Elements module¶
¶
The poliastro.core.elements module contains a set of functions related to orbital elements
conversion.
This module contains a set of functions that can be used to convert between different elements that define the orbit of a body.
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poliastro.core.elements.rv_pqw¶ Returns r and v vectors in perifocal frame.
\[ \begin{align}\begin{aligned}\begin{split}\vec{r} = \frac{h^2}{\mu}\frac{1}{1 + e\cos(\theta)}\begin{bmatrix} \cos(\theta)\\ \sin(\theta)\\ 0 \end{bmatrix} \\\\\\\end{split}\\\begin{split}\vec{v} = \frac{h^2}{\mu}\begin{bmatrix} -\sin(\theta)\\ e+\cos(\theta)\\ 0 \end{bmatrix}\end{split}\end{aligned}\end{align} \]Parameters: Returns: - r (ndarray) – Position. Dimension 3 vector
- v (ndarray) – Velocity. Dimension 3 vector
Examples
>>> from poliastro.constants import GM_earth >>> k = GM_earth #Earth gravitational parameter
>>> ecc = 0.3 #Eccentricity >>> h = 60000e6 #Angular momentum of the orbit [m^2]/[s] >>> nu = np.deg2rad(120) #True Anomaly [rad] >>> p = h**2 / k #Parameter of the orbit >>> r, v = rv_pqw(k, p, ecc, nu)
>>> #Printing the results r = [-5312706.25105345 9201877.15251336 0] [m] v = [-5753.30180931 -1328.66813933 0] [m]/[s]
Note
These formulas can be checked at Curtis 3rd. Edition, page 110. Also the example proposed is 2.11 of Curtis 3rd Edition book.
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poliastro.core.elements.coe_rotation_matrix¶ Create a rotation matrix for coe transformation
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poliastro.core.elements.coe2rv¶ Converts from classical orbital to state vectors.
Classical orbital elements are converted into position and velocity vectors by rv_pqw algorithm. A rotation matrix is applied to position and velocity vectors to get them expressed in terms of an IJK basis.
\[\begin{split}\begin{align} \vec{r}_{IJK} &= [ROT3(-\Omega)][ROT1(-i)][ROT3(-\omega)]\vec{r}_{PQW} = \left [ \frac{IJK}{PQW} \right ]\vec{r}_{PQW}\\ \vec{v}_{IJK} &= [ROT3(-\Omega)][ROT1(-i)][ROT3(-\omega)]\vec{v}_{PQW} = \left [ \frac{IJK}{PQW} \right ]\vec{v}_{PQW}\\ \end{align}\end{split}\]Previous rotations (3-1-3) can be expressed in terms of a single rotation matrix:
\[\left [ \frac{IJK}{PQW} \right ]\]\[\begin{split}\begin{bmatrix} \cos(\Omega)\cos(\omega) - \sin(\Omega)\sin(\omega)\cos(i) & -\cos(\Omega)\sin(\omega) - \sin(\Omega)\cos(\omega)\cos(i) & \sin(\Omega)\sin(i)\\ \sin(\Omega)\cos(\omega) + \cos(\Omega)\sin(\omega)\cos(i) & -\sin(\Omega)\sin(\omega) + \cos(\Omega)\cos(\omega)\cos(i) & -\cos(\Omega)\sin(i)\\ \sin(\omega)\sin(i) & \cos(\omega)\sin(i) & \cos(i) \end{bmatrix}\end{split}\]Parameters: Returns: - r_ijk (np.array) – Position vector in basis ijk
- v_ijk (np.array) – Velocity vector in basis ijk
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poliastro.core.elements.coe2mee¶ Converts from classical orbital elements to modified equinoctial orbital elements.
The definition of the modified equinoctial orbital elements is taken from [Walker, 1985].
The modified equinoctial orbital elements are a set of orbital elements that are useful for trajectory analysis and optimization. They are valid for circular, elliptic, and hyperbolic orbits. These direct modified equinoctial equations exhibit no singularity for zero eccentricity and orbital inclinations equal to 0 and 90 degrees. However, two of the components are singular for an orbital inclination of 180 degrees.
\[\begin{split}\begin{align} p &= a(1-e^2) \\ f &= e\cos(\omega + \Omega) \\ g &= e\sin(\omega + \Omega) \\ h &= \tan(\frac{i}{2})\cos(\Omega) \\ k &= \tan(\frac{i}{2})\sin(\Omega) \\ L &= \Omega + \omega + \theta \\ \end{align}\end{split}\]Parameters: Returns: - p (float) – Semi-latus rectum or parameter
- f (float) – Equinoctial parameter f
- g (float) – Equinoctial parameter g
- h (float) – Equinoctial parameter h
- k (float) – Equinoctial parameter k
- L (float) – Longitude
Note
The conversion equations are taken directly from the original paper.
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poliastro.core.elements.rv2coe¶ Converts from vectors to classical orbital elements.
- First the angular momentum is computed:
- \[\vec{h} = \vec{r} \times \vec{v}\]
- With it the eccentricity can be solved:
- \[\begin{split}\begin{align} \vec{e} &= \frac{1}{\mu}\left [ \left ( v^{2} - \frac{\mu}{r}\right ) \vec{r} - (\vec{r} \cdot \vec{v})\vec{v} \right ] \\ e &= \sqrt{\vec{e}\cdot\vec{e}} \\ \end{align}\end{split}\]
- The node vector line is solved:
- \[\begin{split}\begin{align} \vec{N} &= \vec{k} \times \vec{h} \\ N &= \sqrt{\vec{N}\cdot\vec{N}} \end{align}\end{split}\]
- The rigth ascension node is computed:
- \[\begin{split}\Omega = \left\{ \begin{array}{lcc} cos^{-1}{\left ( \frac{N_{x}}{N} \right )} & if & N_{y} \geq 0 \\ \\ 360^{o} -cos^{-1}{\left ( \frac{N_{x}}{N} \right )} & if & N_{y} < 0 \\ \end{array} \right.\end{split}\]
- The argument of perigee:
- \[\begin{split}\omega = \left\{ \begin{array}{lcc} cos^{-1}{\left ( \frac{\vec{N}\vec{e}}{Ne} \right )} & if & e_{z} \geq 0 \\ \\ 360^{o} -cos^{-1}{\left ( \frac{\vec{N}\vec{e}}{Ne} \right )} & if & e_{z} < 0 \\ \end{array} \right.\end{split}\]
- And finally the true anomaly:
- \[\begin{split}\nu = \left\{ \begin{array}{lcc} cos^{-1}{\left ( \frac{\vec{e}\vec{r}}{er} \right )} & if & v_{r} \geq 0 \\ \\ 360^{o} -cos^{-1}{\left ( \frac{\vec{e}\vec{r}}{er} \right )} & if & v_{r} < 0 \\ \end{array} \right.\end{split}\]
Parameters: Returns: - p (float) – Semi-latus rectum of parameter (km)
- ecc (float) – Eccentricity
- inc (float) – Inclination (rad)
- raan (float) – Right ascension of the ascending nod (rad)
- argp (float) – Argument of Perigee (rad)
- nu (float) – True Anomaly (rad)
Examples
>>> from poliastro.constants import GM_earth >>> from astropy import units as u >>> k = GM_earth.to(u.km ** 3 / u.s ** 2).value # Earth gravitational parameter >>> r = np.array([-6045., -3490., 2500.]) >>> v = np.array([-3.457, 6.618, 2.533]) >>> p, ecc, inc, raan, argp, nu = rv2coe(k, r, v) >>> print("p:", p, "[km]") p: 8530.47436396927 [km] >>> print("ecc:", ecc) ecc: 0.17121118195416898 >>> print("inc:", np.rad2deg(inc), "[deg]") inc: 153.2492285182475 [deg] >>> print("raan:", np.rad2deg(raan), "[deg]") raan: 255.27928533439618 [deg] >>> print("argp:", np.rad2deg(argp), "[deg]") argp: 20.068139973005366 [deg] >>> print("nu:", np.rad2deg(nu), "[deg]") nu: 28.445804984192122 [deg]
Note
This example is a real exercise from Orbital Mechanics for Engineering students by Howard D.Curtis. This exercise is 4.3 of 3rd. Edition, page 200.
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poliastro.core.elements.mee2coe¶ Converts from modified equinoctial orbital elements to classical orbital elements.
The definition of the modified equinoctial orbital elements is taken from [Walker, 1985].
\[\begin{split}\begin{align} p &= a(1 - e^{2})\\ e &= \sqrt{f^{2} + g^{2}}\\ i &= 2\arctan{(\sqrt{h^{2} + k^{2}})}\\ raan &= atan2(k, h) \pmod{2\pi}\\ argp &= (atan2(g, f) - raan) \pmod{2\pi}\\ nu &= (L - atan2(g, f)) \pmod{2\pi}\\ \end{align}\end{split}\]Parameters: Returns: - p (float) – Semi-latus rectum
- ecc (float) – Eccentricity of the orbit
- inc (float) – Inclination of the orbit
- raan (float) – RAAN of orbit
- argp (float) – Argument of the periapsis
- nu (float) – True anomaly
Note
The conversion is always safe because arctan2 works also for 0, 0 arguments.