Formulario di meccanica del volo spaziale

It looks like you've provided a detailed set of orbital mechanics formulas and parameters. Here's an organized summary to help clarify the key points: ### Orbital Mechanics Formulas #### 1. **Keplerian Elements:** - **Mean Motion (n):** \[ n = \sqrt{\frac{\mu}{a^3}} \] - **True Anomaly (ν):** \[ \nu = 2\arctan\left(\sqrt{\frac{1+e}{1-e}} \tan\left(\frac{\theta}{2}\right)\right) \] - **Eccentric Anomaly (E):** \[ E = 2\arctan\left(\sqrt{\frac{1-e}{1+e}} \tan\left(\frac{\nu}{2}\right)\right) \] #### 2. **Kepler's Equation:** - To solve for the eccentric anomaly \(E\) given mean anomaly \(M\): \[ M = E - e \sin(E) \] This is solved iteratively. #### 3. **Orbital Elements from Position and Velocity Vectors:** - **Specific Angular Momentum (\(\mathbf{h}\)):** \[ \mathbf{h} = \mathbf{r} \times \mathbf{v} \] - **Eccentricity Vector (\(\mathbf{e}\)):** \[ \mathbf{e} = \frac{\mathbf{v} \times \mathbf{h}}{\mu} - \frac{\mathbf{r}}{|\mathbf{r}|} \] - **Semilatus Rectum (\(p\)):** \[ p = \frac{h^2}{\mu} \] - **Eccentricity (\(e\)):** \[ e = |\mathbf{e}| \] #### 4. **Keplerian Elements from Orbital Elements:** - **Semi-major axis (\(a\)):** \[ a = \frac{p}{1-e^2} \] - **Argument of Perigee (\(\omega\)):** \[ \omega = \arctan\left(\frac{\mathbf{e}_y}{\mathbf{e}_x}\right) \] - **True Longitude (\(L\)):** \[ L = \nu + \omega \] #### 5. **Orbital Elements from Keplerian Elements:** - **Semi-major axis (\(a\)):** \[ a = \frac{p}{1-e^2} \] - **Eccentricity (\(e\)):** \[ e \] - **Argument of Perigee (\(\omega\)):** \[ \omega \] ### Parametric Formulations #### 6. **Classical Orbital Elements:** - **Inclination (\(i\)):** \[ i = \arccos\left(\frac{h_z}{|\mathbf{h}|}\right) \] - **Right Ascension of the Ascending Node (\(\Omega\)):** \[ \Omega = \arctan\left(\frac{h_y}{h_x}\right) \] #### 7. **Alternative Orbital Elements:** - **Argument of Perigee (\(\omega\)):** \[ \omega