Problem 45800. SatCom #6: Inclination of a Sun-Synchronous Orbit

Created by Mark Posen

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{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Satellite and Space Engineering - Problem #5</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>This is part of a series of problems looking at topics in satellite and space communications and systems engineering.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>A particularly interesting (and useful) orbit is the 'Sun-Synchronous Orbit.' This orbit has the special feature that the plane of the orbit precesses (rotates) in inertial space at exactly the same rate as the earth rotates around the sun. Therefore, the orbit plane always maintains a fixed angle with respect to the sun, which means that the satellite always passes over the same point on the ground at the same local mean solar time. Now, satellite orbits, in the absence of external forces, will not precess, but will remain on a plane fixed with respect to inertial space. However, the unequal forces on the satellite caused by the equatorial bulge of the Earth tends to make inclined orbits precess, and by tuning the orbit inclination and altitude (actually the semi-major axis and eccentricity of the orbit ellipse), the orbit can be made to precess at just the right angular rate to maintain a fixed direction towards the sun. (See:</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Sun-synchronous_orbit\"><w:r><w:t>&lt;https://en.wikipedia.org/wiki/Sun-synchronous_orbit</w:t></w:r></w:hyperlink><w:r><w:t>&gt; for more information about such orbits.)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>You are given the satellite orbit's apogee and perigee altitudes (in km). Calculate the inclination needed to achieve a sun-synchronous orbit.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>You should take the radius of the Earth to be 6371km.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Hint : If you are not sure about how to derive the semi-major axis and eccentricity of the orbit given its apogee altitude, perigee altitude and the Earth's radius, you probably ought to try</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>Problem 45797. SatCom #5: Determine Elliptical Orbit Parameters</w:t></w:r><w:r><w:t> first (</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/45797-satcom-5-determine-elliptical-orbit-parameters\"><w:r><w:t>&lt;https://www.mathworks.com/matlabcentral/cody/problems/45797-satcom-5-determine-elliptical-orbit-parameters</w:t></w:r></w:hyperlink><w:r><w:t>&gt; ).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Example: The CLOUDSAT satellite has an apogee of 710 km and a perigee of 709 km. It's orbit inclination is approximately 98.2 degrees.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>Some future problems in this series will build on work done in previous problems, so if you get a working solution I suggest you hang onto the code!</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 525px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 262.5px; transform-origin: 407px 262.5px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 157.5px 8px; transform-origin: 157.5px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">Satellite and Space Engineering - Problem #5</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 366.5px 8px; transform-origin: 366.5px 8px; unicode-bidi: normal; "><span style="font-style: italic; ">This is part of a series of problems looking at topics in satellite and space communications and systems engineering.</span></span></div><div style="block-size: 189px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 94.5px; text-align: left; transform-origin: 384px 94.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 8px; transform-origin: 384px 8px; unicode-bidi: normal; "><span style="">A particularly interesting (and useful) orbit is the 'Sun-Synchronous Orbit.' This orbit has the special feature that the plane of the orbit precesses (rotates) in inertial space at exactly the same rate as the earth rotates around the sun. Therefore, the orbit plane always maintains a fixed angle with respect to the sun, which means that the satellite always passes over the same point on the ground at the same local mean solar time. Now, satellite orbits, in the absence of external forces, will not precess, but will remain on a plane fixed with respect to inertial space. However, the unequal forces on the satellite caused by the equatorial bulge of the Earth tends to make inclined orbits precess, and by tuning the orbit inclination and altitude (actually the semi-major axis and eccentricity of the orbit ellipse), the orbit can be made to precess at just the right angular rate to maintain a fixed direction towards the sun. (See:</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://en.wikipedia.org/wiki/Sun-synchronous_orbit"><span style=""><span style="">&lt;https://en.wikipedia.org/wiki/Sun-synchronous_orbit</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.5px 8px; transform-origin: 16.5px 8px; unicode-bidi: normal; "><span style="">&gt; for more information about such orbits.)</span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382px 8px; transform-origin: 382px 8px; unicode-bidi: normal; "><span style="">You are given the satellite orbit's apogee and perigee altitudes (in km). Calculate the inclination needed to achieve a sun-synchronous orbit.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 170px 8px; transform-origin: 170px 8px; unicode-bidi: normal; "><span style="">You should take the radius of the Earth to be 6371km.</span></span></div><div style="block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.5px 8px; transform-origin: 372.5px 8px; unicode-bidi: normal; "><span style="">Hint : If you are not sure about how to derive the semi-major axis and eccentricity of the orbit given its apogee altitude, perigee altitude and the Earth's radius, you probably ought to try</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "><span style=""> </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 165.5px 8px; transform-origin: 165.5px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">Problem 45797. SatCom #5: Determine Elliptical Orbit Parameters</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; "><span style=""> first (</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/45797-satcom-5-determine-elliptical-orbit-parameters"><span style=""><span style="">&lt;https://www.mathworks.com/matlabcentral/cody/problems/45797-satcom-5-determine-elliptical-orbit-parameters</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.5px 8px; transform-origin: 10.5px 8px; unicode-bidi: normal; "><span style="">&gt; ).</span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 8px; transform-origin: 384px 8px; unicode-bidi: normal; "><span style="">Example: The CLOUDSAT satellite has an apogee of 710 km and a perigee of 709 km. It's orbit inclination is approximately 98.2 degrees.</span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 8px; transform-origin: 384px 8px; unicode-bidi: normal; "><span style="font-style: italic; ">Some future problems in this series will build on work done in previous problems, so if you get a working solution I suggest you hang onto the code!</span></span></div></div></div>

Satellite and Space Engineering - Problem #5
This is part of a series of problems looking at topics in satellite and space communications and systems engineering.
A particularly interesting (and useful) orbit is the 'Sun-Synchronous Orbit.' This orbit has the special feature that the plane of the orbit precesses (rotates) in inertial space at exactly the same rate as the earth rotates around the sun. Therefore, the orbit plane always maintains a fixed angle with respect to the sun, which means that the satellite always passes over the same point on the ground at the same local mean solar time. Now, satellite orbits, in the absence of external forces, will not precess, but will remain on a plane fixed with respect to inertial space. However, the unequal forces on the satellite caused by the equatorial bulge of the Earth tends to make inclined orbits precess, and by tuning the orbit inclination and altitude (actually the semi-major axis and eccentricity of the orbit ellipse), the orbit can be made to precess at just the right angular rate to maintain a fixed direction towards the sun. (See: <https://en.wikipedia.org/wiki/Sun-synchronous_orbit> for more information about such orbits.)
You are given the satellite orbit's apogee and perigee altitudes (in km). Calculate the inclination needed to achieve a sun-synchronous orbit.
You should take the radius of the Earth to be 6371km.
Hint : If you are not sure about how to derive the semi-major axis and eccentricity of the orbit given its apogee altitude, perigee altitude and the Earth's radius, you probably ought to try Problem 45797. SatCom #5: Determine Elliptical Orbit Parameters first ( <https://www.mathworks.com/matlabcentral/cody/problems/45797-satcom-5-determine-elliptical-orbit-parameters> ).
Example: The CLOUDSAT satellite has an apogee of 710 km and a perigee of 709 km. It's orbit inclination is approximately 98.2 degrees.
Some future problems in this series will build on work done in previous problems, so if you get a working solution I suggest you hang onto the code!

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Matteo Valentinuzzi on 30 Aug 2021

The link for the first hint doesn't work. Using the equations and coefficients in the wikipedia page gives me results close to the solutions, but not close enough for the solution check to pass. Could you please write the values of the coefficients to be used, or provide some info that was contained in the first hint?

Chris on 30 Aug 2021

Yeah, I used real J2, RAAN* and mu with no love. http://www.satobs.org/seesat/Dec-2018/0040.html Use this guy's equation and it works.

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Problem 45800. SatCom #6: Inclination of a Sun-Synchronous Orbit

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