# OK, here the right calculus (I hope)

```Optimal solution for a maser deceleration starship. The starship carries all
the necessary repulsion mass and gets the energy needed to accelerate the
repulsion mass from the maser.

To do this we want to find the solution that consumes the least energy.

Energy needed to decelerate a ship depends on it's empty mass, its start
velocity, the exhaust velocity. So we need the formula E(Mo,Vstart,Vexh)

Definition of g or gamma, also known as the energy factor:

1
------------------
2
(1)  g =            Vexh
Sqrt[1 - ------]
2
c

Momentum mass needed at time t to decelerate with a m/s/s

M[t] (a+b)
(2)  j[t] = ------------
g Vexh

b is the acceleration caused by the maser beam. The value of b depends on
the power needed, this is yet unknown.

Power needed at time t:

P[t]=j[t] c^2 (g-1)

Momentum caused by the maser beam: p=E/c or F=P/c

Since F = M[t] b and P/c = j[t] c (g-1) we get

M[t] b = j[t] c (g-1)

Solving b gives:

b = j[t] c (g-1)/M[t]

Substituting b into (2) gives:

M[t] a      j[t] c (g-1)
(3a)  j[t] = --------- + --------------
g Vexh         g Vexh

Resolving j[t] gives:

M[t] a
(3b)  j[t] = --------------------
g Vexh - c (g - 1)

Definition of the time T that it takes to decelerate from Vstart to 0:

c           Vstart
(4)  T = --- ArcTanh[--------]
a             c

Mass of the ship(Mo) and the repulsion mass(j[t]) at time t:

T
/
(5)  M[t] = Mo + | j[t] dt
/
t

Substitute (3b) in (5):

T
a           /
(6)  M[t] = Mo + -------------------- | M[t] dt
g Vexh - c (g - 1)  /
t

Solving this differential equation gives:

a (T - t)
(7)  M[t] = Mo Exp[---------------------]
g Vexh - c (g - 1)

The total amount of energy needed to accelerate the repulsion mass to Vexh
is defined by:

T
2      /
(9)  Ek = c (g-1) | j[t] dt
/
0

(Oops, forgot the "minus 1" in my previous letter)

2                       a T
(10) Ek = c (g-1) Mo (Exp [--------------------] - 1)
g Vexh + c (1 - g)

2                         c                   Vstart
(11) Ek = c (g-1) Mo (Exp [-------------------- ArcTanh[--------]] - 1)
g Vexh + c (1 - g)             c

Finally... E(Mo,Vstart,Vexh)

Now we only need to find its minimum, that could be done by solving dE/dVexh=0

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

Note that the power of the maser-beam is NOT constant, it is supposed that
it decreases while the ship gets lighter (because it repulses mass).

Timothy

P.S. Goodnight to all

```