Astronomy:Range rate

Range rate defines a signed scalar value describing the time rate of change of the range (distance) between two locations.

Derivation

Given a differentiable vector [math]\displaystyle{ \mathbf{r} \epsilon \mathbb{R}^3 }[/math] defining the instantaneous position of a target relative to an observer.

Let

[math]\displaystyle{ \mathbf{v} = \frac{d\mathbf{r}}{dt} }[/math]

(1)

with [math]\displaystyle{ \mathbf{v} }[/math] [math]\displaystyle{ \epsilon }[/math] [math]\displaystyle{ \mathbb{R}^3 }[/math], the instantaneous velocity of the target with respect to the observer.

The magnitude of the position vector [math]\displaystyle{ \mathbf{r} }[/math] is defined as

[math]\displaystyle{ ||\mathbf{r}|| = \langle \mathbf{r},\mathbf{r} \rangle^{1/2} }[/math]

(2)

The quantity range rate is the time derivative of the magnitude (norm) of [math]\displaystyle{ \mathbf{r} }[/math], expressed as

[math]\displaystyle{ \frac{d||\mathbf{r}||}{dt} }[/math]

(3)

Substituting (2) into (3)

[math]\displaystyle{ \frac{d||\mathbf{r}||}{dt} = \frac{d \langle \mathbf{r},\mathbf{r} \rangle^{1/2} }{dt} }[/math]

Evaluating the derivative of the right-hand-side

[math]\displaystyle{ \frac{d||\mathbf{r}||}{dt} = \frac{1}{2} \frac{d \langle \mathbf{r},\mathbf{r} \rangle}{dt} \frac{1}{\langle \mathbf{r},\mathbf{r} \rangle^{1/2}} }[/math]

[math]\displaystyle{ \frac{d||\mathbf{r}||}{dt} = \frac{1}{2} \frac{\langle \frac{d\mathbf{r}}{dt},\mathbf{r} \rangle + \langle \mathbf{r},\frac{d\mathbf{r}}{dt} \rangle}{\langle \mathbf{r},\mathbf{r} \rangle^{1/2}} }[/math]

using (1) the expression becomes

[math]\displaystyle{ \frac{d||\mathbf{r}||}{dt} = \frac{1}{2} \frac{\langle \mathbf{v},\mathbf{r} \rangle+\langle \mathbf{r},\mathbf{v} \rangle}{\langle \mathbf{r},\mathbf{r} \rangle^{1/2}} }[/math]

Since

[math]\displaystyle{ \langle \mathbf{v},\mathbf{r} \rangle = \langle \mathbf{r},\mathbf{v} \rangle }[/math]

With

[math]\displaystyle{ \hat{\mathbf{r}} =\frac{ \mathbf{r} }{\langle \mathbf{r},\mathbf{r} \rangle^{1/2}} }[/math]

The range rate is simply defined as

[math]\displaystyle{ \frac{d||\mathbf{r}||}{dt} = \frac{\langle \mathbf{r},\mathbf{v} \rangle}{\langle \mathbf{r},\mathbf{r} \rangle^{1/2}} = \langle \hat{\mathbf{r}},\mathbf{v} \rangle }[/math]

the projection of the observer to target velocity vector onto the [math]\displaystyle{ \hat{\mathbf{r}} }[/math] unit vector.

A singularity exists for coincident observer target, i.e. [math]\displaystyle{ \mathbf{r} = \begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix} }[/math]. In this case, range rate does not exist as [math]\displaystyle{ ||\mathbf{r}|| = 0 }[/math].

See also

• inner product
• orbit determination
• Lp space

Sources

• Hoffman, Kenneth M.; Kunzel, Ray (1971), Linear Algebra (Second ed.), Prentice-Hall Inc., ISBN 0135367972, https://archive.org/details/linearalgebra00hoff_0 
• Renze, John; Stover, Christopher; and Weisstein, Eric W. "Inner Product." From MathWorld—A Wolfram Web Resource.http://mathworld.wolfram.com/InnerProduct.html