Energy requirements for an interstellar probe

In this note I will go through the basic physics of launching an interstellar probe. I will assume that the probe is not a rocket, i.e., the energy required for accelerating the probe is supplied externally rather than being carried on board. This assumption is true for at least two proposed launch methods, the large linear accelerator, and the laser cannon.

I will also assume that the objective is to reach nearby stars in times that are no more than centuries or at worst thousands of years. To do so requires speeds that are on the order of .001c -.01c. Thus, at a speed of .001c, it would take 4350 years to reach Alpha Centauri, whereas at a speed of .01c it would take 435 years. In the analysis I use a speed of 1000 km/sec, ~.003c, which would take abou 1300 years. I also ignore the cost of escaping the solar gravity well; it is negligible for the velocities and energies involved.

To simplify the analysis I will assume that the probe is accelerated in a straight line with constant acceleration. Let a be the acceleration, v be velocity of the probe during the acceleration, x be the distance moved during the acceleration, and t be time.

The equations of motion are:

(1) v = a*t
x = a*t**2/2

Eliminating t yields

(2a) v**2 = 2*a*x
(2b) x = v**2/(2*a)

Let V be the launch velocity. Then (assuming an initial velocity of 0) the energy of the probe is:

(3) E = m*V**2/2

neglecting relativistic corrections. Now consider a probe weighing 10 metric tons with a launch velocity of 1000 km/sec (approximately .003c). We have, in mks units,

m = 10**4 kilograms,
V = 10**6 meters/second
E = 5*10**15 joules

Since 1 BTU = 1054 joules the required energy is approximately

E = 5*10**12 BTU.

The total annual energy production of the US is approximately 1 Quad = 10**15 BTU’s (This number may be wrong) so that each launch requires .5% of the current annual energy production of the US.

The distance required for the launch can be determined from equation (2b). The distances required for accelerations of 1g (10 m/sec), 10g (100 m/sec), and 100g (1000 m/sec) are:

1g	50,000,000 km
10g	5,000,000 km
100g	500,000 km

The time required for the is given by T = V/a. For V = 1000 km/sec we have:

1g	100,000 sec
10g	10,000 sec
100g	1,000 sec

There is one other number of interest, the rate at which power is transferred to the probe during the launch. This is given by E/T and is in watts in the mks system. The values for the different values of g are:

1g	50 gigawatts
10g	500 gigawatts
100g	5000 gigawatts

There is an essential difficulty here in that the transfer of energy to the probe cannot be done with perfect efficiency; some portion of the energy must be transferred in the form of heat with, given the high energies involved, the effective thermal destruction of the probe.

In the above calculations it does not matter whether the launch facility is a linear accelerator or a laser cannon; the essential issue is the total amount of kinetic energy that must be transferred to the probe, and the consequences of transferring at various rates. In principle the linear accelerator is preferable since the only inefficiency should be hysteresis losses. In contrast the laser cannon has much a much high conversion of energy supplied to thermal energy. In consequence low accelerations must be used over large distances. The disadvantage of doing this is that the distance at which energy can be efficiently transferred to the probe via laser cannon is limited because of beam spreading.

It has been suggested that various technologies, e.g., sails, ion drives, and sling shot acceleration can reduce the costs. These are weak reeds. The amount of kinetic energy that can be acquired via sail is many orders of magnitude smaller than the required amount of energy. The sling shot effect (stealing momentum from a planet or a star) is falls short of the required level by about two orders of magnitude. The ion drive solves the rocket equation problem; the catch with it is that the probe now has to have a power plant capable of generating all of the kinetic energy required.

The above discussion only covers the launch. The problem of slowing down at the terminus is much worse because all of the kinetic energy must be lost with the further difficulty that there would be no energy transfer facility at the terminus.

In conclusion the prospects of launching an interstellar probe at high velocity are not good, even granting the existence of a civilization with the requisite resources.