Radiation pressure arises from the exchange of momentum between electromagnetic radiation and the spacecraft. In Tudat, radiation pressure accelerations require models for sources (how radiation is emitted) and targets (how the spacecraft is accelerated depending on the incident radiation). Here, the ‘source’ may also be a body that reflects light from another body (e.g. albedo). Both sources and targets may be defined in any number of ways. Regardless of how the source and target models are defined, creating the acceleration model for them is always done in the same manner, using the radiation_pressure(), which takes the source model of the body exerting the acceleration, and the target model of the body undergoing the acceleration, and links these models to set up the specific acceleration model. For extensive details on the mathematical models, see [Stiller2023]

In most orbits, there are two sources of radiation: direct solar radiation, and albedo + thermal radiation of the central body (particularly in low orbits). Both require different treatment. Therefore, there are two source models in Tudat. Settings for a body are defined in the radiation_source_settings attribute of the BodySettings class

### Isotropic point source

The radiation due to an isotropic (point) source depends only on the distance from the source, not on the relative latitude/longitude If the source is far away, all rays hitting the target are virtually parallel. This is, for example, the case for solar radiation at 1 AU. The default source model for the Sun is such a point source with a luminosity of 3.828 × 1026 W. Defining settings for an isotropic source model is done using the isotropic_radiation_source() function, which requires a luminosity model. These may be defined by one of the following models:

Defining the second to last option, for a solar irradiance of 1367 W/m:sup:2 at 1 AU, the settings for the body called ‘Sun’ would be modified as follows:

### Extended source

Planetary radiation is generally not isotropic and the spacecraft is relatively close to the surface. Therefore, the central body is modeled as an extended source, which is discretized into panels. This model was described by [Knocke1988]. Each panel emits radiation as defined by a radiosity model. Typically, these include albedo radiation (reflected solar radiation) and/or thermal radiation (due to surface heating). Defining settings for an extended source model is done using the panelled_extended_radiation_source() function, which requires surface radiosity models, and settings for the surface discretization.

The following options are supported for defining surface radiosity models:

For a number of the above models, a surface distribution of a property has to be defined (e.g. albedo, emissivity). A number of options are available for this:

When using any of the above models to calculate a radiation pressure acceleration on a target, the extended source is panelled and the per-panel contribution to the source’s irradiance at the target is computed. This panelling is done dynamically, in the sense that the panel locations are re-evaluated at every step of the numerical integration such that the panelling is always symmetric about the nadir point. The panelling methods is based on [Knocke1988] and described in more detail by [Stiller2023]. Summarized, the main assumptions are:

• The source body is assumed spherical

• Only the spherical cap of the body that is visible from the target is panelled

• A single spherical panel is put at nadir, with $$N$$ rings around it with $$M_{i}$$ panels in ring $$i$$

• Each panel has equal projected, attenuated area (see Eq. 8 of Stiller)

The fidelity of the results increases with the number of panels (which can be defined by the user). Convergence tests are recommended to find a sufficient number of rings. Commonly used numbers of rings: LAGEOS: 2-3 rings for Earth; LRO: 5-6 rings for the Moon.

Putting the above options together, the above creates a panelled source model for the Earth from both albedo and IR, using the pre-defined Knocke-style surface distribution of both. Three rings are used in the dynamic panelling with 6, 12 and 18 panels in the first, second and third ring, respectively.

original_source_name = "Sun" ),
original_source_name = "Sun" ) ]
earth_surface_radiosity_models, [ 6, 12, 18 ] )

Albedo and thermal radiosity models often require a so-called original source (typically the Sun), the radiation of which is reflected or re-radiated. Thermal radiation defined directly (without reference to the original source), for instance by specifying a global temperature, is not yet implemented and exposed to Python.

The spacecraft acceleration due to radiation pressure depends on the cross-section area, optical properties, and mass. The dependence on the area-to-mass ratio is similar to drag. Optical properties are relevant since reflected radiation imparts more momentum than absorbed radiation. There are two target models in Tudat. Settings for a body are defined in the radiation_pressure_target_settings attribute of the BodySettings class.

### Cannonball target

A cannonball target models the spacecraft as isotropic sphere defined by the cross-section area and a radiation pressure coefficient. This model is useful for applications that do not require high-fidelity radiation pressure modelling, but cannot capture the finer details of the radiation pressure interaction and may therefore not be suited to high-fidelity analysis. Settings for the cannonball model are created using the cannonball_radiation_target() function.

### Paneled target

A panelled radiation pressure target model provides a more realistic representation than the cannonball model. It builds up the spacecraft out of a series of panels, where the interaction of the radiation with each of the panels is computed separately. Each panel may have different optical properties, and may be defined as being either fixed to the spaceraft body (e.g. bus panels) or may be defined to move w.r.t. the spacecraft body-fixed frame (for instance Sun-pointing solar arrays, or Earth-pointing antennas). At the moment, Tudat does not include panel shadowing in the calculations.

Details on defining a panelled spacecraft model are defined by Vehicle shape models. The interaction of each panel is defined by a so-called reflection law. At the moment, Tudat implements two panel reflection laws:

With the body panels defined, the radiation pressure target model settings are created using the panelled_radiation_target() function.

## Dependent variables

There is a number of dependent variables associated with radiation pressure acceleration:

For point source only:

For extended source only:

• Total area of source panels contributing to irradiance at target (e.g. area of spherical cap that is panelled for computing the radiation pressure), visible_radiation_source_area()

## Assumptions

• The paneled target is much smaller than the extended source and far enough away. Therefore, all target panels receive the same irradiance, from the same direction. The source irradiance is evaluated at the target center.

• The extended source far enough away from the original source (e.g., 1 AU for Earth and Sun). Therefore, the panels of the extended source receive the same irradiance, from the same direction. The original source irradiance is evaluated at the source center.

• The extended source is a perfect sphere, and not an oblate spheroid. Panels are distributed on the perfect sphere.

Knocke1988(1,2,3)

Knocke et al., (1988). Earth radiation pressure effects on satellites. American Institute of Aeronautics and Astronautics, Astrodynamics Conference, https://doi.org/10.2514/6.1988-4292.

Stiller2023(1,2)

Knocke et al., (1988). Short-term orbital effects of radiation pressure on the Lunar Reconnaissance Orbiter. TU Delft, Research paper for the Honours Programme Bachelor, http://resolver.tudelft.nl/uuid:8a82400a-2233-4a84-98be-ed37f7eeb620.

## Backwards compatibility

As of tudatpy version 0.8, the radiation pressure implementation has been completely refactored. The code for the old cannonball radiation pressure models will, however, still be supported for some time. You can easily modify your code to start using the new interfaces, and access all the powerful new functionality we provide for radiation pressure!

Source model In version <0.8, only the Sun was supported as a source, with a hard-coded constant luminosity. The default settings for the Sun’s radiation pressure source models are identical to the ones in version >= 0.8, and no action needs to be taken to modify the code.

Target model In version <0.8, the cannonball radiation pressure properties were defined through a ‘radiation pressure interface’, which has been replaced with a more flexible and generic target model.

Creation of radiation pressure settings as follows (in version <0.8):

occulting_bodies = ["Earth"]

Is to be replaced with the creation of radiation_pressure_target_settings (in version >=0.8):

occulting_bodies_dict = dict()
occulting_bodies_dict[ "Sun" ].append( "Earth" )