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# ecef2eci

Position and velocity vectors in Earth-centered inertial mean-equator mean-equinox

## Syntax

``[r_eci] = ecef2eci(utc,r_ecef)``
``[r_eci,v_eci] = ecef2eci(___,v_ecef)``
``[r_eci,v_eci,a_eci] = ecef2eci(___,a_ecef)``
``[r_eci,v_eci,a_eci] = ecef2eci(___,Name,Value)``

## Description

example

````[r_eci] = ecef2eci(utc,r_ecef)` calculates the position vector in the Earth-centered inertial (ECI) coordinate system for a given position vector in the Earth-centered Earth-fixed (ECEF) coordinate system at a specific Coordinated Universal Time (UTC). For more information on the Earth-centered Earth-fixed coordinate system, see Algorithms.`[r_eci,v_eci] = ecef2eci(___,v_ecef)` calculates the position and velocity vectors for given position and velocity vectors. `[r_eci,v_eci,a_eci] = ecef2eci(___,a_ecef)` calculates the position, velocity, acceleration vectors for given position, velocity, and acceleration vectors. `[r_eci,v_eci,a_eci] = ecef2eci(___,Name,Value)` calculates the position, velocity, and acceleration vectors at a higher precision using Earth orientation parameters. ```

## Examples

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Convert the ECEF position and velocity to ECI at 12:00 on January 4, 2019.

```r_ecef = [-5762640 -1682738 3156028]; v_ecef = [3832 -4024 4837]; utc = [2019 1 4 12 0 0]; [r_eci, v_eci] = ecef2eci(utc, r_ecef, v_ecef);```
```r_eci = 1.0e+06 * -2.9818 5.2070 3.1616 v_eci = 1.0e+03 * -3.3837 -4.8870 4.8430```

Convert the ECEF position to ECI at 12:00 on January 4, 2019, including the effects of polar motion.

```r_ecef = [-5762640 -1682738 3156028]; utc = [2019 1 4 12 0 0]; mjd = mjuliandate(utc); pm = polarMotion(mjd, 'action', 'none')*180/pi; r_eci = ecef2eci(utc, r_ecef, 'pm', pm);```
```r_eci = 1.0e+06 * -2.9818 5.2070 3.1616```

## Input Arguments

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UTC in the order year, month, day, hour, minutes, and seconds, specified as 1-by-6 array of UTC values:

Time ValueEnter
YearDouble value that is a whole number greater than 1, such as `2013`.
MonthDouble value that is a whole number greater than 0, within the range `1` to `12`.
DayDouble value that is a whole number greater than 0, within the range `1` to `31`.
HourDouble value that is a whole number greater than 0, within the range `1` to `24`.
Minute and secondDouble value that is a whole number greater than 0, within the range `1` to `60`.

Example: `[2000 1 12 4 52 12.4]`

Data Types: `double`

Array of ECEF position components, specified as a 3-by-1 array.

Data Types: `double`

ECEF velocity components, specified as a 3-by-1 array.

Data Types: `double`

ECEF acceleration components, specified as a 3-by-1 array.

Data Types: `double`

### Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: `'dUT1',0.234`

Difference between International Atomic Time (TAI) and UTC, specified as a scalar, in seconds.

Example: 32

Data Types: `double`

Difference between UTC and Universal Time (UT1), specified as a scalar, in seconds.

Example: 0.234

Data Types: `double`

Polar displacements due to the motion of Earth crust along the x- and y-axis, in degrees.

Tip

To calculate the displacement, use the `polarMotion` function.

Example: `pm = polarMotion(mjd, 'action', 'none')*180/pi;`

Data Types: `double`

Adjustment to the location of the Celestial Intermediate Pole (CIP), in degrees, specified as a comma-separated pair consisting of `dCIP` and an M-by-2 array. This location (dDeltaX, dDeltaY) is along the x- and y- axes. By default, this function assumes a 1-by-2 array of zeroes.

For historical values, see the International Earth Rotation and Reference Systems Service Web site (`https://www.iers.org`) and navigate to the Earth orientation data Data/Products page.

• M-by-2 array

Specify an M-by-2 array of location adjustment values, where M is the number of direction cosine or transformation matrices to be converted. Each row corresponds to one set of dDeltaX and dDeltaY values.

Example: `[-0.2530e-6 -0.0188e-6]`

Data Types: `double`

Excess length of day (difference between astronomically determined duration of day and 86400 SI seconds), specified as a scalar, in seconds.

Example: 32

Data Types: `double`

## Output Arguments

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ECI position components, specified as a 3-by-1 array.

ECI velocity components, specified as a 3-by-1 array.

ECI acceleration components, specified as a 3-by-1 array.

## Algorithms

The `ecef2eci` function uses these Earth-centric coordinate systems:

• Earth Centered Inertial Frame (ECI) — The inertial frame used is the International Celestial Reference Frame (ICRF). This frame can be treated as equal to the ECI coordinate system realized at J2000 (Jan 1 2000 12:00:00 TT). For more information, see ECI Coordinates.

• Earth-centered Earth-fixed Frame (ECEF) — The fixed-frame used is the International Terrestrial Reference Frame (ITRF). This reference frame is realized by the IAU2000/2006 reduction from the ICRF coordinate system. For more information, see ECEF Coordinates.

## References

[1] Vallado, D. A. Fundamentals of Astrodynamics and Applications. alg. 4. New York: McGraw-Hill, 1997.

[2] Gottlieb, R. G., "Fast Gravity, Gravity Partials, Normalized Gravity, Gravity Gradient Torque and Magnetic Field: Derivation, Code and Data," Technical Report NASA Contractor Report 188243, NASA Lyndon B. Johnson Space Center, Houston, Texas, February 1993.

[3] Konopliv, A. S., S. W. Asmar, E. Carranza, W. L. Sjogen, D. N. Yuan., "Recent Gravity Models as a Result of the Lunar Prospector Mission, Icarus", Vol. 150, no. 1, pp 1–18, 2001.

[4] Lemoine, F. G., D. E. Smith, D.D. Rowlands, M.T. Zuber, G. A. Neumann, and D. S. Chinn, "An improved solution of the gravity field of Mars (GMM-2B) from Mars Global Surveyor", Journal Of Geophysical Research, Vol. 106, No. E10, pp 23359-23376, October 25, 2001.

[5] Seidelmann, P.K., Archinal, B.A., A’hearn, M.F. et al. Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2006. Celestial Mech Dyn Astr 98, 155–180 (2007).

## See Also

Introduced in R2019a

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