Eph rmc2 1 J l72

Below is a table that compares gammas with the fraction Eph/mc?

Y

Eph/mc

1.5

0.559

1.6

0.624

1.7

0.687

1.8

0 . 748

1.9

0 . 808

2.0

Equations 8.5 and 8.6 indicate the need to reflect a lot of photons! Assume that this sail craft is powered by some solar collector near Earth. If the collector is near Earth there are about 1500 watts/m2 of power of solar radiation. For a year as ~ 3 x 107 sec a meter would collect 4.5 x 1010 J. So the mc2 equivalent is about 5 x 10"7 kg. So collecting area is important. To reach a 7 = 2 in one year requires an area of around 1.7 x 106 m2 to accelerate 1 kg to this speed. The collector would require a radius of 525 meters, where this assume 100% conversion of solar light energy to a beam that drives the spacecraft. Clearly for a larger spacecraft the area goes up proportionately. For the 107 kg craft mentioned above this would imply a light collecting area with a ~ 1500 km radius.

The photon sail offers the advantage of not requiring exotic engineering and physics. It also clearly has potential performance capabilities of the photon rocket. However, it requires the construction of large solar collectors. Clearly as this photon sail clears the solar system it will have to be driven forward by a directed beam of light energy. This will require some sort of solar concentrator or a solar driven laser beam of truly huge proportions. A 500 m radius collector receives 2.35 gigawatts of power, where in reality the conversion losses will require a ~ 1600 m radius collector. Hence the scale of things becomes colossal and requires large scale fabrication in space.

An examination of the times and distances required to accelerate to a 7 = 1.15, or v ~ .5c, for an average acceleration g = .01 m/sec2 to g = .1 m/sec2

g (m/s2) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

257.7

171.8 128.8 103.1

t (yr) 540.85 270.4 180.3 135.2 108.17 90.14 77.26 67.61 60.09 54.08

17.86

15.87 14.29

The time and distance performance is not very good, but a solar sail with the higher end of these specification may well be able to reach some of the nearer stars within a 5-10 ly distance. This performance might be improved if the light is further concentrated. Assume that the average accelerations are increased by a factor of 2.5. The performance is then

g

T

t

d

0.025

206.1

216.3

57.14

0.050

103.1

108.2

28.57

0.075

68.71

72.11

19.05

0.100

51.53

54.08

14.29

0.125

41.23

43.27

11.43

0.150

34.36

36.06

9.524

0.175

29.45

30.91

8.163

0.200

25.77

27.04

7.143

0.225

22.90

24.04

6.349

0.250

20.61

21.63

This craft for g = .12 m/s2 could reach nearby stars within ~ 12 ly within a 44 year time period and data could be received on Earth within 56 years. This is within the acceptable range of mission times. The star a Centuri is 4.36 ly, and t Ceti is 11.8 ly from Earth. These are the two G-class stars similar to the sun that would be of value exploring. There are in addition 25 other stars, mostly M-class red dwarf type stars, within this range. The star e Eridani, a K-type star 10.4 ly away has two planets, so this star system is a candidate for direct exploration just barely within these performance parameters of a starsail.

Above it was mentioned that these were average accelerations. Given a photon source, whether as collimated solar radiation or a laser, from the perspective of an observer on the sail craft the photons will reach the craft increasingly red shifted. As such the photon pressure will decline. An acceleration of a photon sail could only be made constant if the flux is increased in a manner to compensate for this. To see how the acceleration of a photon sailcraft depends upon its velocity a simple relativistic analysis is required. Consider a mass with a velocity v which reflects a photon send from Earth with a frequency v to a photon with frequency v' and attains a velocity v'. Conservation of relativistic momentum gives the relationships between the spatial momentum and energy as h h (7V - 7«) = —{i/ + !/)(7' - 7) = -^(f' - i>), (8.10)

mc mc2

where h is the Planck unit of action. Eliminating the reflected photon frequency these give the following, y'(c — v') — 7 (c — V ) =--v.

A differential expression for the velocity may be obtained for a variable n as the number of photons incident on the craft on it frame,

The rate at which the photons are incident on the craft is related to the rate photons are sent from Earth by dn = Y-1dN. For photons sent at a rate R = dN/dt this leads to the differential equation

This does predict that the acceleration will drop to zero as the velocity approaches the speed of light. Yet to understand the dynamics of a photon sail this differential equation must be integrated. This is done numerically with the result indicated in Figure 8.2. It is clear that the photon sail is best for 7 < 1.25 or for v < 0.6c, which is sufficient for exploration of the interstellar neighborhood. In this case the acceleration declines to ^ .42 the initial acceleration and the average acceleration is ^ .75 the initial acceleration. These will decline further over time if the sail craft is sent further.

Increasing time, units arbitary depending upon photon flux

Fig. 8.2. Evolution of the velocity and acceleration of a photon sail.

For a y = 1.15 or v = .494c the acceleration declines to .726 the initial acceleration and has an average of .882 its initial acceleration. A photon sail craft which reaches half light speed is capable of reaching stars within the 20 light year range within an acceptable time frame.

The simplest way to concentrate light is with a lens. A lens with a focal length equal to the distance between it and the sun will cause the diverging light to become collimated into a beam. A plano convex lens with a radius of r = 1500 km situated 1.5 x 108 km from the sun will then have a radius of curvature as measured from the enter of the sun given by the thin lens equation

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