Difference between revisions of "Specific energy"
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| − | In [[orbital mechanics]], '''specific energy''' (symbol <math>\epsilon</math>) is the total orbital energy per unit mass of an orbiting body. | + | In [[orbital mechanics]], '''specific energy''' (symbol <math>\epsilon</math>) is the total orbital energy per unit mass of an orbiting body. It is the sum of the kinetic energy and the gravitational potential energy. |
==Circular and elliptical orbits<ref name="SME">J.R. Wertz, D.F. Everett & J.J. Puschell - ''Space mission engineering: The new SMAD''. 2011. pp. 963-970. ISBN 978-1-881883-15-9</ref>== | ==Circular and elliptical orbits<ref name="SME">J.R. Wertz, D.F. Everett & J.J. Puschell - ''Space mission engineering: The new SMAD''. 2011. pp. 963-970. ISBN 978-1-881883-15-9</ref>== | ||
| Line 17: | Line 17: | ||
==Mars circular orbit== | ==Mars circular orbit== | ||
| − | + | For Mars, with <math>\mu</math>= 4.280×10<sup>13</sup> m<sup>3</sup>/s<sup>2</sup>, then for a 1 kg mass at 300 km E= | |
==Earth circular orbit== | ==Earth circular orbit== | ||
| + | For Mars, with <math>\mu</math>= 3.986×10<sup>14</sup> m<sup>3</sup>/s<sup>2</sup>, then for a 1 kg mass at 300 km E= | ||
==References== | ==References== | ||
Latest revision as of 11:50, 26 June 2023
In orbital mechanics, specific energy (symbol Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} ) is the total orbital energy per unit mass of an orbiting body. It is the sum of the kinetic energy and the gravitational potential energy.
Contents
Circular and elliptical orbits[1]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon = - \frac{\mu}{2a} = \frac{V^2}{2} - \frac{\mu}{r} < 0}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a > 0} is the semi-major axis, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is the gravitational parameter for the body being orbited, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is the distance to the body being orbited at some point in time and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is velocity at that time. This relationship comes about because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{V^2}{2}} is the kinetic energy and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \frac{\mu}{2a}} the potential energy of the system.
Parabolic orbits[1]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon = 0}
Since orbital mechanics only concerns itself with changes in orbital energy, the zero could be chosen arbitrarily. It is computationally most convenient to choose the value at escape velocity (i.e. parabolic orbit). This choice makes the semi-major axis inversely proportional to the specific energy and if the mass does not change also to the total orbital energy.
Hyperbolic orbits[1]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon = - \frac{\mu}{2a} = \frac{V^2}{2} - \frac{\mu}{r} > 0}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a < 0} is the semi-transverse axis, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is the gravitational parameter for the body being orbited, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is distance to the body being orbited at some point in time and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is velocity at that time.
Mars circular orbit
For Mars, with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} = 4.280×1013 m3/s2, then for a 1 kg mass at 300 km E=
Earth circular orbit
For Mars, with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} = 3.986×1014 m3/s2, then for a 1 kg mass at 300 km E=
References
- ↑ Jump up to: 1.0 1.1 1.2 J.R. Wertz, D.F. Everett & J.J. Puschell - Space mission engineering: The new SMAD. 2011. pp. 963-970. ISBN 978-1-881883-15-9
