Introduction

Unlike positional information of planes and other aircrafts, satellite positions is not readily available for any timepoint along its orbit. Instead, orbital state vectors at discrete and relatively scarce (compared to, for example, the large information density available for aircrafts) timepoints and generated from multiple observations are generated by entities with access to such information, such as the United States Space Surveillance Network. Unclassified information can be obtained through resources such as Space-Track and CelesTrak, most commonly in the form of TLE (Two/Three Line Elements).

However, due to the periodicity of the trajectories (orbits) followed by satellites, if the state vector of an object orbiting Earth is known at a given time, it is possible to predict their state vectors at future or past times. In order to accurately perform such predictions, known as orbit propagation, it is necessary to use complex models that take into consideration not only the gravitational attraction of Earth, but also atmospheric drag and secular and periodic perturbations of the Moon and the Sun, among other effects.

The most widely applied models are the SGP4 and SDP4 models, whose first implementations were introduced in FORTRAN IV in 1988. asteRisk aims to constitute the basis for astrodynamics analysis in R. To that extent, it provides a native R implementation of the SGP4 and SDP4 models and utilities to parse and read TLE files and to convert coordinates between different reference frames. Additionally, a high-precision orbital propagator is also provided. High-precision propagators model a set of forces that act on satellites to calculate the resulting acceleration, and propagate the orbit by numerically solving the resulting differential equation for the position.

Installation instructions

Before installing asteRisk, make sure you have the latest version of R installed. To install asteRisk, start R and enter:

install.packages("asteRisk")

Once installed, the package can be loaded as shown below:

library(asteRisk)

Reading TLE and RINEX files

TLE (Two-/Three- Line Element) is the standard format for representing orbital state vectors. In short, TLEs have 2 lines that contain the orbital parameters that characterize the state of a satellite at a given time, known as epoch. An additional initial line can be present to indicate the name of the satellite. A detailed description of the TLE format can be found here.

asteRisk provides the utilities to read TLE files (function readTLE()), and to directly parse character vectors containing the lines of a TLE as strings (function parseTLElines()). The resulting lists contain the NORAD catalog number of the satellite, the classification level of the information, its international designator, launch year, launch number, launch piece, the date and time of the state vector, the orbital parameters (angular speed, eccentricity, inclination, argument of the perigee, longitude of the ascending node, anomaly) and the drag coefficient of the satellite.

A file with a set of 29 TLE, provided in Revisiting Space Track Report #3 and typically used to benchmark implementations of the SGP4/SDP4 propagators, is distributed with asteRisk, in file named testTLE.txt:

# Read the provided file with 29 benchmark TLE, which contains objects with a
# variety of orbital parameters

test_TLEs <- readTLE(paste0(path.package("asteRisk"), "/testTLE.txt"))
# TLE number 17 contains a state vectors for Italsat 2
test_TLEs[[17]]

## $NORADcatalogNumber
## [1] "24208"
## 
## $classificationLevel
## [1] "unclassified"
## 
## $internationalDesignator
## [1] "1996-044A"
## 
## $launchYear
## [1] 1996
## 
## $launchNumber
## [1] "044"
## 
## $launchPiece
## [1] "A"
## 
## $dateTime
## [1] "2006-06-26 0:58:29.3433600001845"
## 
## $elementNumber
## [1] 160
## 
## $inclination
## [1] 3.8536
## 
## $ascension
## [1] 80.0121
## 
## $eccentricity
## [1] 0.002664
## 
## $perigeeArgument
## [1] 311.0977
## 
## $meanAnomaly
## [1] 48.3
## 
## $meanMotion
## [1] 1.007781
## 
## $meanMotionDerivative
## [1] -1.88e-06
## 
## $meanMotionSecondDerivative
## [1] 0
## 
## $Bstar
## [1] 1e-04
## 
## $ephemerisType
## [1] "Distributed data (SGP4/SDP4)"
## 
## $epochRevolutionNumber
## [1] 3611
## 
## $objectName
## [1] "#   ITALSAT 2           # 24h resonant GEO, inclination > 3 deg"

# It is also possible to directly parse a character vector with 2 or 3
# elements, where each element is a string representing a line of the TLE, to
# obtain the same result:
italsat2_lines <- c("ITALSAT 2", "1 24208U 96044A   06177.04061740 -.00000094  00000-0  10000-3 0  1600",
    "2 24208   3.8536  80.0121 0026640 311.0977  48.3000  1.00778054 36119")

italsat2_TLE <- parseTLElines(italsat2_lines)
italsat2_TLE

## $NORADcatalogNumber
## [1] "24208"
## 
## $classificationLevel
## [1] "unclassified"
## 
## $internationalDesignator
## [1] "1996-044A"
## 
## $launchYear
## [1] 1996
## 
## $launchNumber
## [1] "044"
## 
## $launchPiece
## [1] "A"
## 
## $dateTime
## [1] "2006-06-26 0:58:29.3433600001845"
## 
## $elementNumber
## [1] 160
## 
## $inclination
## [1] 3.8536
## 
## $ascension
## [1] 80.0121
## 
## $eccentricity
## [1] 0.002664
## 
## $perigeeArgument
## [1] 311.0977
## 
## $meanAnomaly
## [1] 48.3
## 
## $meanMotion
## [1] 1.007781
## 
## $meanMotionDerivative
## [1] -1.88e-06
## 
## $meanMotionSecondDerivative
## [1] 0
## 
## $Bstar
## [1] 1e-04
## 
## $ephemerisType
## [1] "Distributed data (SGP4/SDP4)"
## 
## $epochRevolutionNumber
## [1] 3611
## 
## $objectName
## [1] "ITALSAT 2"

test_TLEs[[17]]$inclination == italsat2_TLE$inclination

## [1] TRUE

RINEX (Receiver Independent Exchange Format), on the other hand, is one of the most widely used formats for providing data of global satellite navigation systems (GNSS). The RINEX standard defines several file types, among which navigation files are used to distribute positional information of the satellites. While the format is mainly limited for GNSS, it is of interest for satellite positioning applications given the large amounts of publicly available, high-precision data for such satellite constellations.

The exact information provided in a RINEX navigation file varies for each GNSS. For example, while GLONASS navigation files provide directly the position, velocity and acceleration in the GCRF frame of coordinates, GPS navigation files provide orbital elements.

asteRisk provides functions to read RINEX navigation files for GPS and GLONASS satellites (functions readGPSNavigationRINEX() and readGLONASSNavigationRINEX() respectively):

# Read the provided test RINEX navigation files for both GPS and GLONASS:

testGPSnav <- readGPSNavigationRINEX(paste0(path.package("asteRisk"), "/testGPSRINEXv2.txt"))
testGLONASSnav <- readGLONASSNavigationRINEX(paste0(path.package("asteRisk"), "/testGLONASSRINEXv2.txt"))

# Count the number of positional messages in each file:

length(testGPSnav$messages)

## [1] 3

length(testGLONASSnav$messages)

## [1] 5

Propagation of orbits

The two orbit propagators currently available in asteRisk are the SGP4 and SDP4 models. They allow the calculation of the position and velocity of the satellite at different times, both before and after the time corresponding to the known state vector (referred to as “epoch”). Kepler’s equation is solved through fixed-point integration. It should be noted that the SGP4 model can only accurately propagate the orbit of objects near Earth (with an orbital period shorter than 225 minutes, corresponding approximately to an altitude lower than 5877.5 km).

For propagation of objects in deep space (with orbital periods larger than 225 minutes, corresponding to altitudes higher than 5877.5 km), the SDP4 model should be used, which contains additions to take into account the secular and periodic perturbations of the Moon and the Sun on the orbit of the satellite. It also considers Earth resonance effects on 24-hour geostationary and 12-hour Molniya orbits.

However, it should be noted that SDP4 employs only simplified drag equations, and lacks corrections for low-perigee orbits. Therefore, it is recommended to apply the standard SGP4 model for satellites that are not in deep space.

asteRisk provides three functions to apply the SGP4/SDP4 propagators. The sgdp4() function automatically determines if the satellite is in deep space or near Earth, and applies the appropriate model. The application of either SGP4 or SDP4 can be forced with the sgp4() and sdp4() functions respectively. However, it is not recommended to apply SGP4 to objects in deep space or SDP4 to objects near Earth. In the following example, we calculate and visualize the trajectory of a satellite with a high-eccentricity Molniya orbit:

# Element 11 of the set of test TLE contains an orbital state vector for
# satellite MOLNIYA 1-83, launched from the USSR in 1992 and decayed in 2007

molniya <- test_TLEs[[11]]
1/molniya$meanMotion

## [1] 0.496845

# From the inverse of the mean motion, we can see that the orbital period is
# approximately half a day, in accordance with a Molniya orbit Let´s use the
# SDP4 model to calculate the position and velocity of the satellite for a full
# orbital period every 10 minutes. It is important to provide the mean motion
# in radians/min, the inclination, anomaly, argument of perigee and longitude
# of the ascending node in radians, and the target time as an increment in
# minutes for the epoch time

targetTimes <- seq(0, 720, by = 10)

results_position_matrix <- matrix(nrow = length(targetTimes), ncol = 3)
results_velocity_matrix <- matrix(nrow = length(targetTimes), ncol = 3)

for (i in 1:length(targetTimes)) {
    new_result <- sgdp4(n0 = molniya$meanMotion * ((2 * pi)/(1440)), e0 = molniya$eccentricity,
        i0 = molniya$inclination * pi/180, M0 = molniya$meanAnomaly * pi/180, omega0 = molniya$perigeeArgument *
            pi/180, OMEGA0 = molniya$ascension * pi/180, Bstar = molniya$Bstar, initialDateTime = molniya$dateTime,
        targetTime = targetTimes[i])
    results_position_matrix[i, ] <- new_result[[1]]
    results_velocity_matrix[i, ] <- new_result[[2]]
}
last_molniya_propagation <- new_result
results_position_matrix = cbind(results_position_matrix, targetTimes)
colnames(results_position_matrix) <- c("x", "y", "z", "time")

# Let´s verify that the SDP4 algorithm was automatically chosen

last_molniya_propagation$algorithm

## [1] "sdp4"

# We can visualize the resulting trajectory using a plotly animation to confirm
# that indeed a full revolution was completed and that the orbit is highly
# eccentric.

library(plotly)
library(lazyeval)
library(dplyr)

# In order to create the animation, we must first define a function to create
# the accumulated dataframe required for the animation

accumulate_by <- function(dat, var) {
    var <- f_eval(var, dat)
    lvls <- plotly:::getLevels(var)
    dats <- lapply(seq_along(lvls), function(x) {
        cbind(dat[var %in% lvls[seq(1, x)], ], frame = lvls[[x]])
    })
    bind_rows(dats)
}

accumulated_df <- accumulate_by(as.data.frame(results_position_matrix), ~time)

orbit_animation <- plot_ly(accumulated_df, x = ~x, y = ~y, z = ~z, type = "scatter3d",
    mode = "marker", opacity = 0.8, line = list(width = 6, color = ~time, reverscale = FALSE),
    frame = ~frame)

orbit_animation <- animation_opts(orbit_animation, frame = 50)

orbit_animation <- layout(orbit_animation, scene = list(xaxis = list(range = c(min(results_position_matrix[,
    1]), max(results_position_matrix[, 1]))), yaxis = list(range = c(min(results_position_matrix[,
    2]), max(results_position_matrix[, 2]))), zaxis = list(range = c(min(results_position_matrix[,
    3]), max(results_position_matrix[, 3])))))

orbit_animation

Conversion between reference frames

The positions and velocities calculated with the SGP4 and SDP4 models are in the TEME (True Equator, Mean Equinox) frame of reference, which is an Earth-centered inertial coordinate frame, where the origin is placed at the center of mass of Earth and the coordinate frame is fixed with respect to the stars (and therefore not fixed with respect to the Earth surface in its rotation).

asteRisk provides the TEMEtoITRF() function, which converts positions and velocities in TEME to the ITRF (International Terrestrial Reference Frame) frame of reference. The ITRF is an ECEF (Earth Centered, Earth Fixed) frame of reference, i.e., a non-inertial frame of reference where the origin is also placed at the center of mass of Earth, and the frame rotates with respect to the stars to remain fixed with respect to the Earth surface as it rotates.

Additionally, the TEMEtoLATLON() and ITRFtoLATLON() functions convert Cartesian coordinates to geodetic latitude, longitude and altitude values, from the TEME and ITRF frames respectively. This can be useful, for example, to visualize the ground track followed by a satellite.

Several of the functions for conversion of systems of coordinates require Earth orientation parameters, which are provided through the asteRiskData accessory package, which can be installed by running “install.packages(‘asteRiskData’, repos=‘https://rafael-ayala.github.io/drat/’)”.

# Let us convert the last propagation previously calculated for the MOLNIYA
# 1-83 satellite into the ITRF frame. In order to do so, it is required to
# provide a date-time string indicating the time for the newly calculated
# position and velocity. Since this was 720 minutes after the epoch for the
# original state vector, we can just add 12 hours to it

molniya$dateTime

new_dateTime <- "2006-06-25 12:33:43"

ITRF_coordinates <- TEMEtoITRF(last_molniya_propagation$position, last_molniya_propagation$velocity,
    new_dateTime)

# Let us now convert the previously calculated set of TEME coordinates to
# geodetic latitude and longitude

geodetic_matrix <- matrix(nrow = nrow(results_position_matrix), ncol = 3)

for (i in 1:nrow(geodetic_matrix)) {
    new_dateTime <- as.character(as.POSIXct(molniya$dateTime, tz = "UTC") + 60 *
        targetTimes[i])
    new_geodetic <- TEMEtoLATLON(results_position_matrix[i, 1:3] * 1000, new_dateTime)
    geodetic_matrix[i, ] <- new_geodetic
}

colnames(geodetic_matrix) <- c("latitude", "longitude", "altitude")

# We can then visualize the ground track of the satellite

library(ggmap)

# Important: note that Stadia Maps requires an API key, which must be obtained
# and registered before running the following. Free options for non-commercial
# use are available

ggmap(get_map(c(left = -180, right = 180, bottom = -80, top = 80), source = "stadia",
    maptype = "stamen_terrain")) + geom_segment(data = as.data.frame(geodetic_matrix),
    aes(x = longitude, y = latitude, xend = c(tail(longitude, n = -1), NA), yend = c(tail(latitude,
        n = -1), NA)), na.rm = TRUE) + geom_point(data = as.data.frame(geodetic_matrix),
    aes(x = longitude, y = latitude), color = "blue", size = 0.3, alpha = 0.8)

High-precision orbital propagator

The SGP4/SDP4 models provide a good accuracy at a low computational cost. However, higher precision can be achieved in orbit propagation by calculating at each instant the acceleration of the satellite resulting from the set of forces that are exerted on it, and solving the second-order ODE that expressed acceleration as the second time-derivative of position through numerical integration. Such propagators are often referred to as high-precision orbital propagators (HPOP). The HPOP implemented in asteRisk takes into consideration Earth gravtitational attraction (using a geopotential model based on spherical harmonics); the effects of Earth ocean and solid tides, the attraction of the Sun, Moon and planets; solar radiation pressure; atmospheric drag, and relativistic effects. The HPOP can be used through the hpop() function. However, it should be kept in mind that, while the HPOP can achieve a much higher precision, it also has much higher computational cost. It should also be noted that the HPOP requires access to data such as Earth orientation parameters, space weather data and solar and geomagnetic storms. Such data is provided in the asteRiskData accessory package, which, as previously mentioned, can be installed by running “install.packages(‘asteRiskData’, repos=‘https://rafael-ayala.github.io/drat/’)”. After having installed the accessory package, it is possible to update the data and coefficients to the latest available versions with the getLatestSpaceData() function.

# The HPOP requires as input the satellite mass, the effective areas subjected
# to solar radiation pressure and atmospheric drag, and the drag and
# reflectivity coefficients.  The mass and cross-section of Molniya satellites
# are approximately 1600 kg and 15 m2, respectively. We will use the
# cross-section to approximate the effective areafor both atmospheric drag and
# radiation pressure.  Regarding the drag and reflectivity coefficients, while
# their values vary for each satellite and orbit, 2.2 and 1.2 respectively can
# be used as approximations.

molniyaMass <- 1600
molniyaCrossSection <- 15
molniyaCd <- 2.2
molniyaCr <- 1.2

# As initial conditions, we will use the initial conditions provided in the
# same TLE for MOLNIYA 1-83 used previously for the SGP4/SDP4 propagator.  We
# first need to calculate the initial position and velocity in the GCRF ECI
# frame of reference from the provided orbital elements.  As an approximation,
# we will use the results obtained for t = 0 with the SGP4/SDP4 propagator. We
# convert those into the GCRF frame of reference.  It should be noted that such
# an approximation introduces an error due to a mismatch between the position
# derivative calculated at each propagation point through SGP4/SDP4 and the
# actual velocity of the satellite.

GCRF_coordinates <- TEMEtoGCRF(results_position_matrix[1, 1:3] * 1000, results_velocity_matrix[1,
    1:3] * 1000, molniya$dateTime)

initialPosition <- GCRF_coordinates$position
initialVelocity <- GCRF_coordinates$velocity

# Let´s use the HPOP to calculate the position each 2 minutes during a period
# of 3 hours

targetTimes <- seq(0, 10800, by = 120)

hpop_results <- hpop(initialPosition, initialVelocity, molniya$dateTime, targetTimes,
    molniyaMass, molniyaCrossSection, molniyaCrossSection, molniyaCd, molniyaCr)

# Now we can calculate and plot the corresponding geodetic coordinates

geodetic_matrix_hpop <- matrix(nrow = nrow(hpop_results), ncol = 3)

for (i in 1:nrow(geodetic_matrix_hpop)) {
    new_dateTime <- as.character(as.POSIXct(molniya$dateTime, tz = "UTC") + targetTimes[i])
    new_geodetic <- GCRFtoLATLON(as.numeric(hpop_results[i, 2:4]), new_dateTime)
    geodetic_matrix_hpop[i, ] <- new_geodetic
}

colnames(geodetic_matrix_hpop) <- c("latitude", "longitude", "altitude")

library(ggmap)

# Important: note that Stadia Maps requires an API key, which must be obtained
# and registered before running the following. Free options for non-commercial
# use are available

ggmap(get_map(c(left = -180, right = 180, bottom = -80, top = 80), source = "stadia",
    maptype = "stamen_terrain")) + geom_segment(data = as.data.frame(geodetic_matrix_hpop),
    aes(x = longitude, y = latitude, xend = c(tail(longitude, n = -1), NA), yend = c(tail(latitude,
        n = -1), NA)), na.rm = TRUE) + geom_point(data = as.data.frame(geodetic_matrix_hpop),
    aes(x = longitude, y = latitude), color = "blue", size = 0.3, alpha = 0.8)

References

https://celestrak.org/NORAD/documentation/spacetrk.pdf

http://www.celestrak.org/publications/aiaa/2006-6753/AIAA-2006-6753.pdf

https://juliapackages.com/p/satellitetoolbox

https://celestrak.org/columns/v04n03/#FAQ01

https://www.space-track.org

Satellite Orbits: Models, Methods and Applications. Oliver Montenbruck and Eberhard Gill.

Fundamentals of Astrodynamics and Applications. David Vallado.