From 5c9d8a43937ea49f3ec42ebd9f60c1265acb4859 Mon Sep 17 00:00:00 2001 From: adrulpz Date: Mon, 6 Jan 2025 13:38:58 -0600 Subject: [PATCH 1/2] Add .gitignore to ignore Jupyter checkpoint files --- .gitignore | 1 + 1 file changed, 1 insertion(+) create mode 100644 .gitignore diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..f481bd0 --- /dev/null +++ b/.gitignore @@ -0,0 +1 @@ +.ipynb_checkpoints/ From abf13f690adf9ce97561572974b4d04e7d2bffc7 Mon Sep 17 00:00:00 2001 From: adrulpz Date: Mon, 6 Jan 2025 13:57:17 -0600 Subject: [PATCH 2/2] Update Lecture7 and add improved images --- orbital-mechanics/Lecture7/Lecture7.ipynb | 1031 ++++++++++---------- orbital-mechanics/Lecture7/images/L7-1.png | Bin 0 -> 50774 bytes orbital-mechanics/Lecture7/images/L7-2.png | Bin 0 -> 31460 bytes orbital-mechanics/Lecture7/images/L7-3.png | Bin 0 -> 39691 bytes orbital-mechanics/Lecture7/images/L7-4.png | Bin 0 -> 47273 bytes orbital-mechanics/Lecture7/images/L7-5.png | Bin 0 -> 38298 bytes orbital-mechanics/Lecture7/images/L7-6.png | Bin 0 -> 46266 bytes 7 files changed, 519 insertions(+), 512 deletions(-) create mode 100644 orbital-mechanics/Lecture7/images/L7-1.png create mode 100644 orbital-mechanics/Lecture7/images/L7-2.png create mode 100644 orbital-mechanics/Lecture7/images/L7-3.png create mode 100644 orbital-mechanics/Lecture7/images/L7-4.png create mode 100644 orbital-mechanics/Lecture7/images/L7-5.png create mode 100644 orbital-mechanics/Lecture7/images/L7-6.png diff --git a/orbital-mechanics/Lecture7/Lecture7.ipynb b/orbital-mechanics/Lecture7/Lecture7.ipynb index 642268d..c0f0cea 100644 --- a/orbital-mechanics/Lecture7/Lecture7.ipynb +++ b/orbital-mechanics/Lecture7/Lecture7.ipynb @@ -1,514 +1,521 @@ { - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Orbital Maneuvers\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "_Prepared by:_ [Mitansh Choksi](https://github.com/mitanshc) and [Angadh Nanjangud](https://www.angadhn.com/)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this lecture we aim to cover the following topics:\n", - "1. [](content:orbit-maneuvers)\n", - "2. [](content:hohmann-transfer)\n", - "3. [](content:bi-elliptic-transfer)\n", - "4. [](content:non-tangential-transfer)\n", - "5. [](content:plane-change-maneuver)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "(content:orbit-maneuvers)=\n", - "## Orbit Maneuvers" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Manuevers are implemented to transfer the spacecraft from one orbit to another.\n", - "Orbital changes can have different objectives:\n", - "- Inject the probe into an interplanetary transfer\n", - "- Move the spacecraft from a parking orbit to an operetive one ( e.g. LEO to GEO)\n", - "- Rendezvous end dock with another spececraft ( e.g. with the ISS)\n", - "- Compensate for the effect of perturbations (e.g. N-S, E-W station keeping of GEO)\n", - "\n", - "Manuevers are implemented by firing the propulsion system, we can see this as an addtional term in the equations of motion:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "```{math}\n", - ":label: L7_1\n", - "\\ddot{\\mathbf{r}} = -\\frac{\\mu}{r^3} \\mathbf{r} + c(\\mathbf{r}, \\dot{\\mathbf{r}}, t)\n", - "```" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "- Firing the propulsion system means need propellant. As launching mass is expensive we need to design control laws that allow the operator to implement the transfer with **minimum propellant**: optimal control theory\n", - "\n", - "- With high thrust rocket engines the firing time is short and the maneuver can be modelled as an instantaneous change in the velocity: impulsive $\\Delta \\mathbf{v}$\n", - "\n", - "- We search for minimum $|{\\Delta \\mathbf{v}}|$ maneuver because they require minimum propellant mass!\n", - "\n", - "- Frequently an orbital maneuver has the objective of changing (increasing) the orbital energy (e.g. moving from a Low-Earth orbit to a GEO, compensate for atmospheric drag,...)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### Effect of a $\\Delta \\mathbf{v}$ on orbital energy" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "![Figure 1](images/L7img1.png)\n", - "\n", - "- _Before_ the maneuver: \n", - "\n", - "```{math}\n", - ":label: L7_2\n", - "E_0 = \\frac{1}{2}v_0^2 - \\frac{\\mu}{r}\n", - "```\n", - "\n", - "- _After_ the maneuver:\n", - "\n", - "```{math}\n", - ":label: L7_3\n", - "E_1 = \\frac{1}{2}(\\mathbf {v_0^2}+\\Delta \\mathbf{v}) \\cdot(\\mathbf{v_0} + \\Delta \\mathbf{v}) - \\frac{\\mu}{r} = \\frac{1}{2} v_0^2 - \\Delta \\mathbf{v} + {\\frac{1}{2} \\Delta v^2 - \\frac{\\mu}{r}}\n", - "```\n", - "\n", - "- _Change_ in Energy\n", - "\n", - "```{math}\n", - ":label: L7_4\n", - "\\Delta E = E_1 - E_0 = \\frac{1}{2}\\Delta v^2 + v_0\\Delta v \\cos(\\alpha)\n", - "```\n", - "\n", - "Here $v_0\\Delta v \\cos(\\alpha)$ is the dominant term.\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For a given $\\Delta v$ (i.e. propellant) how can we maximise the energy change $\\Delta E$?\n", - "\n", - "- $\\Delta E$ is maximised when:\n", - "1. $v_0$ is high $\\mathbf{\\rightarrow}$ applying the maneuver when $v_0$ is high (e.g. at perigee)\n", - "2. $\\cos(\\alpha) = 1$ $ \\rightarrow$ $\\alpha = 0$ $\\rightarrow$ $\\mathbf{v}$ and $\\mathbf{\\Delta v}$ are aligned $\\rightarrow$ tangential burn" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$\\bf Note$: $v_0\\Delta v \\cos(\\alpha)$ can be negative $\\rightarrow$ it is possible to reduce E" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### Effect of a $\\Delta \\mathbf{v}$ on angular momentum" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "![Figure 2](images/L7img2.png)\n", - "\n", - "- _Before_ the maneuver: \n", - "\n", - "```{math}\n", - ":label: L7_5\n", - "\\mathbf{h_0} = \\mathbf{r} \\times \\mathbf{v_0}\n", - "```\n", - "\n", - "- _After_ the maneuver:\n", - "\n", - "```{math}\n", - ":label: L7_6\n", - "\\mathbf{h_1} = \\mathbf{r} \\times (\\mathbf{v_0} + \\Delta \\mathbf{v})\n", - "```\n", - "- _Change in angular momentum_\n", - "\n", - "```{math}\n", - ":label: L7_7\n", - "\\Delta \\mathbf{h} = \\mathbf{h_1} - \\mathbf{h_0} = \\mathbf{r} \\times \\Delta \\mathbf{v}\n", - "```\n", - "\n", - "```{math}\n", - ":label: L7_8\n", - "\\Delta h = r \\Delta v \\sin(\\beta)=r \\Delta V_{\\theta}\n", - "```\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For a given $\\Delta v$ (i.e. propellant) how can we maximise $\\Delta$ $h$?\n", - "1. Execute the maneuver when $r$ is big (e.g. at the apogee)\n", - "2. Change the transverse velocity: $\\Delta \\mathbf{v}$ aligned with $\\mathbf {v_{\\theta}}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**Note**: A radial maneuver does not change the angular momentum as $\\beta = 0^{\\circ}$\n", - "\n", - "```{math}\n", - ":label: L7_9\n", - "\\Delta h = r \\Delta v \\sin(0)=0\n", - "```" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "(content:hohmann-transfer)=\n", - "## Hohmann Transfer" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "![Figure 3](images/L7img3.png)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The problem is that we need the **most efficient**, **bi-impulsive** transfer between **two circular coplanar** orbits. \n", - "\n", - "**Problem**\n", - "\n", - "We need to transfer the spacecraft from an initial orbit with semi-major axis $a_1$ to a final orbit with semi-major axis $a_3$. The most optimal solution for $\\Delta E$ are tangential maneuvers. \n", - "\n", - "**Solution:**\n", - "\n", - "We draw an ellipse tangent to both the circular orbits and compute the $\\Delta \\mathbf{v}_{s}$!\n", - "\n", - "**Transfer Orbit:**\n", - "\n", - "The transfer orbit $2$ has perigee radius $r_p = a_1$ and apogee radius $r_a = a_3$. The semi-major axis is: \n", - "\n", - "\n", - "```{math}\n", - ":label: L7_10\n", - "a_2 = \\frac{r_p+r_a}{2}\n", - "```\n", - "\n", - "Form $a_2$, $r_p$ and $r_a$ the eccentricity of the transfer orbit can be found to be: \n", - "\n", - "```{math}\n", - ":label: L7_11\n", - "e_2 = \\frac{r_a - r_p}{r_a+r_p}\n", - "```" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### Hohmann transfer, $\\Delta V$ and transfer time" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We use the Vis-Viva equation: $\\frac{1}{2}v^2-\\frac{\\mu}{2a}$ (**Hint: We ALWAYS use this to compute** $\\Delta V$). The steps for calcualting $\\Delta V$ are:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "1. Initial circular velocity :\n", - " ```{math}\n", - " :label: L7_12\n", - " v_{A_1} = \\sqrt{\\frac{\\mu}{a1}}\n", - " ```\n", - "\n", - "2. Final circular velocity :\n", - " ```{math}\n", - " :label: L7_13\n", - " v_{B_3} = \\sqrt{\\frac{\\mu}{a3}}\n", - " ```\n", - "\n", - "3. Initial transfer velocity: \n", - "\n", - "$$\n", - "\n", - "\\frac{1}{2}v_{A2}^2-\\frac{\\mu}{a_1} = -\\frac{\\mu}{2a_2} \\\\ \\rightarrow v_{A_2} = \\sqrt{\\frac{2\\mu}{a_1}-\\frac{\\mu}{a_2}} \\\\ \\rightarrow v_{A_2} = \\sqrt{\\frac{\\mu}{a_1}}\\sqrt{2-\\frac{a1} {a_2}} \\\\ \\rightarrow v_{A_2} = v_{A_1}\\sqrt{\\frac{a_3 + a_1 - a_1}{a_2}} \\\\ = v_{A_1}\\sqrt{\\frac{a_3}{a_2}} >v_{A_1}\n", - "\n", - "$$\n", - "\n", - "4. Final transfer velocity: \n", - "\n", - "$$\n", - "\n", - "\\frac{1}{2}v_{B_2}^2-\\frac{\\mu}{a_3} = -\\frac{\\mu}{2a_2} \n", - "\\\\ \\rightarrow v_{B_2} = \\sqrt{\\frac{2\\mu}{a_3}-\\frac{\\mu}{a_2}} \\\\ \\rightarrow v_{B_2} = \\sqrt{\\frac{\\mu}{a_3}}\\sqrt{2-\\frac{a_3} {a_2}} \\\\ \\rightarrow v_{B_2} = v_{B_3}\\sqrt{\\frac{a_1}{a_2}} Plane change manuevers are very expensive as they require a rotation of the entire velocity vector.\n", - "\n", - "**Problem**: Pure Inclination change. \n", - "\n", - "The **Solution**:\n", - "\n", - "To change the inclination without changing the orbit shape we need to rotate $\\mathbf{h}$ while keeping E and $h$ fixed (thus keeping the orbit shape fixed).\n", - "\n", - "* If E is fixed then v is unchanged \n", - "* If h is fixed then $v_\\theta$ is unchanged \n", - "
The above two bullet points shows rotation about **r** of $\\Delta i$ \n", - "\n", - "Let's take a look at a figure to help visualise things." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "![Figure 6](images/L7img6.png)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "* $\\Delta v =2 v_{\\theta}$ sin $\\frac{\\Delta i}{2}$\n", - "* If circular then $v=v_\\theta \\rightarrow \\Delta v = 2v \\sin\\frac{\\Delta i}{2}$ \n", - "* If $\\Delta i$ is small then sin $\\frac{\\Delta i}{2} \\approx \\frac{\\Delta i}{2} \\rightarrow \\Delta v = v \\Delta i$\n", - " * When possible change _i_ when _v_ is low " - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "base", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.9.13" - } - }, - "nbformat": 4, - "nbformat_minor": 2 + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Orbital Maneuvers\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "_Prepared by:_ [Mitansh Choksi](https://github.com/mitanshc), [Adriana Lopez](https://github.com/adrulpz) and [Angadh Nanjangud](https://www.angadhn.com/)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this lecture we aim to cover the following topics:\n", + "1. [](content:orbit-maneuvers)\n", + "2. [](content:hohmann-transfer)\n", + "3. [](content:bi-elliptic-transfer)\n", + "4. [](content:non-tangential-transfer)\n", + "5. [](content:plane-change-maneuver)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "(content:orbit-maneuvers)=\n", + "## Orbit Maneuvers" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Manuevers are implemented to transfer the spacecraft from one orbit to another.\n", + "Orbital changes can have different objectives:\n", + "- Inject the probe into an interplanetary transfer\n", + "- Move the spacecraft from a parking orbit to an operetive one ( e.g. LEO to GEO)\n", + "- Rendezvous and dock with another spececraft ( e.g. with the ISS)\n", + "- Compensate for the effect of perturbations (e.g. N-S, E-W station keeping of GEO)\n", + "\n", + "Manuevers are implemented by firing the propulsion system, we can see this as an addtional term in the equations of motion:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "```{math}\n", + ":label: L7_1\n", + "\\ddot{\\mathbf{r}} = -\\frac{\\mu}{r^3} \\mathbf{r} + c(\\mathbf{r}, \\dot{\\mathbf{r}}, t)\n", + "```" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "- Firing the propulsion system means need propellant. As launching mass is expensive we need to design control laws that allow the operator to implement the transfer with **minimum propellant**: optimal control theory\n", + "\n", + "- With high thrust rocket engines the firing time is short and the maneuver can be modelled as an instantaneous change in the velocity: impulsive $\\Delta \\mathbf{v}$\n", + "\n", + "- We search for minimum $|{\\Delta \\mathbf{v}}|$ maneuver because they require minimum propellant mass!\n", + "\n", + "- Frequently an orbital maneuver has the objective of changing (increasing) the orbital energy (e.g. moving from a Low-Earth orbit to a GEO, compensate for atmospheric drag,...)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Effect of a $\\Delta \\mathbf{v}$ on orbital energy" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![Figure 1](images/L7-1.png)\n", + "\n", + "- _Before_ the maneuver: \n", + "\n", + "```{math}\n", + ":label: L7_2\n", + "E_0 = \\frac{1}{2}v_0^2 - \\frac{\\mu}{r}\n", + "```\n", + "\n", + "- _After_ the maneuver:\n", + "\n", + "```{math}\n", + ":label: L7_3\n", + "E_1 = \\frac{1}{2}(\\mathbf {v_0}+\\Delta \\mathbf{v}) \\cdot(\\mathbf{v_0} + \\Delta \\mathbf{v}) - \\frac{\\mu}{r} = \\frac{1}{2} v_0^2 + \\mathbf {v_0} \\cdot\\Delta \\mathbf{v} + {\\frac{1}{2} \\Delta v^2 - \\frac{\\mu}{r}}\n", + "```\n", + "\n", + "- _Change_ in Energy\n", + "\n", + "```{math}\n", + ":label: L7_4\n", + "\\Delta E = E_1 - E_0 = \\frac{1}{2}\\Delta v^2 + v_0\\Delta v \\cos(\\alpha)\n", + "```\n", + "\n", + "Here $v_0\\Delta v \\cos(\\alpha)$ is the dominant term.\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "For a given $\\Delta v$ (i.e. propellant) how can we maximise the energy change $\\Delta E$?\n", + "\n", + "- $\\Delta E$ is maximised when:\n", + "1. $v_0$ is high $\\mathbf{\\rightarrow}$ applying the maneuver when $v_0$ is high (e.g. at perigee)\n", + "2. $\\cos(\\alpha) = 1$ $ \\rightarrow$ $\\alpha = 0$ $\\rightarrow$ $\\mathbf{v}$ and $\\mathbf{\\Delta v}$ are aligned $\\rightarrow$ tangential burn" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$\\bf Note$: $v_0\\Delta v \\cos(\\alpha)$ can be negative $\\rightarrow$ it is possible to reduce E" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Effect of a $\\Delta \\mathbf{v}$ on angular momentum" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![Figure 2](images/L7-2.png)\n", + "\n", + "- _Before_ the maneuver: \n", + "\n", + "```{math}\n", + ":label: L7_5\n", + "\\mathbf{h_0} = \\mathbf{r} \\times \\mathbf{v_0}\n", + "```\n", + "\n", + "- _After_ the maneuver:\n", + "\n", + "```{math}\n", + ":label: L7_6\n", + "\\mathbf{h_1} = \\mathbf{r} \\times (\\mathbf{v_0} + \\Delta \\mathbf{v})\n", + "```\n", + "- _Change in angular momentum_\n", + "\n", + "```{math}\n", + ":label: L7_7\n", + "\\Delta \\mathbf{h} = \\mathbf{h_1} - \\mathbf{h_0} = \\mathbf{r} \\times \\Delta \\mathbf{v}\n", + "```\n", + "\n", + "```{math}\n", + ":label: L7_8\n", + "\\Delta h = r \\Delta v \\sin(\\beta)=r \\Delta V_{\\theta}\n", + "```\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "For a given $\\Delta v$ (i.e. propellant) how can we maximise $\\Delta$ $h$?\n", + "1. Execute the maneuver when $r$ is big (e.g. at the apogee)\n", + "2. Change the transverse velocity: $\\Delta \\mathbf{v}$ aligned with $\\mathbf {v_{\\theta}}$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Note**: A radial maneuver does not change the angular momentum as $\\beta = 0^{\\circ}$\n", + "\n", + "```{math}\n", + ":label: L7_9\n", + "\\Delta h = r \\Delta v \\sin(0)=0\n", + "```" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "(content:hohmann-transfer)=\n", + "## Hohmann Transfer" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![Figure 3](images/L7-3.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The problem is that we need the **most efficient**, **bi-impulsive** transfer between **two circular coplanar** orbits. \n", + "\n", + "**Problem**\n", + "\n", + "We need to transfer the spacecraft from an initial orbit with semi-major axis $a_1$ to a final orbit with semi-major axis $a_3$. The most optimal solution for $\\Delta E$ are tangential maneuvers. \n", + "\n", + "**Solution:**\n", + "\n", + "We draw an ellipse tangent to both the circular orbits and compute the $\\Delta \\mathbf{v}_{s}$!\n", + "\n", + "**Transfer Orbit:**\n", + "\n", + "The transfer orbit $2$ has perigee radius $r_p = a_1$ and apogee radius $r_a = a_3$. The semi-major axis is: \n", + "\n", + "\n", + "```{math}\n", + ":label: L7_10\n", + "a_2 = \\frac{r_p+r_a}{2}\n", + "```\n", + "\n", + "Form $a_2$, $r_p$ and $r_a$ the eccentricity of the transfer orbit can be found to be: \n", + "\n", + "```{math}\n", + ":label: L7_11\n", + "e_2 = \\frac{r_a - r_p}{r_a+r_p}\n", + "```" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Hohmann transfer, $\\Delta \\mathbf{v}$ and transfer time" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We use the Vis-Viva equation: $\\frac{1}{2}v^2-\\frac{\\mu}{2a}$ (**Hint: We ALWAYS use this to compute** $\\Delta \\mathbf{v}$ ). The steps for calcualting $\\Delta \\mathbf{v}$ are:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "1. Initial circular velocity :\n", + " ```{math}\n", + " :label: L7_12\n", + " v_{A_1} = \\sqrt{\\frac{\\mu}{a1}}\n", + " ```\n", + "\n", + "2. Final circular velocity :\n", + " ```{math}\n", + " :label: L7_13\n", + " v_{B_3} = \\sqrt{\\frac{\\mu}{a3}}\n", + " ```\n", + "\n", + "3. Initial transfer velocity: \n", + "\n", + "$$\n", + "\n", + "\\frac{1}{2}v_{A2}^2-\\frac{\\mu}{a_1} = -\\frac{\\mu}{2a_2} \\\\ \\rightarrow v_{A_2} = \\sqrt{\\frac{2\\mu}{a_1}-\\frac{\\mu}{a_2}} \\\\ \\rightarrow v_{A_2} = \\sqrt{\\frac{\\mu}{a_1}}\\sqrt{2-\\frac{a1} {a_2}} \\\\ \\rightarrow v_{A_2} = v_{A_1}\\sqrt{\\frac{a_3 + a_1 - a_1}{a_2}} \\\\ = v_{A_1}\\sqrt{\\frac{a_3}{a_2}} >v_{A_1}\n", + "\n", + "$$\n", + "\n", + "4. Final transfer velocity: \n", + "\n", + "$$\n", + "\n", + "\\frac{1}{2}v_{B_2}^2-\\frac{\\mu}{a_3} = -\\frac{\\mu}{2a_2} \n", + "\\\\ \\rightarrow v_{B_2}\\sqrt{\\frac{2\\mu}{a_3}-\\frac{\\mu}{a_2}} \\\\ \\rightarrow v_{B_2} = \\sqrt{\\frac{\\mu}{a_3}}\\sqrt{2-\\frac{a3} {a_2}} \\\\ \\rightarrow v_{B_2} = v_{B_3}\\sqrt{\\frac{a_1}{a_2}} Plane change manuevers are very expensive as they require a rotation of the entire velocity vector.\n", + "\n", + "**Problem**: Pure Inclination change. \n", + "\n", + "The **Solution**:\n", + "\n", + "To change the inclination without changing the orbit shape we need to rotate $\\mathbf{h}$ while keeping E and $h$ fixed (thus keeping the orbit shape fixed).\n", + "\n", + "* If E is fixed then $\\mathbf{v}$ is unchanged \n", + "* If $h$ is fixed then $\\mathbf{v_\\theta}$ is unchanged \n", + "
The above two bullet points shows rotation about **r** of $\\Delta i$ \n", + "\n", + "Let's take a look at a figure to help visualise things." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![Figure 6](images/L7-6.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "* $\\Delta \\mathbf{v} =2 \\mathbf{v_{\\theta}}$ sin $\\frac{\\Delta i}{2}$\n", + "* If circular then $\\mathbf{v}=\\mathbf{v_\\theta \\rightarrow} \\Delta \\mathbf{v} = 2\\mathbf{v} \\sin\\frac{\\Delta i}{2}$ \n", + "* If $\\Delta i$ is small then sin $\\frac{\\Delta i}{2} \\approx \\frac{\\Delta i}{2} \\rightarrow \\Delta \\mathbf{v} = \\mathbf{v} \\Delta i$\n", + " * When possible change _i_ when _v_ is low " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + 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