# 2.1. Acceleration Model Set-up¶

Below is a schemaric overview of the various acceleration options in Tudat, and the top-level architecture for how to set up acceleration models.

## 2.1.1. Acceleration settings¶

The settings for accelerations are defined and stored by the user in a SelectedAccelerationMap.

class SelectedAccelerationMap

This is a typedef for:

std::map< std::string, std::map< std::string, std::vector< std::shared_ptr< AccelerationSettings > > > >


This is a double map (with twice a string as a key). The two levels correspond to the names of bodies undergoing an acceleration (first key) , and those for bodies exerting an acceleration (second key). This allows any number of bodies to be propagated, undergoing any number (and type) of accelerations from any number of bodies. In this manner, settings for each required acceleration model are stored in an object of type AccelerationSettings.

For a given environment, most acceleration models are completely defined by:

• Type of acceleration model (a list is provided in the AvailableAcceleration enum in Tudat/Astrodynamics/BasicAstrodynamics/accelerationModelTypes.h).
• Name of body undergoing acceleration
• Name of body exerting acceleration

For instance, when using the following from the Un-guided Capsule Entry:

SelectedAccelerationMap accelerationSettings;
accelerationSettings[ "Apollo" ][ "Earth" ].push_back( std::make_shared< AccelerationSettings >( central_gravity ) );
accelerationSettings[ "Apollo" ][ "Earth" ].push_back( std::make_shared< AccelerationSettings >( aerodynamic ) );


We have defined a point-mass Earth gravity model and an aerodynamic acceleration due to Earth’s atmosphere to be used. In this example, we have only defined the type of the acceleration, without the need for any additional information. All required variables used on the computations of the accelerations are uniquely defined in the Apollo and Earth entries of the body map (provided that the required environment models have been set as discussed in Environment Set-up).

Below a graphical representation of the acceleration setup, with the different model types and top-level architecture.

As was the case for the settings of the environment models, certain types of accelerations require additional information. An important example is the spherical harmonic acceleration. We cannot replace central_gravity with spherical_harmonic_gravity in the above, as there is still an ambiguity in how the acceleration model is defined. In particular, we now also need the maximum degree and order of the gravity field that is to be used in addition to the three properties listed above. Consequently, we have created a dedicated AccelerationSettings derived class for defining spherical harmonic acceleration settings: SphericalHarmonicAccelerationSettings. Updating the above example to use J2, J3 and J4 (maximum degree = 4; maximum order = 0), we now have:

SelectedAccelerationMap accelerationSettings;
accelerationSettings[ "Apollo" ][ "Earth" ].push_back( std::make_shared< SphericalHarmonicAccelerationSettings >( 4, 0 ) );
accelerationSettings[ "Apollo" ][ "Earth" ].push_back( std::make_shared< AccelerationSettings >( aerodynamic ) );


A full list of the available acceleration models, as well as their required input and environment models, is given at the end of this page.

Note

When creating an object of the AccelerationSettings type (or its derived class), you must not provide any of the third body acceleration types (third_body_central_gravity, third_body_spherical_harmonic_gravity, third_body_mutual_spherical_harmonic_gravity) as input. If you wish to use a third-body gravity acceleration (typically from a point mass), simply provide central_gravity as input. Depending on the settings for your central bodies, the code will automatically create the corresponding acceleration object (central or third-body).

Having defined all the required settings for the accelerations in your SelectedAccelerationMap, you can create the actual acceleration models by using the createAccelerationModelsMap function. This function requires four input parameters:

The list of central bodies defines the reference frame origins in which the bodies are propagated. The use of a hierarchical system is perfectly acceptable. For instance, one can propagate the Earth and Mars w.r.t. the Sun, the Sun w.r.t. the barycenter, the Moon w.r.t the Earth, etc. For this case, the central bodies and propagated bodies are defined as:

std::vector< std::string > propagatedBodies;
propagatedBodies.push_back( "Moon" );
propagatedBodies.push_back( "Earth" );
propagatedBodies.push_back( "Mars" );
propagatedBodies.push_back( "Sun" );

std::vector< std::string > centralBodies;
centralBodies.push_back( "Earth" );
centralBodies.push_back( "Sun" );
centralBodies.push_back( "Sun" );
centralBodies.push_back( "SSB" );


There is no hardcoded limit to the number of permitted levels in the frame hierarchy, but it is not allowed to include circular dependencies, i.e. body A w.r.t. body B, body B w.r.t. body C and body C w.r.t. body A. More information of the acceleration models is discussed in Propagator Settings: Basics. The following gives an example on how to create the acceleration model objects:

NamedBodyMap bodyMap;
....
// Create environment here
....
std::vector< std::string > propagatedBodies;
std::vector< std::string > centralBodies;
....
// Set central and propagated bodies here
....
AccelerationMap accelerationModelMap = createAccelerationModelsMap( bodyMap, accelerationMap, propagatedBodies, centralBodies );


Mutual acceleration between bodies being propagated (i.e body A exerting acceleation on body B and vice versa), as is the case for solar system dynamics, is automatically handled by the createAccelerationModelsMap code and requires no specific consideration. Moreover, when creating a gravitational acceleration, the code checks whether it is a direct or a third-body gravitational acceleration and creates the acceleration models accordingly. Similarly, the code automatically checks which value of the gravitational parameter “mu” to use in such computations. For instance, when computing the gravitational acceleration due to the Sun acting on the Earth, mu_Sun is used when propagating w.r.t. the barycenter, whereas mu_Sun + mu_Earth is used when propagating w.r.t. the Sun.

For every acceleration, a model for the current state of the body exerting the acceleration must be available (the state of the body undergoing the acceleration is taken from the numerically propagated state). This means that, in the above example of the Apollo capsule entering Earth’s atmosphere (Un-guided Capsule Entry), we must include one of the following:

• An ephemeris member for Earth.
• Numerically integrate the Earth concurrently with our Apollo vehicle.

For this example, the second option is of course a bit ‘non-standard’. However, for cases where entire planetary systems are propagated, such an approach is typically taken (for certain applications, the numerically propagated body must also have a particular ephemeris member object, as discussed in Propagator Settings: Basics).

## 2.1.2. Available acceleration models¶

As stated above, the createAccelerationModelsMap function uses your environment and settings for the accelerations to automatically retrieve and put together all functions used to calculate the accelerations during each function evaluation of the numerical scheme. For reference, we first provide a bried list of available acceleration models:

• Point-mass gravity (central of third-body)
• Spherical harmonic gravity (central of third-body)
• Mutual spherical harmonic gravity (central of third-body)
• Aerodynamic acceleration
• Thrust acceleration
• Relativistic acceleration correction (IERS 2010 Conventions)
• Empiricical accelerations (constant, sine and cosine of true anomaly components in RSW frame)
• Tidal effect on natural satellites (Lainey et al., 2007, 2012)

Subsequently, we provide details on how to add settings for the model to the SelectedAccelerationMap. In addition, we define the list of environment models required for their creation.

class AccelerationSettings

Base class for setting the accelerations on a body. Settings currently available are the following:

Point mass gravity

Settings for a point mass gravity acceleration. No derived class of AccelerationSettings is required, this acceleration setting are constructed by feeding central_gravity to the constructor. Added to SelectedAccelerationMap as follows, for example of acceleration exerted on “Apollo” by “Earth”:

SelectedAccelerationMap accelerationSettings;
accelerationSettings[ "Apollo" ][ "Earth" ].push_back( std::make_shared< AccelerationSettings >( central_gravity ) );


Requires the following environment models to be defined:

class SphericalHarmonicAccelerationSettings

Settings for the accelerations as set by SphericalHarmonicsGravityFieldSettings. Added to SelectedAccelerationMap as follows, for example of acceleration exerted on “Apollo” by “Earth”:

SelectedAccelerationMap accelerationSettings;
int maximumDegree = 12;
int maximumOrder = 12;
accelerationSettings[ "Apollo" ][ "Earth" ].push_back( std::make_shared< SphericalHarmonicAccelerationSettings >( maximumDegree, maximumOrder ) );


where the gravity field will be expanded up to degree and order 12 in the acceleration model. Requires the following environment models to be defined:

Note

The spherical harmonic acceleration up to degree N and order M includes the point-mass gravity acceleration (which is the degree and order 0 term).

class MutualSphericalHarmonicAccelerationSettings

This model is typically only used for detailed propagation of planetary systems. It is added to SelectedAccelerationMap as follows, for example of acceleration exerted on “Io” by “Jupiter”:

SelectedAccelerationMap accelerationSettings;
int maximumDegreeOfIo = 12;
int maximumOrderOfIo = 12;
int maximumDegreeOfJupiter = 4;
int maximumOrderOfJupiter = 4;
accelerationSettings[ "Io" ][ "Jupiter" ].push_back( std::make_shared< MutualSphericalHarmonicAccelerationSettings >(
maximumDegreeOfJupiter, maximumOrderOfJupiter, maximumDegreeOfIo, maximumOrderOfIo ) );


where the gravity fields of Io and Jupiter will be expanded up to degree and order 12 and 4, respectively, in the acceleration model. Requires the following environment models to be defined:

For the case where a third-body mutual spherical harmonic acceleration (e.g. Ganymede on Io when propagating w.r.t. Jupiter), additional parameters have to be provided that denote the expansion degree/order of the central body, so:

SelectedAccelerationMap accelerationSettings;
int maximumDegreeOfIo = 12;
int maximumOrderOfIo = 12;
int maximumDegreeOfGanymede = 4;
int maximumOrderOfGanymede = 4;
int maximumDegreeOfJupiter = 4;
int maximumOrderOfJupiter = 4;
accelerationSettings[ "Io" ][ "Jupiter" ].push_back( std::make_shared< MutualSphericalHarmonicAccelerationSettings >(
maximumDegreeOfJupiter, maximumOrderOfJupiter, maximumDegreeOfGanymede, maximumOrderOfGanymede, maximumDegreeOfIo, maximumOrderOfIo ) );


where Jupiter now takes the role of central body, instead of body exerting the acceleration.

Aerodynamic acceleration

No derived class of AccelerationSettings required, accessed by feeding aerodynamic to the constructor. Added to SelectedAccelerationMap as follows, for example of acceleration exerted on “Apollo” by “Earth” (e.g. atmosphere model belonging to Earth):

SelectedAccelerationMap accelerationSettings;
accelerationSettings[ "Apollo" ][ "Earth" ].push_back( std::make_shared< AccelerationSettings >( aerodynamic ) );


Requires the following environment models to be defined:

• Atmosphere model for body exerting acceleration (set by AtmosphereSettings).
• Shape model for body exerting acceleration (set by BodyShapeSettings).
• Aerodynamic coefficient interface for body undergoing acceleration (set by AerodynamicCoefficientSettings). NOTE: In the case that the aerodynamic coefficients are defined as a function of the vehicle orientation (e.g. angle of attack and sideslip angle), these angles can be manually or automatically defined.
• Mass model for body undergoing acceleration.
• Current state of body undergoing acceleration and body with atmosphere.

Warning

Defining settings for a vehicle’s orientation, which may influence your aerodynamic force, is done after creating the acceleration models, as discused here.

Cannonball radiation pressure

No derived class of AccelerationSettings required, accessed by feeding cannon_ball_radiation_pressure to the constructor. Added to SelectedAccelerationMap as follows, for example of acceleration exerted on “Apollo” by “Sun”:

SelectedAccelerationMap accelerationSettings;
accelerationSettings[ "Apollo" ][ "Sun" ].push_back( std::make_shared< AccelerationSettings >( cannon_ball_radiation_pressure ) );


Requires the following environment models to be defined:

class ThrustAccelerationSettings

Used to define the resulting accerelations of a thrust force, requiring:

• Mass of body undergoing acceleration.
• Settings for both the direction and magnitude of the thrust force (set by ThrustMagnitudeSettings). These models may in turn have additional environmental dependencies.

Setting up a thrust acceleration is discussed in more detail on the page Thrust Guidance.

class RelativisticAccelerationCorrectionSettings

A first-order (in $$1/c^{2}$$) correction to the acceleration due to the influence of relativity. It implements the model of Chapter 10, Section 3 of the IERS 2010 Conventions. It requires a specific derived class of AccelerationSettings. Added to SelectedAccelerationMap as follows, for example that includes all three contributions (Schwarzschild, Lense-Thirring and de Sitter)

SelectedAccelerationMap accelerationSettings;
bool calculateSchwarzschildCorrection = true;
bool calculateLenseThirringCorrection = true;
bool calculateDeSitterCorrection = true;
std::string primaryBody = "Sun";
const Eigen::Vector3d marsAngularMomentum = ...
accelerationSettings[ "Orbiter" ][ "Mars" ] = std::make_shared< RelativisticAccelerationCorrectionSettings >(
calculateSchwarzschildCorrection, calculateLenseThirringCorrection,  calculateDeSitterCorrection, primaryBody,
centralBodyAngularMomentum )


Here, the ‘primary body’ for a planetary orbiter should always be set as the Sun (only relevant for de Sitter correction). The angular momentum vector of the orbited body is only relevant for Lense-Thirring correction.

class EmpiricalAccelerationSettings

A constant/once-per-orbit acceleration, expressed in the RSW frame, for which the mangitude is determined empirically (typically during an orbit determination process). The acceleration components are defined according to Montenbruck and Gill (2000), with a total of 9 components: a constant, sine and cosine term (with true anomaly as argument) for each of the three independent directions of the RSW frame. The settings object (for a vehicle called “Orbiter” around Mars) is created as:

SelectedAccelerationMap accelerationSettings;
Eigen::Vector3d constantAcceleration = ( Eigen::Vector3d( ) << 0.4, -0.1, 0.05 ).finished( );
Eigen::Vector3d sineAcceleration = ( Eigen::Vector3d( ) << 0.0, 0.02, 0.0 ).finished( );
Eigen::Vector3d cosineAcceleration = ( Eigen::Vector3d( ) << -0.01, 0.0, 0.0 ).finished( );
accelerationSettings[ "Orbiter" ][ "Mars" ] = std::make_shared< EmpiricalAccelerationSettings >(
constantAcceleration, sineAcceleration, cosineAcceleration );


Where the three input variables represent:

• Vector containing the constant terms of the accelerations in the R, S and W directions.
• Vector containing the sine terms of the accelerations in the R, S and W directions.
• Vector containing the cosine terms of the accelerations in the R, S and W directions.
class DirectTidalDissipationAccelerationSettings

The direct of tidal effects in a satellite system, applied directly as an acceleration (as opposed to a modification of spherical harmonic coefficients). The model is based on Lainey et al. (2007,2012). It can compute either the acceleration due to tides, and in particular tidal dissipation, on a planetary satellites. The accelertion can compute either the effect of tide raised on the satellite by the planet, or on the planet by the satellite. The satellite is assumed to be tidally locked to the planet.

double loveNumber = 0.1;
double timeLag = 100.0;

SelectedAccelerationMap accelerationSettings;
accelerationSettings[ "Io" ][ "Jupiter" ] = std::make_shared< DirectTidalDissipationAccelerationSettings >(
loveNumber, timeLag, false, false );


Where the three input variables represent:

• Value of the k2 Love number (real value) that is used.
• Value of the tidal time lag (in seconds) that is used.
• Boolean denoting whether the term independent of the time lag is to be computed (default true)
• Boolean denoting whether the tide raised on the planet is to be modelled (if true), or the tide raised on the satellite (if false). Default is true.