{"id":101,"date":"2024-10-17T16:36:14","date_gmt":"2024-10-17T15:36:14","guid":{"rendered":"https:\/\/blogs.imperial.ac.uk\/parikshit\/?p=101"},"modified":"2024-10-17T16:36:14","modified_gmt":"2024-10-17T15:36:14","slug":"long-term-orbital-perturbations-of-satellites-due-to-solar-radiation-pressure","status":"publish","type":"post","link":"https:\/\/blogs.imperial.ac.uk\/parikshit\/2024\/10\/17\/long-term-orbital-perturbations-of-satellites-due-to-solar-radiation-pressure\/","title":{"rendered":"Long-Term Orbital Perturbations of Satellites Due to Solar Radiation Pressure"},"content":{"rendered":"<figure id=\"attachment_120\" aria-describedby=\"caption-attachment-120\" style=\"width: 1024px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" class=\"size-full wp-image-120\" src=\"http:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Final_PSPAlfvenWave_Version2_NoTransitions_H264.00400_print.jpg\" alt=\"\" \/><figcaption id=\"caption-attachment-120\" class=\"wp-caption-text\">CREDIT: NASA\/Johns Hopkins APL\/Ben Smith<\/figcaption><\/figure>\n<p>Hello again! Welcome to the follow-up blog from the previous one where we discussed How Solar Radiation Pressure Affects Satellites: Calculating Instantaneous Force and Acceleration. As the title suggests, this time we&#8217;ll dive deeper into understanding how to calculate long-term orbital perturbations due to solar radiation pressure.<\/p>\n<p>This is a vast topic, but my goal is to simplify it as much as possible for everyone. These blogs are aimed at providing a foundational introduction to the fascinating realm of orbital mechanics.<\/p>\n<p>Now, I have a confession\u2014if you&#8217;re a mathematician or physicist, can we have a video call? Sometimes I wonder if I should have pursued Astrophysics instead of diving into AI and Robotics. ? Just kidding! I&#8217;m much better in this field, and I can&#8217;t imagine myself knee-deep in equations with Einstein-style hair. At least, I like to think I&#8217;m cool&#8230; Please don\u2019t get offended by my nerdy jokes!<\/p>\n<p>Alright, enough yapping\u2014let\u2019s get back to the topic. To quickly recap what we discussed in the last blog:<\/p>\n<p><strong>Solar radiation pressure<\/strong> is the force exerted by photons emitted by the Sun when they strike a satellite&#8217;s surface. The magnitude of this pressure depends on:<\/p>\n<ul>\n<li>The satellite\u2019s cross-sectional area exposed to sunlight,<\/li>\n<li>The reflectivity of its surfaces,<\/li>\n<li>Its orientation relative to the Sun,<\/li>\n<li>And its distance from the Sun (though for low Earth orbit, this is usually assumed constant).<\/li>\n<\/ul>\n<p>While the force from solar radiation pressure is small, over time it can cause significant changes in a satellite&#8217;s orbital elements. In the last blog, we focused on calculating the immediate effect of this pressure, but not how it accumulates and affects the spacecraft over time.<\/p>\n<p>This blog will cover exactly that.<\/p>\n<p>To make it more realistic and understandable, we&#8217;ll use <strong>EIRSAT-1<\/strong> as a case study\u2014one of the most well-known student-led CubeSat missions, launched by the students of <strong>University College Dublin<\/strong> as part of the <strong>European Space Agency&#8217;s Fly Your Satellite!<\/strong> programme.<\/p>\n<p>I didn\u2019t find extensive details about the mission (Wikipedia had some info, but we all know how trustworthy that is), so I gathered what I could from official publications and real-time data from the <strong>n2yo<\/strong> website. As always, I\u2019ll provide links below for anyone who wants to fact-check.<\/p>\n<p>Now, let\u2019s get all the information down properly and begin!<\/p>\n<table>\n<thead>\n<tr>\n<th colspan=\"2\">EIRSAT-1 Satellite Data<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Latitude<\/td>\n<td>-41.18\u00b0<\/td>\n<\/tr>\n<tr>\n<td>Longitude<\/td>\n<td>74.18\u00b0<\/td>\n<\/tr>\n<tr>\n<td>Altitude<\/td>\n<td>481.19 km (299 mi)<\/td>\n<\/tr>\n<tr>\n<td>Speed<\/td>\n<td>7.62 km\/s (4.73 mi\/s)<\/td>\n<\/tr>\n<tr>\n<td>Azimuth<\/td>\n<td>232.0\u00b0 SW<\/td>\n<\/tr>\n<tr>\n<td>Elevation<\/td>\n<td>-55\u00b0<\/td>\n<\/tr>\n<tr>\n<td>Right Ascension<\/td>\n<td>07h 51m 50s<\/td>\n<\/tr>\n<tr>\n<td>Declination<\/td>\n<td>-60\u00b0 20&#8242; 10&#8221;<\/td>\n<\/tr>\n<tr>\n<td>Launch Mass<\/td>\n<td>2.305 kg<\/td>\n<\/tr>\n<tr>\n<td>Dimensions<\/td>\n<td>10.67 cm \u00d7 10.67 cm \u00d7 22.7 cm (4.20 in \u00d7 4.20 in \u00d7 8.94 in)<\/td>\n<\/tr>\n<tr>\n<td>Satellite Period<\/td>\n<td>94 minutes<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Now we have the data let\u2019s try to understand how and what steps are we going to take also before that let\u2019s list down the factors to consider when calculating the effect of solar radiation pressure on an object:<\/p>\n<p>Factors to Consider for Solar Radiation Pressure Calculations:<\/p>\n<p><strong>Orbital Elements:<\/strong><\/p>\n<ul>\n<li>Semi-major axis<\/li>\n<li>Eccentricity<\/li>\n<li>Inclination<\/li>\n<li>Right Ascension of Ascending Node (RAAN)<\/li>\n<li>Argument of Perigee<\/li>\n<li>True Anomaly<\/li>\n<li>Mean Anomaly<\/li>\n<li>Eccentric Anomaly<\/li>\n<\/ul>\n<p><strong>Satellite Geometry:<\/strong><\/p>\n<ul>\n<li>Cross-sectional area<\/li>\n<li>Reflectivity data<\/li>\n<\/ul>\n<p><strong>Material Properties:<\/strong><\/p>\n<ul>\n<li>Absorptivity and emissivity<\/li>\n<li>Degradation rates over time<\/li>\n<\/ul>\n<p><strong>Orientation Data:<\/strong><\/p>\n<ul>\n<li>The satellite\u2019s attitude and orientation over time<\/li>\n<\/ul>\n<p><strong>Distance to the Sun:<\/strong><\/p>\n<ul>\n<li>Updated regularly throughout the year to account for orbital variations<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><strong>Step-by-Step Calculation<\/strong><\/p>\n<h5><strong>Solar Radiation Pressure Calculation<\/strong><\/h5>\n<p>We already know the formula for calculating solar radiation pressure:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"134\" class=\"size-medium wp-image-107 aligncenter\" src=\"http:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Screenshot-2024-10-17-160042-1-300x134.png\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Determine the Cross-Sectional Area<\/strong>: Using the dimensions of<\/p>\n<p>EIRSAT-1 (10.67 cm \u00d7 10.67 cm \u00d7 22.7 cm), the cross-sectional area facing the Sun (depending on its orientation) could be approximated. For the largest face:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"28\" class=\"size-medium wp-image-108 aligncenter\" src=\"http:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Screenshot-2024-10-17-160220-300x28.png\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Force on the Satellite Due to Solar Radiation Pressure<\/strong><\/p>\n<p>We\u2019ll calculate the force exerted by solar radiation pressure using the following formula:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"75\" class=\"size-medium wp-image-109 aligncenter\" src=\"http:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Screenshot-2024-10-17-160407-300x75.png\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Instantaneous Acceleration on the Satellite Due to Solar Radiation Pressure<\/strong><\/p>\n<p>We can calculate the acceleration using Newton\u2019s second law:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"178\" height=\"209\" class=\"size-full wp-image-110 aligncenter\" src=\"http:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Screenshot-2024-10-17-160447.png\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Atmospheric Drag Force Formula<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"221\" height=\"54\" class=\"size-full wp-image-111 aligncenter\" src=\"http:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Screenshot-2024-10-17-160600.png\" alt=\"\" \/><\/p>\n<p><i>F<sub>D<\/sub><\/i> is the drag force in Newtons,<\/p>\n<p><i>C<sub>D<\/sub><\/i> is the drag coefficient (typically between 2.0 and 2.5 for satellites),<\/p>\n<p><i>A<\/i> is the cross-sectional area of the satellite (0.00024228 m<sup>2<\/sup>),<\/p>\n<p><i>\u03c1<\/i> is the atmospheric density at the satellite&#8217;s altitude,<\/p>\n<p><i>v<sup>2<\/sup><\/i> is the satellite\u2019s velocity relative to the atmosphere (7.69 km\/s).<\/p>\n<p>&nbsp;<\/p>\n<p>Now we don\u2019t know what is <strong>Atmospheric Density<\/strong> for that point of time so I used<\/p>\n<p>NRLMSISE 2.0 is an empirical, global reference atmospheric model of the Earth from ground to space. It models the temperatures and densities of the atmosphere&#8217;s components.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"985\" height=\"689\" class=\"alignnone size-full wp-image-113\" src=\"http:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Screenshot-2024-10-17-143642.png\" alt=\"\" \/><\/p>\n<p>Now we get,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"206\" height=\"29\" class=\"wp-image-116 aligncenter\" src=\"http:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Screenshot-2024-10-17-161145.png\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>let\u2019s substitute it in the formula<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"440\" height=\"192\" class=\"size-full wp-image-115 aligncenter\" src=\"http:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Screenshot-2024-10-17-161237.png\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Acceleration due to Drag <\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"202\" height=\"182\" class=\"size-full wp-image-117 aligncenter\" src=\"http:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Screenshot-2024-10-17-161435.png\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>Now Finally,<\/p>\n<h4><strong> Lagrange\u2019s Planetary Equations:<\/strong><\/h4>\n<p>You might wonder why we still need to use <strong>Lagrange\u2019s planetary equations<\/strong> when we\u2019ve already calculated the forces acting on the satellite. These equations allow us to determine <strong>how the orbital elements<\/strong> of a satellite (such as semi-major axis, eccentricity, inclination, and others) <strong>change over time<\/strong> due to external forces.<\/p>\n<p>Since the focus of this blog is to calculate <strong>long-term orbital perturbations<\/strong>, Lagrange\u2019s planetary equations provide a structured way to analyse the cumulative effects of small forces like <strong>solar radiation pressure (SRP)<\/strong> and <strong>atmospheric drag<\/strong>. They give us a clear picture of how a satellite\u2019s orbit evolves over extended periods.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"713\" height=\"479\" class=\"aligncenter wp-image-119 \" src=\"http:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Screenshot-2024-10-17-150832.png\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Above are the formulas for the\u00a0 Lagrange\u2019s Planetary Equations where<\/p>\n<p style=\"text-align: left\"><i>a\u0307<\/i> is the Semi-major Axis<\/p>\n<p style=\"text-align: left\"><i>e\u0307<\/i> is the Eccentricity<\/p>\n<p style=\"text-align: left\"><i>i\u0307<\/i> is the Inclination<\/p>\n<p style=\"text-align: left\"><i>\u03c9\u0307<\/i> is the Argument of Perigee<\/p>\n<p style=\"text-align: left\"><i>\u03a9\u0307<\/i> is the Right Ascension of the Ascending Node<\/p>\n<p style=\"text-align: left\"><i>T\u0307<\/i> is the Time (Orbital Period)<\/p>\n<p>I will be creating a separate blog dedicated to solving these equations in detail. If you have the necessary data for your own satellite, you\u2019ll be able to input it into the equations and follow along to solve for long-term orbital changes.<\/p>\n<p><strong>Disclaimer:\u00a0<\/strong>I am not a mathematician or physicist; the information presented here is sourced from various online references, which I have linked below. If you have any suggestions or feedback, feel free to reach out to me at\u00a0<strong>ppadole@imperial.ac.uk<\/strong>\u00a0?<\/p>\n<p><strong>References<\/strong><\/p>\n<p><a href=\"https:\/\/phys.libretexts.org\/Bookshelves\/Astronomy__Cosmology\/Celestial_Mechanics_(Tatum)\/14%3A_General_Perturbation_Theory\/14.04%3A_Lagrange's_Planetary_Equations#title\" target=\"_blank\" rel=\"noopener\"><br \/>\nGeneral Perturbation Theory &#8211; Lagrange&#8217;s Planetary Equations, LibreTexts<\/a><a href=\"https:\/\/control.asu.edu\/Classes\/MAE462\/462Lecture12.pdf\" target=\"_blank\" rel=\"noopener\"><br \/>\n<\/a><\/p>\n<p><a href=\"https:\/\/control.asu.edu\/Classes\/MAE462\/462Lecture12.pdf\" target=\"_blank\" rel=\"noopener\">Lagrange&#8217;s Planetary Equations, ASU Control Systems<\/a><a href=\"https:\/\/www.eirsat1.ie\/conferences\" target=\"_blank\" rel=\"noopener\"><br \/>\n<\/a><\/p>\n<p><a href=\"https:\/\/www.eirsat1.ie\/conferences\" target=\"_blank\" rel=\"noopener\">EIRSAT-1 Conferences, EIRSAT-1 Official Website<\/a><\/p>\n<p><a href=\"https:\/\/www.spiedigitallibrary.org\/journals\/Journal-of-Astronomical-Telescopes-Instruments-and-Systems\/volume-9\/issue-03\/037001\/Development-description-and-validation-of-the-operations-manual-for-EIRSAT\/10.1117\/1.JATIS.9.3.037001.full?SSO=1#_=_\" target=\"_blank\" rel=\"noopener\">Development, Description, and Validation of the Operations Manual for EIRSAT-1, Journal of Astronomical Telescopes, Instruments, and Systems<\/a><\/p>\n<p><a href=\"https:\/\/kauai.ccmc.gsfc.nasa.gov\/instantrun\/nrlmsis\/\">NRLMSIS Atmosphere Model<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Hello again! Welcome to the follow-up blog from the previous one where we discussed How Solar Radiation Pressure Affects Satellites: Calculating Instantaneous Force and Acceleration. As the title suggests, this time we&#8217;ll dive deeper into understanding how to calculate long-term orbital perturbations due to solar radiation pressure. This is a vast topic, but my goal [&hellip;]<\/p>\n","protected":false},"author":1743,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-101","post","type-post","status-publish","format-standard","hentry","category-maths-and-physics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Long-Term Orbital Perturbations of Satellites Due to Solar Radiation Pressure - Parikshit Padole<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.imperial.ac.uk\/parikshit\/2024\/10\/17\/long-term-orbital-perturbations-of-satellites-due-to-solar-radiation-pressure\/\" \/>\n<meta property=\"og:locale\" content=\"en_GB\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Long-Term Orbital Perturbations of Satellites Due to Solar Radiation Pressure - Parikshit Padole\" \/>\n<meta property=\"og:description\" content=\"Hello again! Welcome to the follow-up blog from the previous one where we discussed How Solar Radiation Pressure Affects Satellites: Calculating Instantaneous Force and Acceleration. As the title suggests, this time we&#8217;ll dive deeper into understanding how to calculate long-term orbital perturbations due to solar radiation pressure. 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Parikshit Padole","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/blogs.imperial.ac.uk\/parikshit\/2024\/10\/17\/long-term-orbital-perturbations-of-satellites-due-to-solar-radiation-pressure\/","og_locale":"en_GB","og_type":"article","og_title":"Long-Term Orbital Perturbations of Satellites Due to Solar Radiation Pressure - Parikshit Padole","og_description":"Hello again! Welcome to the follow-up blog from the previous one where we discussed How Solar Radiation Pressure Affects Satellites: Calculating Instantaneous Force and Acceleration. As the title suggests, this time we&#8217;ll dive deeper into understanding how to calculate long-term orbital perturbations due to solar radiation pressure. This is a vast topic, but my goal [&hellip;]","og_url":"https:\/\/blogs.imperial.ac.uk\/parikshit\/2024\/10\/17\/long-term-orbital-perturbations-of-satellites-due-to-solar-radiation-pressure\/","og_site_name":"Parikshit Padole","article_published_time":"2024-10-17T15:36:14+00:00","og_image":[{"width":1024,"height":576,"url":"https:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Final_PSPAlfvenWave_Version2_NoTransitions_H264.00400_print.jpg","type":"image\/jpeg"}],"author":"Parikshit Padole","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Parikshit Padole","Estimated reading time":"7 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/blogs.imperial.ac.uk\/parikshit\/2024\/10\/17\/long-term-orbital-perturbations-of-satellites-due-to-solar-radiation-pressure\/#article","isPartOf":{"@id":"https:\/\/blogs.imperial.ac.uk\/parikshit\/2024\/10\/17\/long-term-orbital-perturbations-of-satellites-due-to-solar-radiation-pressure\/"},"author":{"name":"Parikshit Padole","@id":"https:\/\/blogs.imperial.ac.uk\/parikshit\/#\/schema\/person\/3446e4644294882b7a20ed3ffcdf67fe"},"headline":"Long-Term Orbital Perturbations of Satellites Due to Solar Radiation Pressure","datePublished":"2024-10-17T15:36:14+00:00","mainEntityOfPage":{"@id":"https:\/\/blogs.imperial.ac.uk\/parikshit\/2024\/10\/17\/long-term-orbital-perturbations-of-satellites-due-to-solar-radiation-pressure\/"},"wordCount":1054,"image":{"@id":"https:\/\/blogs.imperial.ac.uk\/parikshit\/2024\/10\/17\/long-term-orbital-perturbations-of-satellites-due-to-solar-radiation-pressure\/#primaryimage"},"thumbnailUrl":"http:\/\/blogs.imperial.ac.uk\/parikshit\/files\/2024\/10\/Final_PSPAlfvenWave_Version2_NoTransitions_H264.00400_print.jpg","articleSection":["Maths and Physics"],"inLanguage":"en-GB"},{"@type":"WebPage","@id":"https:\/\/blogs.imperial.ac.uk\/parikshit\/2024\/10\/17\/long-term-orbital-perturbations-of-satellites-due-to-solar-radiation-pressure\/","url":"https:\/\/blogs.imperial.ac.uk\/parikshit\/2024\/10\/17\/long-term-orbital-perturbations-of-satellites-due-to-solar-radiation-pressure\/","name":"Long-Term Orbital Perturbations of Satellites Due to Solar Radiation Pressure - 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