Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles
This papers introduces an analytic method to define multistage launcher trajectories to determine the payload mass that can be inserted in orbits of different semimajor axes and inclinations. This method can evaluate the gravity loss, which is the main term to be subtracted to the Tziolkowski evalua...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-06-01
|
| Series: | Applied Sciences |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2076-3417/12/11/5685 |
| _version_ | 1797494041652232192 |
|---|---|
| author | Paolo Teofilatto Stefano Carletta Mauro Pontani |
| author_facet | Paolo Teofilatto Stefano Carletta Mauro Pontani |
| author_sort | Paolo Teofilatto |
| collection | DOAJ |
| description | This papers introduces an analytic method to define multistage launcher trajectories to determine the payload mass that can be inserted in orbits of different semimajor axes and inclinations. This method can evaluate the gravity loss, which is the main term to be subtracted to the Tziolkowski evaluation of the velocity provided by the thrust of a launcher. In the method, the trajectories are dependent on two parameters only: the final flight-path angle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>f</mi></msub></semantics></math></inline-formula> at the end of the gravity-turn arc of the launcher trajectory and the duration <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>c</mi></msub></semantics></math></inline-formula> of the coasting arc following the gravity-turn phase. The analytic formulas for the gravity-turn phase, being solutions of differential equations with a singularity, allow us to identify the trajectory with a required final flight-path angle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>f</mi></msub></semantics></math></inline-formula> in infinite solutions with the same initial vertical launch condition. This can also drive the selection of the parameters of the pitch manoeuvre needed to turn the launcher from the initial vertical arc. For any pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>f</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>c</mi></msub></semantics></math></inline-formula>, a launcher trajectory is determined. A numerical solver is used to identify the values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>f</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>c</mi></msub></semantics></math></inline-formula>, allowing for the insertion of the payload mass into the required orbit. The analytic method is compared with a numerical code including the drag effect, which is the only effect overlooked in the analytic formulas. The analytical method is proven to predict the payload mass with an error never exceeding the 10% of the actual payload mass, found through numerical propagation. |
| first_indexed | 2024-03-10T01:28:37Z |
| format | Article |
| id | doaj.art-4d47016e1ef74d58abfceeaa753ef75a |
| institution | Directory Open Access Journal |
| issn | 2076-3417 |
| language | English |
| last_indexed | 2024-03-10T01:28:37Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Applied Sciences |
| spelling | doaj.art-4d47016e1ef74d58abfceeaa753ef75a2023-11-23T13:46:08ZengMDPI AGApplied Sciences2076-34172022-06-011211568510.3390/app12115685Analytic Derivation of Ascent Trajectories and Performance of Launch VehiclesPaolo Teofilatto0Stefano Carletta1Mauro Pontani2School of Aerospace Engineering, Sapienza University of Rome, Via Salaria 851, 00138 Rome, ItalySchool of Aerospace Engineering, Sapienza University of Rome, Via Salaria 851, 00138 Rome, ItalyDepartment of Astronautical, Electrical, and Energy Engineering, Sapienza University of Rome, Via Salaria 851, 00138 Rome, ItalyThis papers introduces an analytic method to define multistage launcher trajectories to determine the payload mass that can be inserted in orbits of different semimajor axes and inclinations. This method can evaluate the gravity loss, which is the main term to be subtracted to the Tziolkowski evaluation of the velocity provided by the thrust of a launcher. In the method, the trajectories are dependent on two parameters only: the final flight-path angle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>f</mi></msub></semantics></math></inline-formula> at the end of the gravity-turn arc of the launcher trajectory and the duration <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>c</mi></msub></semantics></math></inline-formula> of the coasting arc following the gravity-turn phase. The analytic formulas for the gravity-turn phase, being solutions of differential equations with a singularity, allow us to identify the trajectory with a required final flight-path angle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>f</mi></msub></semantics></math></inline-formula> in infinite solutions with the same initial vertical launch condition. This can also drive the selection of the parameters of the pitch manoeuvre needed to turn the launcher from the initial vertical arc. For any pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>f</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>c</mi></msub></semantics></math></inline-formula>, a launcher trajectory is determined. A numerical solver is used to identify the values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>f</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>c</mi></msub></semantics></math></inline-formula>, allowing for the insertion of the payload mass into the required orbit. The analytic method is compared with a numerical code including the drag effect, which is the only effect overlooked in the analytic formulas. The analytical method is proven to predict the payload mass with an error never exceeding the 10% of the actual payload mass, found through numerical propagation.https://www.mdpi.com/2076-3417/12/11/5685multistage launch vehiclesascent trajectory optimizationanalytical performance evaluationrocket staging |
| spellingShingle | Paolo Teofilatto Stefano Carletta Mauro Pontani Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles Applied Sciences multistage launch vehicles ascent trajectory optimization analytical performance evaluation rocket staging |
| title | Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles |
| title_full | Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles |
| title_fullStr | Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles |
| title_full_unstemmed | Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles |
| title_short | Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles |
| title_sort | analytic derivation of ascent trajectories and performance of launch vehicles |
| topic | multistage launch vehicles ascent trajectory optimization analytical performance evaluation rocket staging |
| url | https://www.mdpi.com/2076-3417/12/11/5685 |
| work_keys_str_mv | AT paoloteofilatto analyticderivationofascenttrajectoriesandperformanceoflaunchvehicles AT stefanocarletta analyticderivationofascenttrajectoriesandperformanceoflaunchvehicles AT mauropontani analyticderivationofascenttrajectoriesandperformanceoflaunchvehicles |