Orbital vocabulary confusion! How can the tangential velocity of an elliptical Kepler orbit not be tangent to the orbit?

Question

I'm now officially confused about the usage of "tangential" when breaking down orbital velocity components. It started with edits and comments on this answer to Orbital speed is (vector) sum of tangential and normal speed?

I've (possibly/probably incorrectly) used "tangential" to refer to the velocity component perpendicular to radial in Low-thrust spiraling to escape, is the flight path angle (gamma) at C3=0 always 39 degrees? and also in How to calculate the flight path angle, γ, from a state vector?. I say possibly/probably inccorectly because the the velocity should always be tangent to the orbit. But rather than correct me, @MarkAdler's answer to the first question continues the distinction between tangential velocity and the direction of motion:

Below is the same plot for when accelerating tangentially, as opposed to in the velocity direction.

and @TomSpilker's answer to the second question does likewise:

In addition to $\gamma$, the angle between the tangential direction and the velocity vector, there is $\beta$, the angle between the radial direction and the velocity vector.

However, the diagram below from Julio@'s answer to ** suggests the component perpendicular to the radial direction might be called normal velocity.

Question: How can the tangential velocity of an elliptical Kepler orbit not be tangent to the orbit, but instead be perpendicular to the radial component? Help me Mr. Wizzard!

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Source

Answer 1

This is a disagreement about how words are used, as in , How to calculate the flight path angle, γ, from a state vector? , Are LVLH and RSW coordinate systems the same thing? , and What is different between Hill frame and LVLH frame in satellite? . The mathematical relationships are clear and consistent, but the names we assign to the elements of the diagram are open to creative interpretation and thus inconsistency.

The Consultative Committee for Space Data Systems's "Recommended Standard for Conjunction Data Messages" (CCSDS 508.0-B-1), discussed in more detail in What is the probability of impact? , defines two different frames the words "along-track" or "in-track" might mean. CCSDS calls them "RTN" and "TVN", while other sources give different names for the same two things. For example, Vallado calls them RSW and NTW (page 157 of the 4th edition), and the SGP4 documentation uses UVW and PTW.

"RTN" stands for "Radial, Transverse, Normal". Normal means unit vector parallel to the angular momentum, which points toward position $\times$ velocity. Radial means parallel to the vector pointing from the central body to the orbiting object, or equivalently, from the object away from the central body. Transverse means the unit vector that completes the right-handed system, which points in the plane of the orbit somewhere close to but not exactly coinciding with the object's velocity, except at apogee and perigee, or if eccentricity is zero.

"TVN" stands for "Transverse, Velocity, Normal". Normal is the same as before. Velocity exactly coincides with the actual instantaneous velocity direction. Transverse still means complete the right-handed system, but now that means it points in the plane of the orbit somewhere close to but not exactly coinciding with the object's outward radial position vector, except at apogee and perigee, or if eccentricity is zero.

Note the phrase "if eccentricity is zero": for purely circular orbits, the two definitions coincide, and there is no difference. The ellipse pictured in the question has an eccentricity of roughly 0.7, which is in the range usually called highly elliptical orbits, so the differences are especially large and obvious.

The TVN definition is the only one I've seen used in mathematics, where it is called the Frenet-Serret frame, or "TNB" for "Tangent, Normal, Binormal". However, those words do not have the same meanings as in CCSDS. Instead, Tangent means in the direction of the velocity. Normal means perpendicular to velocity in the direction of the curvature of the path, matching the "Transverse" of TVN. Binormal means Tangent $\times$ Normal, which coincides with what RTN and TVN call "Normal".

This choice is perfectly natural in classical differential geometry, where the intrinsic curvature and torsion of the path, parameterized by length, are independent of what you choose the origin of your vector space to be. There's no point in distinguishing a "radial" direction, because there's nothing special about the zero position.

In orbital mechanics, however, the opposite is true: the whole point is to figure out what the curve must be based on the dominant gravitation of the massive object (or system barycenter) whose location provides a natural and obvious definition of the zero position.

With regard to the components depicted in the figure, I agree that most likely $\mathbf{v}_r$ means "radial component of velocity" and $\mathbf{v}_n$ means "normal component of velocity", but the meaning of normal is already overloaded, so I think it does as much harm as good. If I were writing a textbook, I would probably use $\mathbf{v}_{||}$ (pronounced "v-par") to mean "component of velocity parallel to the radial direction" and $\mathbf{v}_{\perp}$ (pronounced "v-perp") to mean "component of velocity perpendicular to the radial direction".

I personally would not have used "tangential" in either of the quotations shown, because I happen to think that velocity is inherently tangential. However, that's probably because I was taught differential geometry long before I became an astrodynamicist, and early exposure to mathematics warped my vocabulary in a variety of ways.

Answer 2

The terms radial and tangential are relative to the central body NOT the instantaneous vector, so if the radial velocity is nonzero, then the tangential component is orthogonal to that and parallel to the planet's horizon. It is Vtangent * R that is the angular momentum that is conserved in a Kepleran orbit.

Momentarily at apogee and perigee Vtan = V.

Answer 3

It is not easy to find the mathematical expressions for the radial and normal components of the velocity in elliptical orbits. If we put the words

elliptical orbit radial velocity component

in an internet search engine, you will not find the mathematical expressions you are looking for in the first entries. But the drawing of the elliptical orbit that illustrates this StackExchange thread does appear among the first images of the search result.

And in this good drawing it is clear that the mathematical expressions we are looking for are those corresponding to the $v_r$ and $v_n$ that appear in it.

I therefore provide the expressions for the radial component and perpendicular to the radius vector component of the velocities in elliptical orbits, so that they can be used by future "seekers" of these expressions who come to this thread.

With the usual nomenclature:

$G$ = universal gravitational constant

$M$ = mass of primary

$m$ = mass of secondary

$a$ = semi-major axis of the elliptical orbit

$e$ = eccentricity of the elliptical orbit

$\theta$ = true anomaly

$r$ = radius vector (position vector from de focus)

The mathematical expression to calculate the radial velocity is:

$$v_r=\sqrt{\frac{G (M+m)}{a(1-e^2)}} \ \ e \sin \theta$$

And the expression for the velocity component perpendicular to the radius vector is:

$$v_n=\sqrt{\frac{G (M+m)}{a(1-e^2)}} \ \ (1+e \cos \theta)$$

Naturally, it follows that:

$$v=\sqrt{v_r^2+v_n^2}$$

That operating can be converted into the familiar expression:

$$v=\sqrt{2 G (M+m)\left ( \frac 1 r - \frac 1{2a} \right )}$$

$$v=\sqrt{2 \mu \left ( \frac 1 r - \frac 1{2a} \right )}$$

Where $\mu=G(M+m)$ is the gravitational parameter.

Best regards