Gravity Models

Specific Force and Gravity Models

Specific force is defined as the non-gravitational force per unit mass and is measured in meters per seconds squared. Specific force is the quantity measured by accelerometers and is defined as:

\[ \mathbf{f}^{b}_{ib} = \mathbf{a}^b_{ib} - \boldsymbol{\gamma}^{b}_{ib}, \label{3.7.1} \]

where \(\mathbf{f}\) is the specific force, \(\mathbf{a}\) is the acceleration, and \(\mathbf{\gamma}\) is the acceleration due to the gravitaitonal force.

Consider an object that is stationary with respect to ECEF frame. Then:

\[ \begin{align} \mathbf{v}^e_{eb} &= \dot{\mathbf{r}}^{e}_{eb} = \dot{\mathbf{r}}^{e}_{ib} = 0 \\ \mathbf{a}^e_{eb} &= \ddot{\mathbf{r}}^{e}_{eb} = \ddot{\mathbf{r}}^{e}_{ib} = 0. \label{3.7.2} \\ \end{align} \]

The inertially referenced acceleration resolved about ECEF would be:

\[ \begin{align} \mathbf{a}^{e}_{ib} &= \ddot{\mathbf{r}}^{e}_{ib} + \boldsymbol{\Omega}^{e}_{ib} \boldsymbol{\Omega}^{e}_{ib} \mathbf{r}^{e}_{ib} + 2 \boldsymbol{\Omega}^{e}_{ib} \dot{\mathbf{r}^{e}_{ib}} + \dot{\boldsymbol{\Omega}}^{e}_{ib} \mathbf{r}^{e}_{ib} \\ &= \boldsymbol{\Omega}^{e}_{ib} \boldsymbol{\Omega}^{e}_{ib} \mathbf{r}^{e}_{ib}. \label{3.7.3} \end{align} \]

Substituting equation (\(\ref{3.7.3}\)) into (\(\ref{3.7.1}\)) and using the identity \(\mathbf{r}^\gamma_{eb} = \mathbf{r}^\gamma_{ib}\) yields to:

\[ \mathbf{f}^{e}_{ib} = \boldsymbol{\Omega}^{e}_{ib} \boldsymbol{\Omega}^{e}_{ib} \mathbf{r}^{e}_{eb} - \boldsymbol{\gamma}^{e}_{ib}. \label{3.7.4} \]

The specific force sensed when an object is stationary with respect to ECEF frame is the reaction to what is known as the acceleration due to gravity:

\[ \begin{align} \mathbf{g}^e_b &= \left. -f^{e}_{ib} \right|_{\mathbf{v}^e_{eb}=0, \ \mathbf{a}^{e}_{eb} = 0} \\ &= \boldsymbol{\gamma}^{e}_{ib} - \boldsymbol{\Omega}^{e}_{ib} \boldsymbol{\Omega}^{e}_{ib} \mathbf{r}^{e}_{eb} \\ &= \boldsymbol{\gamma}^{e}_{ib} + \omega^2_{ie} \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right] \mathbf{r}^e_{eb}. \label{3.7.5} \end{align} \]

In navigation frame:

\[ \begin{align} \mathbf{g}^n_b &= \boldsymbol{\gamma}^{n}_{ib} + \boldsymbol{\Omega}^{n}_{ib} \boldsymbol{\Omega}^{n}_{ib} \mathbf{r}^{n}_{eb} \\ &= \boldsymbol{\gamma}^{n}_{ib} + \mathbf{R}^{n}_e \boldsymbol{\Omega}^{e}_{ib} \mathbf{R}^{e}_n \mathbf{R}^{n}_e \boldsymbol{\Omega}^{e}_{ib} \mathbf{R}^{e}_n \mathbf{r}^{n}_{eb} \\ &= \boldsymbol{\gamma}^{n}_{ib} + \mathbf{R}^{n}_e \boldsymbol{\Omega}^{e}_{ib} \boldsymbol{\Omega}^{e}_{ib} \mathbf{R}^{e}_n \mathbf{r}^{n}_{eb} \\ &= \boldsymbol{\gamma}^{n}_{ib} + \omega^2_{ie} \mathbf{R}^{n}_e \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right] \mathbf{R}^{e}_n \mathbf{r}^{n}_{eb} \\ &= \boldsymbol{\gamma}^{n}_{ib} + \omega^2_{ie} \left[ \begin{array}{ccc} \text{sin}^2 L_b & 0 & \text{cos} L_b \text{sin} L_b \\ 0 & 1 & 0 \\ \text{cos}L_b \text{sin}L_b & 0 & \text{cos}^2 L_b \end{array} \right] \mathbf{r}^n_{eb}. \label{3.7.6} \end{align} \]

The first term is the gravitational acceleration or the acceleration due to gravitational force. The second term is the outward centrifugal acceleration due to the Earth's rotation. Figure 3.7.1 illustrates this.

At earth's surface, the total acceleration due to gravity is about \(\mathbf{g} = 9.8\) m/s^2, with centrifugal component contributing up to 0.034 m/s^2.

When working with ECI, only the gravitation acceleration is required and can be directly calculated as:

\[ \begin{align} \boldsymbol{\gamma}^i_{ib} &= -\frac{\mu}{|\mathbf{r}^i_{ib}|^3} \left\{ \mathbf{r}^i_{ib} + \frac{3}{2}J_2 \frac{R_0^2}{|\mathbf{r}^i_{ib}|^2} \left\{ \begin{array}{c} \left[ 1 - 5(r^i_{ib,z} / |\mathbf{r}^i_{ib}|)^2 \right] r^i_{ib, x} \\ \left[ 1 - 5(r^i_{ib,z} / |\mathbf{r}^i_{ib}|)^2 \right] r^i_{ib, y} \\ \left[ 3 - 5(r^i_{ib,z} / |\mathbf{r}^i_{ib}|)^2 \right] r^i_{ib, z} \end{array} \right\} \right\} \\ \boldsymbol{\gamma}^e_{ib} &= -\frac{\mu}{|\mathbf{r}^e_{eb}|^3} \left\{ \mathbf{r}^e_{eb} + \frac{3}{2}J_2 \frac{R_0^2}{|\mathbf{r}^e_{eb}|^2} \left\{ \begin{array}{c} \left[ 1 - 5(r^e_{eb,z} / |\mathbf{r}^e_{eb}|)^2 \right] r^e_{eb, x} \\ \left[ 1 - 5(r^e_{eb,z} / |\mathbf{r}^e_{eb}|)^2 \right] r^e_{eb, y} \\ \left[ 3 - 5(r^e_{eb,z} / |\mathbf{r}^e_{eb}|)^2 \right] r^e_{eb, z} \end{array} \right\} \right\} \label{3.7.7} \\ \end{align}, \]

where \(J_2\) is the Earth's second harmonic gravitational constant and is equal to \(1.082627 \times 10^{-3}\) and \(\mu\) is the Earth's gravitational constant and its WGS 84 value is \(3.986004418 \times 10^{14} \ m^3 s^{-2}\).