per kg of ship is the total amount of energy needed, to calculate the
mean power, you could devide the Energy by T, the time it takes to
decelerate the ship.

Here are a few of these power numbers, 

Vstart    P[0]      P[T]
0.1      5.01E08    9.35E07
0.5      4.19E09    5.34E08
0.9      2.73E10    1.54E09
0.99     1.55E11    2.98E09
0.9996   1.23E12    5.14E09

P[0] is the power needed at the beginning of the deceleration, P[T] is the
power needed at the end of the deceleration. The difference between P[0] and
P[T] is  caused only by the decrease of the total mass of the ship.

Oh yes, the power and energy mentioned here are all relative to the
starship, not to Earth. This means that the doppler shift increases all
numbers when one takes Earth as reference point. (which is important because
it has to generate the power)

The influence of the doppler shift is shown here:

Vstart  Energy per kg of ship (in Joules)
 0.1       7.45E14    7.73E14
 0.2       3.25E15    3.50E15
 0.3       8.11E15    9.08E15
 0.4       1.64E16    1.91E16
 0.5       2.99E16    3.64E16
 0.6       5.23E16    6.69E16
 0.7       9.21E16    1.25E17
 0.8       1.73E17    2.53E17
 0.9       4.04E17    6.62E17
 0.99      3.12E18    7.02E17
 0.9996    2.91E19    9.26E19

The first row is the ENERGY relative to the starship, the second row is the
energy relative to Earth. I won't show the power numbers here, but I think
you can make a rough guess.

The influence of the doppler shift is calculated as follows:

                 2
       Sqrt[1 - b ]
f'= f ---------------  where b=v/c
          1 - b 

v[t]=c Tanh[a t/c]

U = h*f  so just linear dependence

    T                 2
    /       Sqrt[1 - b ]
E = | P[t] --------------- dt      where b = v[t]/c
    /          1 - b 
    0


Timothy

P.S. If someone likes a complete document (instead of the fractured ones)
just sent me a mail.