Low Thrust Trajectories

setup & notation

State: $\vec{\mathbf{r}}\in {\mathfrak{Re}}^3$, $\vec{\mathbf{v}}=\frac{\mathrm{d}\vec{\mathbf{r}}}{\mathrm{d}t}$, $\mu$ (primary’s GM) Thrust: magnitude $T$, mass $m$, specific acceleration $\alpha \equiv T/m$. Unit triad along the orbit: $\{\widehat{\vec{\mathbf{r}}},\widehat{\vec{\mathbf{t}}},\widehat{\vec{\mathbf{n}}}\}$ (radial, along-track/circumferential, out-of-plane).
Decompose thrust acceleration:

$${\vec{\mathbf{a}}}_T={\alpha }_R\widehat{\vec{\mathbf{r}}}+{\alpha }_T\widehat{\vec{\mathbf{t}}}+{\alpha }_N\widehat{\vec{\mathbf{n}}},\alpha =\sqrt{{\alpha }_R^2+{\alpha }_T^2+{\alpha }_N^2}.$$

Equations of motion:

$$\frac{{\mathrm{d}}^2\vec{\mathbf{r}}}{\mathrm{d}{t}^2}=-\frac{\mu }{r^3}\vec{\mathbf{r}}+{\vec{\mathbf{a}}}_T.$$

Conserved quantities without thrust: specific energy $ϵ=\frac{\vec{\mathbf{v}}\cdot \vec{\mathbf{v}}}{2}-\frac{\mu }{r}$, specific angular momentum $\vec{\mathbf{h}}=\vec{\mathbf{r}}\times \vec{\mathbf{v}}$. With thrust:

$$\frac{\mathrm{d}ϵ}{\mathrm{d}t}=\vec{\mathbf{v}}\cdot {\vec{\mathbf{a}}}_T,\frac{\mathrm{d}\vec{\mathbf{h}}}{\mathrm{d}t}=\vec{\mathbf{r}}\times {\vec{\mathbf{a}}}_T.$$

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trajectories (LTT)

The “trajectory shaping” intuition:

• Energy change comes from the component of thrust along velocity (circumferential/tangential):

  $$\frac{\mathrm{d}ϵ}{\mathrm{d}t}=v{\alpha }_T+v_r{\alpha }_R.$$

  For near-circular motion, $v_r\approx 0\Rightarrow \frac{\mathrm{d}ϵ}{\mathrm{d}t}\approx v{\alpha }_T$.

• Plane/orientation change comes from out-of-plane thrust:

  $$\frac{\mathrm{d}\vec{\mathbf{h}}}{\mathrm{d}t}=\vec{\mathbf{r}}\times ({\alpha }_N\widehat{\vec{\mathbf{n}}})=r{\alpha }_N\widehat{\vec{\mathbf{t}}}\Rightarrow \text{tilt of the orbit plane (}\frac{\mathrm{d}i}{\mathrm{d}t},\frac{\mathrm{d}\Omega }{\mathrm{d}t}\text{)}.$$

• Eccentricity is excited by radial thrust (most strongly near peri/apoapsis). A purely tangential control law can keep $e\approx 0$ to produce smooth spirals.

Rule of thumb: to raise $a$ efficiently, point thrust along $\widehat{\vec{\mathbf{t}}}$; to shape $e$, inject a radial component timed with true anomaly; to tilt the plane, use ${\alpha }_N$.

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spirals (LTS)

Assume a planar, near-circular orbit ($e\approx 0$) with thrust held tangential ($\alpha ={\alpha }_T$).

semimajor-axis growth

Specific energy $ϵ=-\frac{\mu }{2a}\Rightarrow \frac{\mathrm{d}ϵ}{\mathrm{d}t}=\frac{\mu }{2a^2}\frac{\mathrm{d}a}{\mathrm{d}t}$. For circular motion $v=\sqrt{\mu /a}$ and $\frac{\mathrm{d}ϵ}{\mathrm{d}t}\approx v\alpha$. Hence

$$\boxed{\frac{\mathrm{d}a}{\mathrm{d}t}=\frac{2a^{3/2}}{\sqrt{\mu }}\alpha }$$

(valid while $e\approx 0$).

time to spiral from $a_0$ to $a_1$

$$t={\int }_{a_0}^{a_1}\frac{\mathrm{d}a}{\frac{\mathrm{d}a}{\mathrm{d}t}}=\frac{\sqrt{\mu }}{\alpha }(\frac{1}{\sqrt{a_0}}-\frac{1}{\sqrt{a_1}}).$$

accumulated $\Delta v$

For constant $\alpha$, $\Delta v=\int \alpha \mathrm{d}t=\alpha t\Rightarrow$

$$\boxed{\Delta v_{\text{LTS}}=\sqrt{\mu }(\frac{1}{\sqrt{a_0}}-\frac{1}{\sqrt{a_1}}).}$$

As $a_1\to \infty$ (escape), $\Delta v_{\text{LTS}}\to \sqrt{\mu /a_0}=v_{\text{circ},0}$.

spiral in $r(\theta )$ form (useful for geometry/coverage)

With $n=\sqrt{\mu /a^3}$ and $a\approx r$ for circular paths,

$$\frac{\mathrm{d}r}{\mathrm{d}\theta }=\frac{\frac{\mathrm{d}r}{\mathrm{d}t}}{\frac{\mathrm{d}\theta }{\mathrm{d}t}}=\frac{\frac{\mathrm{d}a}{\mathrm{d}t}}{n}=\frac{2a^{3/2}\alpha }{\sqrt{\mu }}\cdot \frac{a^{3/2}}{\sqrt{\mu }}=\boxed{\frac{2\alpha }{\mu }{r}^3}.$$

Integrating: $-\frac{1}{2r^2}=\frac{2\alpha }{\mu }\theta +C\Rightarrow r(\theta )$ increases monotonically under prograde thrust.

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impulsive escape (from circular orbit)

From a circular orbit of radius $r_0$, $v_{\text{circ},0}=\sqrt{\mu /r_0}$. Parabolic (escape) speed at $r_0$: $v_{\text{esc},0}=\sqrt{2\mu /r_0}$. A single impulsive burn to escape requires

$$\boxed{\Delta v_{\text{imp,esc}}=(\sqrt{2}-1)v_{\text{circ},0}\approx 0.4142v_{\text{circ},0}}.$$

Compare with continuous tangential thrust to escape (letting $a_1\to \infty$):

$$\Delta v_{\text{LTS,esc}}=v_{\text{circ},0}\text{(larger due to gravity/finite-burn losses)}.$$

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trajectory equations for LTS (workhorse forms)

For low-thrust design, two robust, implementable differential relations are:

energy & angular momentum

$$\boxed{\frac{\mathrm{d}ϵ}{\mathrm{d}t}=\vec{\mathbf{v}}\cdot {\vec{\mathbf{a}}}_T},\boxed{\frac{\mathrm{d}\vec{\mathbf{h}}}{\mathrm{d}t}=\vec{\mathbf{r}}\times {\vec{\mathbf{a}}}_T}.$$

• For tangential thrust in near-circular orbits: $$\frac{\mathrm{d}a}{\mathrm{d}t}=\frac{2a^{3/2}}{\sqrt{\mu }}{\alpha }_T,\frac{\mathrm{d}e}{\mathrm{d}t}\approx 0,\frac{\mathrm{d}i}{\mathrm{d}t}\approx 0.$$
• If a small radial component ${\alpha }_R$ is introduced, it excites $e$ roughly proportionally to ${\alpha }_R$ with strongest effect near $f\in \{0,\pi \}$.

planar spiral in anomaly

Using $r=a(1-e^2)/(1+e\cos \left(f\right))$ and keeping $e\approx 0$, the ODE

$$\frac{\mathrm{d}r}{\mathrm{d}\theta }=\frac{2\alpha }{\mu }{r}^3$$

gives an analytic spiral useful for coarse trajectory sketches and coverage calculations.

Note: For full generality (time-optimal or constrained pointing laws), use the Gauss variational equations in $(a,e,i,\Omega ,\omega ,f)$ with $({\alpha }_R,{\alpha }_T,{\alpha }_N)$. In these notes we emphasize the circular/tangential specializations actually used for spiral design; see also the Week 6 Part 2 notes for the full element-rate set.

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efficiency comparison

We compare ideal impulsive maneuvers to continuous tangential (circumferential) thrust at the same $I_{sp}$ (so $\Delta v$ is a proxy for propellant).

circular $a_0\to a_1$

• Impulsive (Hohmann): $$\Delta v_H=\sqrt{\frac{\mu }{a_0}}(\sqrt{\frac{2a_1}{a_0+a_1}}-1)+\sqrt{\frac{\mu }{a_1}}(1-\sqrt{\frac{2a_0}{a_0+a_1}}).$$
• Continuous tangential (LTS) (near-circular throughout): $$\Delta v_{\text{LTS}}=\sqrt{\mu }(\frac{1}{\sqrt{a_0}}-\frac{1}{\sqrt{a_1}}).$$

Observation: $\Delta v_{\text{LTS}}>\Delta v_H$ for raises of practical interest. The gap is the well-known gravity/steering loss of finite-thrust spirals. (High $I_{sp}$ can still reduce mass vs. chemical impulsive, despite larger $\Delta v$.)

escape from circular

$$\Delta v_{\text{imp,esc}}=(\sqrt{2}-1)v_{\text{circ},0}\text{vs.}\Delta v_{\text{LTS,esc}}=v_{\text{circ},0}.$$

Impulsive escape is most $\Delta v$–efficient; LTS trades $\Delta v$ for feasibility with low thrust and high $I_{sp}$.

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design notes & controls

• To hold $e\approx 0$ during a raise, keep thrust nearly tangential and modulate a small ${\alpha }_R$ in anti-phase with $v_r$.
• For time-optimal spirals under thrust/pointing constraints, use optimal-control (Pontryagin) with the above dynamics; bang-bang in-plane pointing between near-tangential and slightly inward-radial can reduce time at the cost of modest $e$ excursions.
• Out-of-plane targets (inclination change) are cheapest near high $r$ where $v$ is low; combine with raises to “pay” for tilts at apoapsis.

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quick reference (circular, tangential thrust)

$$\begin{array}{rl}\frac{\mathrm{d}a}{\mathrm{d}t}& =\frac{2a^{3/2}}{\sqrt{\mu }}\alpha \\ t(a_0\to a_1)& =\frac{\sqrt{\mu }}{\alpha }(\frac{1}{\sqrt{a_0}}-\frac{1}{\sqrt{a_1}})\\ \Delta v(a_0\to a_1)& =\sqrt{\mu }(\frac{1}{\sqrt{a_0}}-\frac{1}{\sqrt{a_1}})\\ \Delta v_{\text{esc}}^{\text{LTS}}& =v_{\text{circ},0}.\end{array}$$