Satellite Orbits

Abstract

The principle of the GPS system is to measure the signal transmission paths from the satellites to the receivers. Therefore, the satellite orbits are very important topics in GPS theory. In this chapter, the basic orbits theory is briefly described.

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The principle of the GPS system is to measure the signal transmission paths from the satellites to the receivers. Therefore, the satellite orbits are very important topics in GPS theory. In this chapter, the basic orbits theory is briefly described. For the GPS applications in orbits correction and orbits determination, the advanced orbits perturbation theory will be discussed in Chap. 11.

3.1 Keplerian Motion

The simplified satellite orbiting is called Keplerian motion , and the problem is called the two-body problem. The satellite is supposed to move in a central force field. The equation of satellite motion is described by Newton’s second law of motion by

$$ \vec{f} = m \cdot a = m \cdot \ddot{\vec{r}}, $$

(3.1)

where $\vec{f}$ is the attraction force, m is the mass of the satellite, a, or alternatively, $\ddot{\vec{r}}$ is the acceleration of the motion (second-order differentiation of vector $\vec{r}$ with respect to the time), and according to Newton’s law,

$$ \vec{f} = - \frac{GMm}{{r^{2} }}\frac{{\vec{r}}}{r}, $$

(3.2)

where G is the universal gravitational constant , M is the mass of the earth, r is the distance between the mass centre of the earth and the mass centre of the satellite. The equation of satellite motion is then

$$ \ddot{\vec{r}} = - \frac{\mu }{{r^{2} }}\frac{{\vec{r}}}{r}, $$

(3.3)

where μ (=GM) is called the earth’s gravitational constant.

Equation 3.3 of satellite motion is valid only in an inertial coordinate system , so the ECSF coordinate system discussed in Chap. 2 will be used for describing the orbit of the satellite. The vector form of the equation of motion can be rewritten through three x, y, and z components $(\vec{r} = (x,y,z))$ as

$$ \begin{aligned} {\ddot{\it{x}}} = & - \frac{\mu }{{r^{3} }}x \\ {\ddot{\it{y}}} = & - \frac{\mu }{{r^{3} }}y \\ {\ddot{\it{z}}} = & - \frac{\mu }{{r^{3} }}z. \\ \end{aligned} $$

(3.4)

Multiplying y, z to the first equation of 3.4, and x, z to the second, x, y to the third, and then forming differences of them, one gets

$$ \begin{aligned} y{\ddot{\it{z}}} - z{\ddot{\it{y}}} & = 0 \\ z{\ddot{\it{x}}} - x{\ddot{\it{z}}} & = 0 \\ x{\ddot{\it{y}}} - y{\ddot{\it{x}}} & = 0, \\ \end{aligned} $$

(3.5)

or in vector form:

$$ \vec{r} \times \ddot{\vec{r}} = 0. $$

(3.6)

Equations 3.5 and 3.6 are equivalent to

$$ \begin{aligned} \frac{{{\text{d}}(y\dot{z} - z\dot{y})}}{{{\text{d}}t}} & = 0 \\ \frac{{{\text{d}}(z\dot{x} - x\dot{z})}}{{{\text{d}}t}} & = 0 \\ \frac{{{\text{d}}(x\dot{y} - y\dot{x})}}{{{\text{d}}t}} & = 0, \\ \end{aligned} $$

(3.7)

$$ \frac{{{\text{d}}(\vec{r} \times \dot{\vec{r}})}}{{{\text{d}}t}} = 0. $$

(3.8)

Integrating Eqs. 3.7 and 3.8 lead to

$$ \begin{aligned} y\dot{z} - z\dot{y} & = A \\ z\dot{x} - x\dot{z} & = B \\ x\dot{y} - y\dot{x} & = C, \\ \end{aligned} $$

(3.9)

$$ \vec{r} \times \dot{\vec{r}} = \vec{h} = \left( {\begin{array}{*{20}c} A \\ B \\ C \\ \end{array} } \right), $$

(3.10)

where A, B, C are integration constants; they form the integration constant vector $\vec{h}.$ That is:

$$ h = \sqrt {A^{2} + B^{2} + C^{2} } = \left| {\vec{r} \times \dot{\vec{r}}} \right|. $$

(3.11)

The constant h is two times of the area that the radius vector sweeps during a unit time. This is indeed Kepler’s second law. Then h/2 is called the area velocity of the radius of the satellite.

Multiplying x, y, and z to the three equations of 3.9 and adding them together, one has

$$ Ax + By + Cz = 0. $$

(3.12)

That is, the satellite motion fulfils the equation of a plane, and the origin of the coordinate system is in the plane. In other words, the satellite moves in a plane in the central force field of the earth. The plane is called the orbital plane of the satellite.

The angle between the orbital plane and the equatorial plane is called inclination of the satellite (denoted by i, cf. Fig. 3.1). Alternatively, the inclination i is the angle between the vector $\vec{z} = (0,0,1)$ and $\vec{h} = (A,B,C)$ i.e.

$$ \cos i = \frac{{\vec{z} \cdot \vec{h}}}{{\left| {\vec{z}} \right| \cdot \left| {\vec{h}} \right|}} = \frac{C}{h}. $$

(3.13)

Fig. 3.1

Orbital plane

The orbital plane cuts the equator at two points. They are called ascending node N and descending node. (See the next section for details). Vector $\vec{s}$ denotes the vector from the earth’s centre pointed to the ascending point. The angle between the ascending node and the x-axis (vernal equinox ) is called the right ascension of the ascending node (denoted by Ω). Thus,

$$ \vec{s} = \vec{z} \times \vec{h}, $$

and

$$ \cos \varOmega = \frac{{\vec{s} \cdot \vec{x}}}{{\left| {\vec{s}} \right| \cdot \left| {\vec{x}} \right|}} = \frac{ - B}{{\sqrt {A^{2} + B^{2} } }}, $$

$$ \sin \varOmega = \frac{{\vec{s} \cdot \vec{y}}}{{\left| {\vec{s}} \right| \cdot \left| {\vec{y}} \right|}} = \frac{A}{{\sqrt {A^{2} + B^{2} } }}. $$

(3.14)

Parameters i and Ω uniquely defined the place of orbital plane and therefore are called orbital plane parameters. Ω, i, and h are then selected as integration constants, which have significant geometric meanings of the satellite orbits.

3.1.1 Satellite Motion in the Orbital Plane

In the orbital plane, a two-dimensional rectangular coordinate system is given in Fig. 3.2. The coordinates can be represented in polar coordinate r and ϑ as

$$ \begin{aligned} p = r\cos \vartheta \\ q = r\sin \vartheta . \\ \end{aligned} $$

(3.15)

Fig. 3.2

Polar coordinates in the orbital plane

The equation of motion in pq-coordinates is similar to the Eq. 3.4 as

$$ \begin{aligned} {\ddot{\it{p}}} = & - \frac{\mu }{{r^{3} }}p \\ {\ddot{\it{q}}} = & - \frac{\mu }{{r^{3} }}q. \\ \end{aligned} $$

(3.16)

From Eq. 3.15, one has

$$ \begin{aligned} \dot{p} & = \dot{r}\cos \vartheta - r\dot{\vartheta }\sin \vartheta \\ \dot{q} & = \dot{r}\sin \vartheta + r\dot{\vartheta }\cos \vartheta \\ {\ddot{\it{p}}} & = ({\ddot{\it{r}}} - r\dot{\vartheta }^{2} )\cos \vartheta - (r{\ddot{\it{\vartheta }}} + 2\dot{r}\dot{\vartheta })\sin \vartheta \\ {\ddot{\it{q}}} & = ({\ddot{\it{r}}} - r\dot{\vartheta }^{2} )\sin \vartheta + (r{\ddot{\it{\vartheta }}} + 2\dot{r}\dot{\vartheta })\cos \vartheta . \\ \end{aligned} $$

(3.17)

Substituting Eqs. 3.17 and 3.15 into Eq. 3.16, one gets

$$ \begin{aligned} ({\ddot{\it{r}}} - r\dot{\vartheta }^{2} )\cos \vartheta - (r{\ddot{\it{\vartheta }}} + 2\dot{r}\dot{\vartheta })\sin \vartheta = & - \frac{\mu }{{r^{2} }}\cos \vartheta \\ ({\ddot{\it{r}}} - r\dot{\vartheta }^{2} )\sin \vartheta + (r{\ddot{\it{\vartheta }}} + 2\dot{r}\dot{\vartheta })\cos \vartheta = & - \frac{\mu }{{r^{2} }}\sin \vartheta . \\ \end{aligned} $$

(3.18)

The point from which the polar angle ϑ is measured is arbitrary. So setting ϑ as zero, the equation of motion is then

$$ \begin{aligned} {\ddot{\it{r}}} - r\dot{\vartheta }^{2} & = - \frac{\mu }{{r^{2} }} \hfill \\ r{\ddot{\it{\vartheta }}} + 2\dot{r}\dot{\vartheta } & = 0. \hfill \\ \end{aligned} $$

(3.19)

Multiplying r to the second equation of 3.19, it turns out to be

$$ \frac{{{\text{d}}(r^{2} \dot{\vartheta })}}{{{\text{d}}t}} = 0. $$

(3.20)

Because $r\dot{\vartheta }$ is the tangential velocity, $r^{2} \dot{\vartheta }$ is the two times of the area velocity of the radius of the satellite. Integrating Eq. 3.20 and comparing it with the discussion in Sect. 3.1, one has

$$ r^{2} \dot{\vartheta } = h. $$

(3.21)

h/2 is the area velocity of the radius of the satellite.

For solving the first differential equation of 3.19, the equation must be transformed into a differential equation of r with respect to variable f. Let

$$ u = \frac{1}{r}, $$

(3.22)

then from Eq. 3.21, one gets

$$ \frac{{{\text{d}}\vartheta }}{{{\text{d}}t}} = hu^{2} $$

(3.23)

and

$$ \begin{aligned} \frac{{{\text{d}}r}}{{{\text{d}}t}} = & \frac{{{\text{d}}r}}{{{\text{d}}\vartheta }}\frac{{{\text{d}}\vartheta }}{{{\text{d}}t}} = \frac{\text{d}}{{{\text{d}}\vartheta }}\left( {\frac{1}{u}} \right)hu^{2} = - h\frac{{{\text{d}}u}}{{{\text{d}}\vartheta }} \\ \frac{{{\text{d}}^{2} r}}{{{\text{d}}t^{2} }} = & - h\frac{{{\text{d}}^{2} u}}{{{\text{d}}\vartheta^{2} }}\frac{{{\text{d}}\vartheta }}{{{\text{d}}t}} = - h^{2} u^{2} \frac{{{\text{d}}^{2} u}}{{{\text{d}}\vartheta^{2} }}. \\ \end{aligned} $$

(3.24)

Substituting Eqs. 3.22 and 3.24 into the first equation of 3.19, the equation of motion is then

$$ \frac{{{\text{d}}^{2} u}}{{{\text{d}}\vartheta^{2} }} + u = \frac{\mu }{{h^{2} }}, $$

(3.25)

and its solution is

$$ u = d_{1} \cos \vartheta + d_{2} \sin \vartheta + \frac{\mu }{{h^{2} }}, $$

where d ₁ and d ₂ are constants of integration . The above equation may be simplified as

$$ u = \frac{\mu }{{h^{2} }}\left( {1 + e\cos (\vartheta - \omega )} \right), $$

(3.26)

where

$$ d_{1} = \frac{\mu }{{h^{2} }}e\cos \omega ,\quad d_{2} = \frac{\mu }{{h^{2} }}e\sin \omega . $$

Thus the moving equation of satellite in the orbital plane is

$$ r = \frac{{h^{2} /\mu }}{1 + e\cos (\vartheta - \omega )}. $$

(3.27)

Comparing Eq. 3.27 with a standard polar equation of conic:

$$ r = \frac{{a(1 - e^{2} )}}{1 - e\cos \phi }, $$

(3.28)

orbit Eq. 3.27 is obviously a polar equation of conic section with the origin at one of the foci. Where parameter e is the eccentricity , for e = 0, e < 1, e = 1, e > 1, the conic is a circle, an ellipse, a parabola, and a hyperbola, respectively. For the satellite orbiting around the earth, generally, e < 1. Thus, the satellite orbit is an ellipse, and this is indeed the Kepler’s first law. Parameter a is the semimajor axis of the ellipse, and

$$ \frac{{h^{2} }}{\mu } = a(1 - e^{2} ). $$

(3.29)

It is obvious that parameter a has greater significance in a geometric sense than h, so the use of a is preferred. Parameters a and e define the size and shape of the ellipse and are called ellipse parameters. The ellipse cuts the equator at the ascending and descending nodes. Polar angle φ is counted from the apogee of the ellipse. This can be seen by letting φ = 0; thus r = a(1 + e). φ has a 180° difference with the angle ϑ − ω. Letting f = ϑ − ω, where f is called the true anomaly of the satellite counted from the perigee , then the orbit Eq. 3.27 can be written as

$$ r = \frac{{a(1 - e^{2} )}}{1 + e\cos f}. $$

(3.30)

In the case of f = 0, i.e. the satellite is in the point of perigee , ω = ϑ, ϑ is the polar angle of the perigee counted from the p-axis. Supposing the p-axis is an axis in the equatorial plane and points to the ascending node N, then ω is the angle of perigee counted from the ascending node (cf. Fig. 3.3) and is called the argument of perigee . The argument of perigee defines the axis direction of the ellipse related to the equatorial plane.

Fig. 3.3

Ellipse of the satellite motion

3.1.2 Keplerian Equation

Thus far, five integration constants have been derived. They are inclination angle i, right ascension of ascending node Ω, semimajor axis a, eccentricity e of the ellipse, and argument of perigee ω. Parameters i and Ω decide the place of the orbital plane, a and e decide the size and shape of the ellipse and ω decides the direction of the ellipse (cf. Fig. 3.4). To describe the satellite position in the ellipse, velocity of the motion must be discussed.

Fig. 3.4

Orbital geometry

The period T of the satellite motion is the area of ellipse divided by area velocity:

$$ T = \frac{\pi ab}{{\tfrac{h}{2}}} = \frac{2\pi ab}{{\sqrt {\mu a(1 - e^{2} )} }} = 2\pi a^{3/2} \mu^{ - 1/2} . $$

(3.31)

The average angular velocity n is then

$$ n = \frac{2\pi }{T} = a^{ - 3/2} \mu^{1/2} . $$

(3.32)

Equation 3.32 is the Kepler’s third law. It is obvious that it is easier to describe the angular motion of the satellite under the average angular velocity n in the geometric centre of the ellipse (than in the geocentre). To simplify the problem, an angle called the eccentric anomaly is defined (denoted by E, cf. Fig. 3.5). $S^{\prime}$ is the vertical projection of the satellite S on the circle with a radius of a (semimajor axis of the ellipse). The distance between the geometric centre $O^{\prime}$ of the ellipse and the geocentre O is ae. Thus,

$$ \begin{aligned} x & = r\cos f = a\cos E - ae \\ y & = r\sin f = b\sin E = a\sqrt {1 - e^{2} } \sin E, \\ \end{aligned} $$

(3.33)

where the second equation can be obtained by substituting the first into the standard ellipse equation ((x + ae)²/a ² + y ²/b ² = 1), where b is the semiminor axis of the ellipse. The orbit equation can then be represented by variable E as

$$ r = a(1 - e\cos E). $$

(3.34)

Fig. 3.5

Mean anomaly of satellite

The relation between true and eccentric anomalies can be derived by using Eqs. 3.33 and 3.34:

$$ \tan \frac{f}{2} = \frac{\sin f}{1 + \cos f} = \frac{\sin E}{1 + \cos E}\frac{{\sqrt {1 - e^{2} } }}{1 - e} = \frac{{\sqrt {1 + e} }}{{\sqrt {1 - e} }}\tan \frac{E}{2}. $$

(3.35)

If the xyz-coordinates are rotated such that the xy-plane coincides with the orbital plane , then the area velocity formulas of Eqs. 3.9 and 3.10 have only one component along the z-axis, i.e.

$$ x\dot{y} - y\dot{x} = h = \sqrt {\mu a(1 - e^{2} )} . $$

(3.36)

From Eq. 3.33, one has

$$ \begin{aligned} \dot{x} & = - a\sin E\frac{{{\text{d}}E}}{{{\text{d}}t}} \\ \dot{y} & = a\sqrt {1 - e^{2} \cos E} \frac{{{\text{d}}E}}{{{\text{d}}t}} \\ \end{aligned} $$

(3.37)

Substituting Eqs. 3.33 and 3.37 into Eq. 3.36 and taking Eq. 3.32 into account, a relation between E and t is obtained

$$ (1 - e\cos E){\text{d}}E = \sqrt \mu a^{ - 3/2} {\text{d}}t = n{\text{d}}t. $$

(3.38)

Suppose at the time t _p satellite is at the point perigee , i.e. E(t _p) = 0, and at any time t, E(t) = E, then integration of Eq. 3.38 from 0 to E, namely from t _p to t is

$$ E - e\sin E = M, $$

(3.39)

where

$$ M = n(t - t_{p} ). $$

(3.40)

Equation 3.39 is the Keplerian equation . E is given as a function of M, namely t. Because of Eq. 3.34, the Keplerian equation indirectly assigns r as a function of t. M is called the mean anomaly . M describes the satellite as orbiting the earth with a mean angular velocity n. t _p is called the perigee passage and is the sixth integration constant of the equation of satellite motion in a centre-force field.

Knowing M to compute E, the Keplerian Eq. 3.39 may be solved iteratively. Because of the small e, the convergence can be achieved very quickly.

Three anomalies ( true anomaly f, eccentric anomaly E and mean anomaly M) are equivalent through the relations of Eqs. 3.35 and 3.39. They are functions of time t (including the perigee passage t _p), and they describe the position changes of the satellite with the time in the ECSF coordinates.

3.1.3 State Vector of the Satellite

Consider the orbital right-handed coordinate system: if the xy-plane is the orbital plane , the x-axis points to the perigee, the z-axis is in the direction of vector $\vec{h}$, and the origin is in the geocentre, the position vector $\vec{q}$ of the satellite is then (cf. Eq. 3.33)

$$ \vec{q} = \left( {\begin{array}{*{20}c} {a(\cos E - e)} \\ {a\sqrt {1 - e^{2} } \sin E} \\ 0 \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {r\cos f} \\ {r\sin f} \\ 0 \\ \end{array} } \right). $$

(3.41)

Differentiating Eq. 3.41 with respect to time t and taking Eq. 3.38 into account, the velocity vector of the satellite is then

$$ \dot{\vec{q}} = \left( {\begin{array}{*{20}c} { - \sin E} \\ {\sqrt {1 - e^{2} } \cos E} \\ 0 \\ \end{array} } \right)\frac{na}{1 - e\cos E} = \left( {\begin{array}{*{20}c} { - \sin f} \\ {e + \cos f} \\ 0 \\ \end{array} } \right)\frac{na}{{\sqrt {1 - e^{2} } }}. $$

(3.42)

The second part of the above equation can be derived from the relation between E and f. The state vector of the satellite in the orbital coordinate system can be rotated to the ECSF coordinate system by three successive rotations. First, a clockwise rotation around the 3rd-axis from the perigee to the node is given by (cf. Fig. 3.4)

$$ R_{3} ( - \omega ). $$

Next, a clockwise rotation around the 1st-axis with the angle of inclination i is given by

$$ R_{1} ( - i). $$

Finally, a clockwise rotation around the 3rd-axis from the node to the vernal equinox is given by

$$ R_{3} ( - \varOmega ). $$

So the state vector of the satellite in the ECSF coordinate system is then

$$ \left( {\begin{array}{*{20}c} {\vec{r}} \\ {\dot{\vec{r}}} \\ \end{array} } \right) = R_{3} ( - \varOmega )R_{1} ( - i)R_{3} ( - \omega )\left( {\begin{array}{*{20}c} {\vec{q}} \\ {\dot{\vec{q}}} \\ \end{array} } \right), $$

(3.43)

where

$$ \vec{r} = \left( {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right),\quad \dot{\vec{r}} = \left( {\begin{array}{*{20}c} {\dot{x}} \\ {\dot{y}} \\ {\dot{z}} \\ \end{array} } \right). $$

For the six given Keplerian elements (Ω, i, ω, a, e, M ₀) of t ₀, where M ₀ = n(t ₀ − t _p), the satellite state vector of time t can be computed, e.g., as follows:

• 1. Using Eq. 3.32 to compute the mean angular velocity n;

• 2. Using Eqs. 3.40, 3.39, 3.33, and 3.30 to compute the three anomalies M, E, f, and r;

• 3. Using Eqs. 3.41 and 3.42 to compute the state vector $\vec{q}$ and $\dot{\vec{q}}$ in orbital coordinates;

• 4. Using Eq. 3.43 to rotate state vector $\vec{q}$ and $\dot{\vec{q}}$ to the ECSF coordinates.

Keplerian elements can be given in practice at any time. For example, with t ₀, where only f is a function of t ₀, other parameters are constants. In this case, the related E and M can be computed by Eqs. 3.35 and 3.39; thus t _p can be computed by Eq. 3.40.

From Eq. 3.42, one has

$$ v^{2} = \frac{{a^{2} n^{2} }}{{(1 - e\cos E)^{2} }}[\sin^{2} E + (1 - e^{2} )\cos^{2} E] = \frac{{a^{2} n^{2} (1 + e\cos E)}}{1 - e\cos E}. $$

(3.44)

Taking Eqs. 3.32 and 3.34 into account leads to

$$ v^{2} = \frac{\mu (1 + e\cos E)}{r} = \frac{\mu (2 - r/a)}{r} = \mu (\frac{2}{r} - \frac{1}{a}), $$

(3.45)

where v ²/2 is the kinetic energy scaled by mass, μ/r is the potential energy, and a is the semimajor axis of the ellipse. This is the total energy conservative law of mechanics.

Rotate the vector $\vec{q}$ and $\dot{\vec{q}}$ in Eqs. 3.41 and 3.42 by R ₃(–ω) and denote by $\vec{p}$ and $\dot{\vec{p}}$, i.e.

$$ \vec{p} = \left( {\begin{array}{*{20}c} {p_{1} } \\ {p_{2} } \\ {p_{3} } \\ \end{array} } \right) = R_{3} ( - \omega )\left( {\begin{array}{*{20}c} {r\cos f} \\ {r\sin f} \\ 0 \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {r\cos (\omega + f)} \\ {r\sin (\omega + f)} \\ 0 \\ \end{array} } \right), $$

(3.46)

and

$$ \dot{\vec{p}} = \left( {\begin{array}{*{20}c} {\dot{p}_{1} } \\ {\dot{p}_{2} } \\ {\dot{p}_{3} } \\ \end{array} } \right) = R_{3} ( - \omega )\left( {\begin{array}{*{20}c} { - {\kern 1pt} \sin f} \\ {e + \cos f} \\ 0 \\ \end{array} } \right)\frac{na}{{\sqrt {1 - e^{2} } }} = \left( {\begin{array}{*{20}c} { - \sin (\omega + f) - e\sin \omega } \\ {\cos (\omega + f) + e\cos \omega } \\ 0 \\ \end{array} } \right)\frac{na}{{\sqrt {1 - e^{2} } }}. $$

(3.47)

The reverse problem of Eq. 3.43, i.e. for given rectangular satellite state vector $(\vec{r},\dot{\vec{r}})^{\text{T}}$ to compute the Keplerian elements , can be carried out as follows. ω + f is called argument of latitude and denoted by u.

• 1. Using the given state vector to compute the modulus r and v $(r = \left| {\vec{r}} \right|,\;v = \left| {\dot{\vec{r}}} \right|);$

• 2. Using Eqs. 3.10 and 3.11 to compute vector $\vec{h}$ and its modulus h;

• 3. Using Eqs. 3.13 and 3.14 to compute inclination i and the right ascension of ascending node Ω;

• 4. Using Eqs. 3.45, 3.29, and 3.32 to compute semimajor axis a, eccentricity e and average angular velocity n;

• 5. Rotating $\vec{r}$ by $\vec{p} = R_{ 1} (i)R_{ 3} (\varOmega )\vec{r}$ and then using Eq. 3.46 to compute ω + f;

• 6. Rotating $\dot{\vec{r}}$ by $\dot{\vec{p}} = R_{ 1} (i)R_{ 3} (\varOmega )\dot{\vec{r}}$ and then using Eq. 3.47 to compute ω and f;

• 7. Using Eqs. 3.33, 3.39, and 3.40 to compute E, M and t _p.

To transform the GPS state vector from the ECSF coordinate system to other coordinate systems , the formulas discussed in Chap. 2 can be used.

3.2 Disturbed Satellite Motion

Keplerian motion of a satellite is a motion under the assumption that the satellite is only attracted by the central force of the earth. This is, of course, an approximation. For a satellite problem, the earth cannot be considered a mass point or a homogenous sphere. The earth’s total force of attraction can be considered the central force plus the non-central force. The latter is called the earth’s disturbing force , which has an order of 10⁻⁴ compared with the central force. The other attraction forces, which are simply called disturbing forces, are the attraction forces of the sun and the moon, the earth and ocean tide , and surface forces such as solar radiation pressure and atmospheric drag . The satellite motion can then be considered a nominal motion (e.g., Keplerian motion) plus a disturbed motion.

If we further use the Keplerian elements to describe the disturbed motion of the satellite, all elements should be functions of time. Keplerian elements (Ω(t), i(t), ω(t), a(t), e(t), M(t)) can be represented by σ_j(t), j = 1, …, 6, thus the polynomial approximations are

$$ \sigma_{j} (t) = \sigma_{j} (t_{0} ) + \left. {\frac{{{\text{d}}\sigma_{j} (t)}}{{{\text{d}}t}}} \right|_{{t = t_{0} }} (t - t_{0} ) + \ldots \quad j = 1, \ldots ,6. $$

(3.48)

In other words, the disturbed orbit can be further represented by Keplerian elements ; however, all elements are time variables. If the initial elements and their changing rates are given, the instantaneous elements can be obtained. This principle is used in the broadcast ephemerides .

Detailed disturbing theory and orbit correction as well as orbit determination will be discussed in Chap. 11 later.

3.3 GPS Broadcast Ephemerides

GPS broadcast ephemerides are forecasted, predicted or extrapolated satellite orbits data, which are transmitted from the satellite to the receiver in the navigation message . Because of the nature of the extrapolation, broadcast ephemerides do not have enough high qualities for precise applications. The predicted orbits are curve fitted to a set of relatively simple disturbed Keplerian elements and transmitted to the users.

The broadcast messages are

SV-id:

satellite number;

t _c :

reference epoch of the satellite clock;

a ₀, a ₁, a ₂ :

polynomial coefficients of the clock error;

t _oe :

reference epoch of the ephemerides;

$\sqrt a$ :

square root of the semimajor axis of the orbital ellipse;

e :

numerical eccentricity of the ellipse;

M ₀ :

mean anomaly at the reference epoch t _e;

ω ₀ :

argument of perigee;

i ₀ :

inclination of the orbital plane;

Ω ₀ :

longitude of the ascending node at the weekly epoch;

Δn :

mean motion difference;

idot :

rate of inclination angle;

$\dot{\varOmega }$ :

rate of node’s right ascension ;

C _uc, C _us :

correction coefficients (of argument of latitude) ;

C _rc, C _rs :

correction coefficients (of geocentric distance );

C _ic, C _is :

correction coefficients (of inclination).

The satellite position at epoch t can be computed as follows:

$$ \begin{aligned} M & = M_{0} + (\sqrt {\frac{\mu }{{a^{3} }}} +\Delta n)(t - t_{oe} ), \\ \varOmega & = \varOmega_{0} + \dot{\varOmega } (t - t_{oe} ), \\ \omega & = \omega_{0} + C_{\text{uc}} \cos (2u_{0} ) + C_{\text{us}} \sin (2u_{0} ), \\ r & = r_{0} + C_{\text{rc}} \cos (2u_{0} ) + C_{\text{rs}} \sin (2u_{0} ),\quad {\text{and}} \\ i & = i_{0} + C_{\text{ic}} \cos (2u_{0} ) + C_{\text{is}} \sin (2u_{0} ) + idot(t - t_{\text{oe}} ), \\ \end{aligned} $$

(3.49)

where

$$ \begin{aligned} E & = M + e\sin E, \\ r_{0} & = a(1 - e\cos E), \\ f & = 2\tan^{ - 1} \left( {\frac{{\sqrt {1 + e} }}{{\sqrt {1 - e} }}\tan \frac{E}{2}} \right),\quad {\text{and}} \\ u_{0} & = \omega_{0} + f. \\ \end{aligned} $$

(3.50)

μ is the earth’s gravitational constant (which can be read from the IERS Conventions, cf. table of constants). The satellite position in the orbital plane coordinate system (the 1st-axis points to the ascending node, the 3rd-axis is vertical to the orbital plane, and the 2nd-axis completes a right-handed system) is then

$$ \left( {\begin{array}{*{20}c} {x^{\prime}} \\ {y^{\prime}} \\ {z^{\prime}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {r\cos u} \\ {r\sin u} \\ 0 \\ \end{array} } \right), $$

where u = ω + f. The position vector can be rotated to the ECSF coordinate system by $R_{ 3} ({-}\varOmega )R_{ 1} \left( {{-}i} \right)$ and then rotated to the ECEF coordinate system by $R_{ 3} (\varTheta )$, where $\varTheta$ is Greenwich Sidereal Time and

$$ \varTheta = \omega_{\text{e}} (t - t_{\text{oe}} ) + \omega_{\text{e}} t_{\text{oe}} , $$

(3.51)

where ω _e is the angular velocity of the earth (can be read from the IERS Conventions, cf. table of constants). The satellite position vector in the ECEF coordinate system is then

$$ \left( {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right)_{\text{ECEF}} = R_{3} ( - \varOmega + \varTheta )R_{1} ( - i)\left( {\begin{array}{*{20}c} {r\cos u} \\ {r\sin u} \\ 0 \\ \end{array} } \right). $$

(3.52)

The first equation of 3.50 is the Keplerian equation , which may be solved iteratively. It is notable that the time t above should be the signal transmission time . (t – t _oe) should be the actual total time difference of the two time epochs and must account for the beginning and end of week crossovers (cf. Spilker 1996). That is, if the difference is greater (or less) than 302,400 s, subtract (or add) 604,800 s. The satellite clock error can be computed by (denoting k as the satellite’s id)

$$ \delta t_{ k} = a_{0} + a_{1} (t - t_{c} ) + a_{2} (t - t_{c} )^{2} . $$

(3.53)

Unit seconds are used for the time variable; the computed clock error has units of 10⁻⁶ s.

3.4 IGS Precise Ephemerides

GPS satellite precise orbits are available through the International GPS Service (IGS) in the form of post-processed results. Such orbits data are called IGS precise ephemerides. They can be downloaded for free from several internet homepages (e.g., www.gfz-potsdam.de).

IGS data are given in the ECEF coordinate system . For all possible satellites, the position vectors are given in x, y, z three components (units: km), and the related clock errors are also given (units: 10⁻⁶ s). The data are given in a suitable time interval (15 min).

To obtain the ephemerides of any interested epoch, a Lagrange polynomial is used to fit the given data and then to interpolate the data at the needed epoch. The general Lagrange polynomial is (e.g., Wang et al. 1979):

$$ y(t) = \sum\limits_{j = 0}^{m} {L_{j} (t)} \cdot y(t_{j} ), $$

(3.54)

where

$$ L_{j} (t) = \prod\limits_{k = 0}^{m} {\frac{{(t - t_{k} )}}{{(t_{j} - t_{k} )}},\quad k \ne j} , $$

(3.55)

where symbol $\Pi$ is a multiplying operator from k = 0 to k = m, m is the order of the polynomial, y(t _j ) are given data at the time t _j , L _j (t) is called the base function of order m, and t is the time on which data will be interpolated. Generally speaking, t should be placed around the middle of the time duration (t ₀, t _m ) if possible. Therefore, m is usually selected as an odd number. For IGS orbit interpolation , a standard m is selected as 7 or 9 from experience.

For the equal distance Lagrange interpolation there is

$$ \begin{aligned} t_{k} & = t_{0} + k\Delta t \hfill \\ t - t_{k} & = t - t_{0} - k\Delta t \hfill \\ t_{j} - t_{k} & = (j - k)\Delta t, \hfill \\ \end{aligned} $$

then

$$ L_{j} (t) = \prod\limits_{k = 0}^{m} {\frac{{(t - t_{0} - k\Delta t)}}{{(j - k)\Delta t}},\quad k \ne j} , $$

(3.56)

where Δt is the data interval.

In order to deal with the broadcast ephemerides in a manner similar to IGS precise ephemerides, the broadcast orbit may be first computed and then transformed to IGS-like data for use.

The forecasted IGS ephemerides are now also available to download for free.

3.5 GLONASS Ephemerides

GLONASS broadcast ephemerides are forecasted, predicted, or extrapolated satellite orbit data, which are transmitted from the satellite to the receiver in the navigation message . The broadcast messages include the following: satellite number, reference epoch of the ephemerides, relative frequency offset , satellite clock offset, satellite position, satellite velocity, satellite acceleration, time system correction with respect to UTC_SU, the time difference between GLONASS time, and GPS time.

The satellite position and velocity at desired epoch t can be interpolated by using the Lagrange polynomial discussed in Sect. 3.4, or alternatively, by a five-order polynomial discussed in Sect. 5.4.2 where the position, velocity, and acceleration data are used.

The precise GLONASS ephemerides are similarly available. The data has nearly the same format as that of GPS and includes the message of the time differences of the GLONASS time and GPS time.

3.6 Galileo Ephemerides

The Galileo Open Service allows access to two navigation message types: F/NAV (freely accessible navigation) and I/NAV (integrity navigation). The content of the two messages differs in various items; however, in general it is very similar to the content of the GPS navigation message . But there are items in the navigation message that depend on the origin of the message (F/NAV or I/NAV): The SV clock parameters actually define the satellite clock for the dual-frequency ionosphere-free linear combination . F/NAV reports the clock parameters valid for the E5a-E1 combination; the I/NAV reports the parameters for the E5b-E1 combination.

3.7 BDS Ephemerides

The BDS open service broadcast navigation message is similar in content to the GPS navigation message . The header section and the first data record (epoch, satellite clock information) are equal to the GPS navigation file. The following six records are similar to GPS. Details can be referred to IGS RINEX format (2015).