Range rate

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

Let

${\displaystyle \mathbf {v} ={\frac {d\mathbf {r} }{dt}}}$

(1)

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

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

${\displaystyle ||\mathbf {r} ||=\langle \mathbf {r} ,\mathbf {r} \rangle ^{1/2}}$

(2)

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

${\displaystyle {\frac {d||\mathbf {r} ||}{dt}}}$

(3)

Substituting (2) into (3)

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

Evaluating the derivative of the right-hand-side

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

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

using (1) the expression becomes

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

Since^[1]

${\displaystyle \langle \mathbf {v} ,\mathbf {r} \rangle =\langle \mathbf {r} ,\mathbf {v} \rangle }$

With

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

The range rate is simply defined as

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

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

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

See also[]

• inner product
• orbit determination
• Lp space

References[]

Sources[]

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