# ¶ Ship engines

A ship must have an engine in order to move. Ships without engines can still be moved between docking bay, hangar and shipyard facilities using Logistics Drones via a Logistics Hub, but cannot move on their own.

Ship engines can be installed from any connected Cargo Bay. You can uninstall a ship's engine to a Cargo Bay on the ship while it is launched, as long as it is not currently moving.

## ¶ Flight mechanics

As of v?, a flight consists of two symmetrical phases: a continuous maximal acceleration for the first half, followed by a maximal decceleration for the second half. There is no maximum speed for ships; no relativistic effects are implemented.

The maximum acceleration a Ship Engine can provide is the lower of two factors:

• The force-limited acceleration ${a}_{f}=\frac{F}{m}a_f=\frac\left\{F\right\}\left\{m\right\}$, where $FF$ is the maximum force the engine mounting brackets or the ship superstructure can sustain, and $mm$ is the gross mass (the mass of the ship, equipment and cargo).
• The power-limited acceleration ${a}_{p}a_p$, which is the rate at which the engine can deliver energy to increase the ship's kinetic energy. ${a}_{p}=\frac{\stackrel{˙}{E}}{m}{v}^{-1}a_p = \frac\left\{\dot\left\{E\right\}\right\}\left\{m\right\}v^\left\{-1\right\}$ where $\frac{\stackrel{˙}{E}}{m}\frac\left\{\dot\left\{E\right\}\right\}\left\{m\right\}$ (the engine power divided by the gross mass) is a constant, and $vv$ is the ship's current velocity.

All flights are initially force-limited, then become power-limited as the velocity of the ship increases. The transition velocity ${v}_{\ast }=v\left({a}_{p}={a}_{f}\right)=\frac{\stackrel{˙}{E}}{F}v_* = v\left(a_p=a_f\right) = \frac\left\{\dot\left\{E\right\}\right\}\left\{F\right\}$ is usually less than and the corresponding distance ${s}_{\ast }s_*$ less than ; so for all but the most powerful engines flight can be approximated as purely power-limited.

The distance covered by the ship during an acceleration (or decceleration) burn is $s\left(\tau \right)\approx \sqrt{\frac{8\stackrel{˙}{E}}{9m}}{\tau }^{\frac{3}{2}}s\left(\tau\right) \approx \sqrt\frac\left\{8\dot\left\{E\right\}\right\}\left\{9m\right\}\tau^\frac\left\{3\right\}\left\{2\right\}$. Since the maximum allowed flight is 3 weeks (), the maximum range ${s}_{\mathrm{m}\mathrm{a}\mathrm{x}}s_\left\{\mathrm\left\{max\right\}\right\}$ of a ship with a given engine is or $\approx 0.0154\sqrt{\frac{\stackrel{˙}{E}}{m}}\approx \mathrm\left\{0.0154\right\} \sqrt\left\{\frac\left\{\dot\left\{E\right\}\right\}\left\{m\right\}\right\}$ astronomical units.

### ¶ Derivations

#### ¶${a}_{p}a_p$

$E=\frac{m{v}^{2}}{2}\phantom{\rule{0ex}{0ex}}\frac{\mathrm{d}E}{\mathrm{d}t}=\stackrel{˙}{E}=mv\stackrel{˙}{v}=mv{a}_{p}\phantom{\rule{0ex}{0ex}}{a}_{p}=\frac{\stackrel{˙}{E}}{m}{v}^{-1}E = \frac\left\{mv^2\right\}\left\{2\right\} \\ \frac\left\{\mathrm\left\{d\right\}E\right\}\left\{\mathrm\left\{d\right\}t\right\} = \dot\left\{E\right\} = mv\dot\left\{v\right\} = mva_p \\ a_p = \frac\left\{\dot\left\{E\right\}\right\}\left\{m\right\}v^\left\{-1\right\}$

#### ¶${v}_{\ast }v_*$

${v}_{\ast }=v\left({a}_{p}={a}_{f}\right)\phantom{\rule{0ex}{0ex}}\frac{\stackrel{˙}{E}}{m}{v}_{\ast }^{-1}=\frac{F}{m}\phantom{\rule{0ex}{0ex}}{v}_{\ast }=\frac{\stackrel{˙}{E}}{m}\frac{m}{F}=\frac{\stackrel{˙}{E}}{F}v_* = v\left(a_p = a_f\right) \\ \frac\left\{\dot\left\{E\right\}\right\}\left\{m\right\}v_*^\left\{-1\right\} = \frac\left\{F\right\}\left\{m\right\} \\ v_* = \frac\left\{\dot\left\{E\right\}\right\}\left\{m\right\} \frac\left\{m\right\}\left\{F\right\} = \frac\left\{\dot\left\{E\right\}\right\}\left\{F\right\}$

#### ¶${s}_{\ast }s_*$

{v}_{\ast }^{2}=2{a}_{f}{s}_{\ast }\phantom{\rule{0ex}{0ex}}\begin{array}{cc}{s}_{\ast }& =\frac{{v}_{\ast }^{2}}{2{a}_{\ast }}={\left(\frac{\stackrel{˙}{E}}{F}\right)}^{2}\frac{m}{2F}\\ & =\frac{m{\stackrel{˙}{E}}^{2}}{2{F}^{3}}\end{array}v_*^2 = 2a_fs_* \\ \begin\left\{aligned\right\} s_* &= \frac\left\{v_*^2\right\}\left\{2a_*\right\} = \left\left( \frac\left\{\dot\left\{E\right\}\right\}\left\{F\right\}\right\right)^2 \frac\left\{m\right\}\left\{2F\right\} \\ &= \frac\left\{m\dot\left\{E\right\}^2\right\}\left\{2F^3\right\} \\ \end\left\{aligned\right\}

#### ¶${t}_{\ast }t_*$

{s}_{\ast }=\frac{{a}_{\ast }{t}_{\ast }^{2}}{2}\phantom{\rule{0ex}{0ex}}\begin{array}{cc}{t}_{\ast }& ={\left(\frac{2{s}_{\ast }}{{a}_{\ast }}\right)}^{\frac{1}{2}}\\ & ={\left(2\frac{m}{F}\frac{m{\stackrel{˙}{E}}^{2}}{2{F}^{3}}\right)}^{\frac{1}{2}}\\ & =\frac{m\stackrel{˙}{E}}{{F}^{2}}\end{array}s_* = \frac\left\{a_*t_*^2\right\}\left\{2\right\} \\ \begin\left\{aligned\right\} t_* &= \left\left( \frac\left\{2s_*\right\}\left\{a_*\right\} \right\right)^\frac\left\{1\right\}\left\{2\right\} \\ &= \left\left( 2 \frac\left\{m\right\}\left\{F\right\} \frac\left\{m\dot\left\{E\right\}^2\right\}\left\{2F^3\right\} \right\right)^\frac\left\{1\right\}\left\{2\right\} \\ &= \frac\left\{m\dot\left\{E\right\}\right\}\left\{F^2\right\} \end\left\{aligned\right\}

#### ¶$s\left(t\right)s\left(t\right)$

$s\left(\tau +\delta \right)>s\left(\tau \right)\text{ }⟹\text{ }ϵ=+1s\left(\tau + \delta\right) > s\left(\tau\right) \implies \epsilon = +1$

$s\left(\tau \right)={s}_{\ast }+\frac{2}{3}\sqrt{\frac{2\stackrel{˙}{E}}{m}}\left({\left(\tau +{c}_{1}^{\mathrm{\prime }}\right)}^{\frac{3}{2}}-{\left({t}_{\ast }+{c}_{1}^{\mathrm{\prime }}\right)}^{\frac{3}{2}}\right)s\left(\tau\right) = s_* + \frac\left\{2\right\}\left\{3\right\}\sqrt\frac\left\{2\dot\left\{E\right\}\right\}\left\{m\right\}\left\left(\left\left(\tau + c^\left\{\prime\right\}_1\right\right)^\left\{\frac\left\{3\right\}\left\{2\right\}\right\} - \left\left(t_* + c^\left\{\prime\right\}_1\right\right)^\left\{\frac\left\{3\right\}\left\{2\right\}\right\}\right\right)$

Differentiating again:

$v\left(\tau \right)=\sqrt{\frac{2\stackrel{˙}{E}}{m}}{\left(\tau +{c}_{1}^{\mathrm{\prime }}\right)}^{\frac{1}{2}}v\left(\tau\right) = \sqrt\frac\left\{2\dot\left\{E\right\}\right\}\left\{m\right\}\left\left(\tau + c^\left\{\prime\right\}_1\right\right)^\left\{\frac\left\{1\right\}\left\{2\right\}\right\}$

Now $v\left({t}_{\ast }\right)=v\left(\frac{m\stackrel{˙}{E}}{{F}^{2}}\right)={v}_{\ast }v\left(t_*\right) = v\left(\frac\left\{m\dot\left\{E\right\}\right\}\left\{F^2\right\}\right) = v_*$ so:

$\sqrt{\frac{2\stackrel{˙}{E}}{m}}{\left(\frac{m\stackrel{˙}{E}}{{F}^{2}}+{c}_{1}^{\mathrm{\prime }}\right)}^{\frac{1}{2}}=\frac{\stackrel{˙}{E}}{F}\phantom{\rule{0ex}{0ex}}\left(\frac{2\stackrel{˙}{E}}{m}\right)\left(\frac{m\stackrel{˙}{E}}{{F}^{2}}+{c}_{1}^{\mathrm{\prime }}\right)={\left(\frac{\stackrel{˙}{E}}{F}\right)}^{2}\phantom{\rule{0ex}{0ex}}\frac{m\stackrel{˙}{E}}{{F}^{2}}+{c}_{1}^{\mathrm{\prime }}=\frac{m\stackrel{˙}{E}}{2{F}^{2}}\phantom{\rule{0ex}{0ex}}{c}_{1}^{\mathrm{\prime }}=-\frac{m\stackrel{˙}{E}}{2{F}^{2}}=\frac{{t}_{\ast }}{2}\sqrt\frac\left\{2\dot\left\{E\right\}\right\}\left\{m\right\}\left\left(\frac\left\{m\dot\left\{E\right\}\right\}\left\{F^2\right\} + c^\left\{\prime\right\}_1\right\right)^\left\{\frac\left\{1\right\}\left\{2\right\}\right\} = \frac\left\{\dot\left\{E\right\}\right\}\left\{F\right\} \\ \left\left(\frac\left\{2\dot\left\{E\right\}\right\}\left\{m\right\}\right\right)\left\left(\frac\left\{m\dot\left\{E\right\}\right\}\left\{F^2\right\} + c^\left\{\prime\right\}_1\right\right) = \left\left(\frac\left\{\dot\left\{E\right\}\right\}\left\{F\right\}\right\right)^2 \\ \frac\left\{m\dot\left\{E\right\}\right\}\left\{F^2\right\} + c^\left\{\prime\right\}_1 = \frac\left\{m\dot\left\{E\right\}\right\}\left\{2F^2\right\} \\ c^\left\{\prime\right\}_1 = - \frac\left\{m\dot\left\{E\right\}\right\}\left\{2F^2\right\} = \frac\left\{t_*\right\}\left\{2\right\}$

Hence:

\begin{array}{cc}s\left(\tau \right)& ={s}_{\ast }+\frac{2}{3}\sqrt{\frac{2\stackrel{˙}{E}}{m}}\left({\left(\tau -\frac{m\stackrel{˙}{E}}{2{F}^{2}}\right)}^{\frac{3}{2}}-{\left({t}_{\ast }-\frac{m\stackrel{˙}{E}}{2{F}^{2}}\right)}^{\frac{3}{2}}\right)\\ & ={s}_{\ast }+\frac{2}{3}\sqrt{\frac{2\stackrel{˙}{E}}{m}}\left({\left(\tau -\frac{{t}_{\ast }}{2}\right)}^{\frac{3}{2}}-{\left(\frac{{t}_{\ast }}{2}\right)}^{\frac{3}{2}}\right)\end{array}\begin\left\{aligned\right\} s\left(\tau\right) &= s_* + \frac\left\{2\right\}\left\{3\right\}\sqrt\frac\left\{2\dot\left\{E\right\}\right\}\left\{m\right\}\left\left(\left\left(\tau - \frac\left\{m\dot\left\{E\right\}\right\}\left\{2F^2\right\}\right\right)^\left\{\frac\left\{3\right\}\left\{2\right\}\right\} - \left\left(t_* - \frac\left\{m\dot\left\{E\right\}\right\}\left\{2F^2\right\}\right\right)^\left\{\frac\left\{3\right\}\left\{2\right\}\right\}\right\right) \\ &= s_* + \frac\left\{2\right\}\left\{3\right\}\sqrt\frac\left\{2\dot\left\{E\right\}\right\}\left\{m\right\}\left\left(\left\left(\tau - \frac\left\{t_*\right\}\left\{2\right\}\right\right)^\left\{\frac\left\{3\right\}\left\{2\right\}\right\} - \left\left(\frac\left\{t_*\right\}\left\{2\right\}\right\right)^\left\{\frac\left\{3\right\}\left\{2\right\}\right\}\right\right) \end\left\{aligned\right\}

For $\tau \gg {t}_{\ast }\tau \gg t_*$ (hence $s\left(\tau \right)\gg {s}_{\ast }s\left(\tau\right) \gg s_*$), this gives:

$s\left(\tau \right)\approx \sqrt{\frac{8\stackrel{˙}{E}}{9m}}{\tau }^{\frac{3}{2}}s\left(\tau\right) \approx \sqrt\frac\left\{8\dot\left\{E\right\}\right\}\left\{9m\right\}\tau^\frac\left\{3\right\}\left\{2\right\}$