AAWSAP DIRD, Aneutronic Fusion Propulsion I, November 2010

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3. Compute the Rocket Performance

a. Time required to reach Proxima Centauri in earth time (inertial reference frame) and in rocket time..

$$
t_{\text{earth}} := 2 \cdot \sqrt{\left[ \frac{\left(\frac{D}{2}\right)}{c} \right]^2 + 2 \cdot \frac{\left(\frac{D}{2}\right)}{a}} \qquad t_{\text{earth}} = 127.903 \, \text{yr}
$$

$$
t_{\text{rocket}} := 2 \cdot \frac{c}{a} \cdot \operatorname{acosh} \left[ a \cdot \frac{\left(\frac{D}{2}\right)}{c^2} + 1 \right] \qquad t_{\text{rocket}} = 127.81 \, \text{yr}
$$

b. Peak velocity.

$$V_{\text{midway}} := c \cdot \tanh \left[ a \cdot \left( \frac{t_{\text{rocket}}}{\frac{2}{c}} \right) \right] \qquad V_{\text{midway}} = 1.975 \times 10^7 \frac{\text{m}}{\text{s}}$$
$$\frac{V_{\text{midway}}}{c} = 6.587 \%$$

c. Compute the required kinetic energy that must be supplied by the fuel.

$$
\gamma := \frac{1}{\sqrt{1 - \left(\frac{V_{\text{midway}}}{c}\right)^2}} \qquad \text{Lorentz factor at the midway point (peak velocity).}
$$

$$
\gamma = 1.002
$$

$$
\text{KE} := 2 \cdot m_{\text{rocket}} \cdot c^2 \cdot (\gamma - 1) \qquad \text{KE} = 1.497 \times 10^{20} \, \text{J}
$$

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