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 $\mathbf{𝐫}ϵ{\mathbb{ℝ}}^3$ defining the instantaneous position of a target relative to an observer.

Let

$\mathbf{𝐯}=\frac{d\mathbf{𝐫}}{dt}$

(1)

with $\mathbf{𝐯}$ $ϵ$ ${\mathbb{ℝ}}^3$, the instantaneous velocity of the target with respect to the observer.

The magnitude of the position vector $\mathbf{𝐫}$ is defined as

$||\mathbf{𝐫}||=⟨\mathbf{𝐫},\mathbf{𝐫}{⟩}^{1/2}$

(2)

The quantity range rate is the time derivative of the magnitude (norm) of $\mathbf{𝐫}$, expressed as

$\frac{d||\mathbf{𝐫}||}{dt}$

(3)

Substituting (2) into (3)

$\frac{d||\mathbf{𝐫}||}{dt}=\frac{d⟨\mathbf{𝐫},\mathbf{𝐫}{⟩}^{1/2}}{dt}$

Evaluating the derivative of the right-hand-side

$\frac{d||\mathbf{𝐫}||}{dt}=\frac{1}{2}\frac{d⟨\mathbf{𝐫},\mathbf{𝐫}⟩}{dt}\frac{1}{⟨\mathbf{𝐫},\mathbf{𝐫}{⟩}^{1/2}}$

$\frac{d||\mathbf{𝐫}||}{dt}=\frac{1}{2}\frac{⟨\frac{d\mathbf{𝐫}}{dt},\mathbf{𝐫}⟩+⟨\mathbf{𝐫},\frac{d\mathbf{𝐫}}{dt}⟩}{⟨\mathbf{𝐫},\mathbf{𝐫}{⟩}^{1/2}}$

using (1) the expression becomes

$\frac{d||\mathbf{𝐫}||}{dt}=\frac{1}{2}\frac{⟨\mathbf{𝐯},\mathbf{𝐫}⟩+⟨\mathbf{𝐫},\mathbf{𝐯}⟩}{⟨\mathbf{𝐫},\mathbf{𝐫}{⟩}^{1/2}}$

Since

$⟨\mathbf{𝐯},\mathbf{𝐫}⟩=⟨\mathbf{𝐫},\mathbf{𝐯}⟩$

With

$\widehat{\mathbf{𝐫}}=\frac{\mathbf{𝐫}}{⟨\mathbf{𝐫},\mathbf{𝐫}{⟩}^{1/2}}$

The range rate is simply defined as

$\frac{d||\mathbf{𝐫}||}{dt}=\frac{⟨\mathbf{𝐫},\mathbf{𝐯}⟩}{⟨\mathbf{𝐫},\mathbf{𝐫}{⟩}^{1/2}}=⟨\widehat{\mathbf{𝐫}},\mathbf{𝐯}⟩$

the projection of the observer to target velocity vector onto the $\widehat{\mathbf{𝐫}}$ unit vector.

A singularity exists for coincident observer target, i.e. $\mathbf{𝐫}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$. In this case, range rate does not exist as $||\mathbf{𝐫}||=0$.

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