Radial trajectory

Key takeaways

Radial trajectory belongs to astronomy; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Radial trajectory to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Radial trajectory from memory before moving on to harder problems.

Reference excerpt

In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line.

Classification There are three types of radial trajectories (orbits).

Radial elliptic trajectory: an orbit corresponding to the part of a degenerate ellipse from the moment the bodies touch each other and move away from each other until they touch each other again. The relative speed of the two objects is less than the escape velocity. This is an elliptic orbit with semi-minor axis = 0 and eccentricity = 1. Although the eccentricity is 1, this is not a parabolic orbit. If the coefficient of restitution of the two bodies is 1 (perfectly elastic) this orbit is periodic. If the coefficient of restitution is less than 1 (inelastic) this orbit is non-periodic. Radial parabolic trajectory, a non-periodic orbit where the relative speed of the two objects is always equal to the escape velocity. There are two cases: the bodies move away from each other or towards each other. Radial hyperbolic trajectory: a non-periodic orbit where the relative speed of the two objects always exceeds the escape velocity. There are two cases: the bodies move away from each other or towards each other. This is a hyperbolic orbit with semi-minor axis = 0 and eccentricity = 1. Although the eccentricity is 1 this is not a parabolic orbit. Unlike standard orbits which are classified by their orbital eccentricity, radial orbits are classified by their specific orbital energy, the constant sum of the total kinetic and potential energy, divided by the reduced mass:

ε = v 2 2 − μ x {\displaystyle \varepsilon ={\frac {v^{2}}{2}}-{\frac {\mu }{x}}}

where x is the distance between the centers of the masses, v is the relative velocity, and μ = G ( m 1 + m 2 ) {\displaystyle \mu =G\left(m_{1}+m_{2}\right)} is the standard gravitational parameter. Another constant is given by:

w = 1 x − v 2 2 μ = − ε μ {\displaystyle w={\frac {1}{x}}-{\frac {v^{2}}{2\mu }}={\frac {-\varepsilon }{\mu }}}

For elliptic trajectories, w is positive. It is the inverse of the apoapsis distance (maximum distance). For parabolic trajectories, w is zero. For hyperbolic trajectories, w is negative, It is − v ∞ 2 2 μ {\displaystyle \textstyle {\frac {-v_{\infty }^{2}}{2\mu }}} where v ∞ {\displaystyle \textstyle v_{\infty }} is the velocity at infinite distance.

Time as a function of distance Given the separation and velocity at any time, and the total mass, it is possible to determine the position at any other time. The first step is to determine the constant w. Use the sign of w to determine the orbit type.

w = 1 x 0 − v 0 2 2 μ {\displaystyle w={\frac {1}{x_{0}}}-{\frac {v_{0}^{2}}{2\mu }}}

where x 0 {\textstyle x_{0}} and v 0 {\textstyle v_{0}} are the separation and relative velocity at any time.

Parabolic trajectory

t ( x ) = 2 x 3 9 μ {\displaystyle t(x)={\sqrt {\frac {2x^{3}}{9\mu }}}}

where t is the time from or until the time at which the two masses, if they were point masses, would coincide, and x is the separation. This equation applies only to radial parabolic trajectories, for general parabolic trajectories see Barker's equation.

Elliptic trajectory

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Radial trajectory

Start with the simplest possible case. Write down what Radial trajectory claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In astronomy, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Radial trajectory before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Radial trajectory ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Radial trajectory

In research: Radial trajectory appears in astronomy research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Radial trajectory in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Radial trajectory is common in secondary-school and first-year university syllabi. It links to neighbouring topics Astrodynamics, Johannes Kepler, Orbits, so understanding it makes those chapters shorter.
In everyday life: Look for Radial trajectory outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Radial trajectory in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Radial trajectory means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Radial trajectory out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Radial trajectory in simple terms?

In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line.

Why does Radial trajectory matter?

Because it connects several astronomy ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Radial trajectory?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Radial trajectory.