Astronomy

Parabolic or hyperbolic trajectories

Parabolic or hyperbolic trajectories


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According to Newton's laws, the trajectories for the two-body problem are conics: either ellipses, or parabolas or hyperbolas. Of course periodic motions require Ellipses and in the Solar system these elliptic curves are a pretty good approximation of the trajectory of each planet.

However, it is possible that the trajectory of a very fast comet could be an hyperbola or a parabola with the Sun at a focus; in that case the motion would not be periodic and the comet would be visible near the Sun only once.

My question: was there ever any observation of a comet on a hyperbolic (or parabolic) trajectory (with the Sun at a focus)? Same question for different stellar objects. It would be certainly very difficult to observe, since the comet would be visible only one time at its perihelion and would go at infinity after this with no return.


Yes, and it is not uncommon for an orbit have an eccentricity close to one. The wikipedia site, linked in a comment above, notes C/1980 E1, which entered the inner solar system with an eccentricity close to one, but had a close encounter with jupiter and was accelerated. It left the inner solar system with a eccentricity of 1.05, and so is on a hyperbolic trajectory, and will escape from the sun's gravity

Orbits that are highly hyperbolic are very unlikely. Comets formed as part of the solar system.

They are not really harder to spot than any other comet. A comet takes many months to make its passage through the inner solar system. There is plenty of time for them to be spotted, especially if you have probes like SOHO or NEAT


Hyperbolic Kepler equation

In orbital mechanics, Kepler’s equation relates various geometric properties of the orbit of a body subject to a central force.

It was first derived by Johannes Kepler in 1609 in Chapter 60 of his Astronomia nova, and in book V of his Epitome of Copernican Astronomy (1621) Kepler proposed an iterative solution to the equation.The equation has played an important role in the history of both physics and mathematics, particularly classical celestial mechanics.

The hyperbolic Kepler equation is used for hyperbolic trajectories (e ≫ 1).
When e = 0, the orbit is circular. Increasing e causes the circle to become elliptical. When e = 1, there are three possibilities: a parabolic trajectory, a trajectory going in or out along an infinite ray emanating from the centre of attraction, or a trajectory that goes back and forth along a line segment from the centre of attraction to a point at some distance away.

A slight increase in e above 1 results in a hyperbolic orbit with a turning angle of just under 180 degrees. Further increases reduce the turning angle, and as e goes to infinity, the orbit becomes a straight line of infinite length.


Contents

Under standard assumptions the body traveling along hyperbolic trajectory will attain in infinity an orbital velocity called hyperbolic excess velocity () that can be computed as:

  • is the standard gravitational parameter,
  • is the (negative) semi-major axis of orbit's hyperbola.

The hyperbolic excess velocity is related to the specific orbital energy or characteristic energy by

is commonly used in planning interplanetary missions.


(we're talking about gravitational trajectories?)

If the planet or comet returns, it has an elliptic trajectory.

If it passes the star only once, and doesn't return, it has a parabolic or hyperbolic trajectory.

A parabolic trajectory has only one asymptotic direction … it both comes from and returns to that direction. It "reaches infinity" with zero speed and energy (and an object launched non-vertically with escape velocity goes into a parabolic trajectory).

A hyperbolic trajectory has two (non-parallel) asymptotes … it comes from one and returns to the other. It "reaches infinity" with positive speed and energy.

I'm trying to wrap my head around the different kinds of trajectories. Firstly, what kind of objects have hyperbolic trajectories? And how do we differentiate between a parabolic trajectory and a hyperbolic one?

When a spacecraft does a fly-by maneuver past a planet in order to gain velocity, it has a hyperbolic trajectory relative to the planet. It is going too fast to be captured by the planet, but close enough to have its course changed.

If you take that same spacecraft and aim for the same altitude above the planet's surface, but slow it down enough, it will get captured in orbit (its energy becomes 'bound' to the gravity field). This is of course an elliptical trajectory.

A parabolic trajectory is at the balance point between hyperbolic and elliptical. Using that same spacecraft, any faster than parabolic is hyperbolic any slower is elliptical.

Hyperbolic orbits are rare in nature but do exist. When a Near Earth Object passes closely by Earth, it has a hyperbolic trajectory relative to Earth. Of course, it also is in an elliptical orbit relative to the sun. That elliptical orbit will be shifted because of the energy transfer between Earth's gravity field and the NEO during the brief time while the NEO is in the hyperbolic "orbit".

An asymptotic trajectory is unfamiliar to me and seems like more of a math professor problem than a real-world situation.


Energy

Under standard assumptions, specific orbital energy () of parabolic trajectory is zero, so the orbital energy conservation equation for this trajectory takes form:

  • is orbital velocity of orbiting body,
  • is radial distance of orbiting body from central body,
  • is the standard gravitational parameter.

This is entirely equivalent to the characteristic energy (square of the speed at infinity) being 0:


'We Don't Planet' Episode 6: Kinds of Orbits

When Johannes Kepler made the intuitive leap in the early 1600s to realize that planetary orbits were not circular, as they had been assumed to be for millennia, he stuck to the trusty ellipse. Indeed, the ellipse accurately and precisely describes the motions of all the major planets.

In this view, a circle is just a special case of the ellipse, where the eccentricity (a term which, for lack of a better phrase, measures the "stretchiness" of an ellipse) is exactly 0. In our solar system, Venus and Neptune have nearly circular orbits with eccentricities of 0.007 and 0.009, respectively, while Mercury has the most elliptical orbit with an eccentricity of 0.206.

But circles and ellipses are themselves just special cases of a more general class of curves called conic sections. If the ellipticity of an orbit is 0, it&rsquos circular. If it's between 0 and 1, it's a standard ellipse. If the ellipticity is equal to 1, it&rsquos a parabolic orbit, and if greater than 1, hyperbolic.

Elliptical and circular orbits are stable, so of course all the planets are characterized by these kinds of eccentricities. In a parabolic or hyperbolic orbit, however, an object approaches a central gravitational body from a distance, swings close just once, and escapes out to infinity. Many comets follow such trajectories: they may orbit peacefully in the Oort Cloud for billions of years, but if perturbed get knocked into a hyperbolic orbit, falling into the inner solar system and then jetting back off into the emptiness of interstellar space.

A spacecraft&rsquos mission profile will also include several kinds of orbits. At launch, a rocket may be in a hyperbolic trajectory before correcting itself and settling into a stable elliptical orbit around the Earth, followed by a boost to a hyperbolic trajectory that sends it off to a distant target.

"We Don't Planet" is hosted by Ohio State University astrophysicist and COSI chief scientist Paul Sutterwith undergraduate student Anna Voelker. Produced by Doug Dangler, ASC Technology Services. Supported by The Ohio State University Department of Astronomyand Center for Cosmology and AstroParticle Physics. You can follow Paul on Twitterand Facebook.


Parabolas, Gravity and Archimedes Death Ray Mirror

Let’s move on to parabolas. Historically, parabolas have played a key role in the understanding of physics even before ellipses. Indeed, a century before Newton’s masterpiece, Galileo discovered that free falling objects all describe parabolic trajectories. Similarly, Marcus du Sautoy applies this principle in the following extract from BBC’s documentary The Code:

At Galileo’s time, understanding the mathematics of free falling objects was essential for designing military weapons, as it enabled to accurately predict the trajectories of canon balls!

The best way to visualize the trajectories of free falling objects is to look at water fountains. Let’s consider for instance the fountain on the right, with the mighty Eiffel tower in the back!

Thanks. Now, try to keep your eyes away from the Eiffel tower to stare at the water spray. The water drops describe parabolic trajectories!

That’s because of these annoying air frictions! Still, you can see that, at least until air frictions don’t become too important, the trajectories match those of the parabolas. And if you don’t believe that trajectories of free falling objects are those of parabolas, check wikipedia’s picture of a nice parabola-looking fountain!

Humm… Once again, the only explanation I have involves the power of calculus. But I’m not going to dwell on this. One thing you should know is that, in the right coordinate system, the coordinates of the points of the parabola are given by the equation $y=x^2$. All other parabolas are obtained by homothety and classical symmetries of this parabola, just like ellipses are obtained by deformation of the circle. This means that by stretching and rotating a parabola along axes, you can make any parabola! In fact, if you play Angry Birds, you probably have a good sense of all possible downwards parabolas!

Now, there’s a pretty awesome corollary to that. If you’re in a plane which follows the curve of a parabola, then you’ll feel as if the force of gravity has changed! In particular, if the plane takes the parabolic trajectory of a free-falling object, you’ll be feeling weightless in the plane! Isn’t it amazing!

It’s actually the exact same phenomenon that makes astronauts weightless! Check Derek Muller’s awesome explanation on Veritasium:

And what trajectory are astronauts following? (Hint: They’re in free fall!)

Yes there is! A parabola is actually an ellipse for which one of the focus has gone to infinity!

Exactly! Now, in fact, you could notice that $f$ becomes nearly equal to $a$. At the limit, we actually have $e=f/a=1$! This means that a parabola is sort of an ellipse of eccentricity 1. In other words, a parabola is an infinitely flattened ellipse. That’s why trajectories of free falling objects on Earth look like parabolas to us: They are ellipses which are so flattened that they are indistinguishable from parabolas!

You are being very perspicacious! Indeed, as we increase the inclination of an ellipse, we obtain more and more flattened ellipses, until, suddenly, we obtain a parabola! This is described below.

Hehe… That’s a great question, but way beyond the scope of this article. You’d need to dig into the awesomeness of topology to answer that!

Those which still make sense do! In particular, this is the case of the equal angle property!

As the other focus goes to infinity, lines which come from it and hit the part of the ellipse near the first focus get less and less inclined. In fact, it’s relevant and common to consider that lines coming from a point at infinity are all parallel.

Yes! More precisely, lines which are parallel to the axis of symmetry get reflected by the parabola towards the focus. This corresponds to the figure below:

Now, take this parabola, and make it a solid of revolution by rotating it along the axis of symmetry. You then obtain what’s known as a circular paraboloid. A great way to get one is to have a fluid rotating in a cup, like in this video by Arvind Gupta’s awesome Toys from Trash initiative.

Circular paraboloids have the properties that all intersections with a plane which contains the axis of symmetry are parabolas with the same focus. The consequence of that is that, when a message is sent from far away, since received signals are parallel, we can capture it very well by using a circular parabolic mirror and placing the receptor at the focus! This is why antennas in telecommunications are parabolic!

But this idea isn’t new. In fact, it was invented and exploited by Isaac Newton (yes, him again!), who designed a telescope to better observe the motions of the stars and come up with his laws of mechanics!

Well, actually, he wasn’t the first one to think of using parabolic mirrors! 2,000 years before him, the genius Archimedes from Syracuse used the same idea to defend his land from the Roman invaders: He used these mirrors to focus the parallel burning sunlights on a single point of an enemy ship to burn it down! The following extract is a video I took from the Musei Capitolini in Roma, in a temporary exposition on the Greek scholar:

I know! Unfortunately, this wasn’t enough for Syracuse to resist to the emerging Roman empire. This enabled Rome to discover Archimedes’ uncountable inventions, which I believe to be a key step towards its domination of the Mediterranean sea! In fact, it seems to me that Archimedes has then been at the top of human knowledge for the next 2,000 years, until Galileo finally rediscovered some of his brilliant achievements!

Well, the Muslims are actually the ones who first rediscovered Archimedes’ work, which have probably helped them be the most developed civilization at their apogee… But enough of History, let’s get back to maths!


Contents

At any position the orbiting body has the escape velocity for that position.

If a body has an escape velocity with respect to the Earth, this is not enough to escape the Solar System, so near the Earth the orbit resembles a parabola, but further away it bends into an elliptical orbit around the Sun.

This velocity ( v ) is closely related to the orbital velocity of a body in a circular orbit of the radius equal to the radial position of orbiting body on the parabolic trajectory:


Topics similar to or like Parabolic trajectory

Elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1 this includes the special case of a circular orbit, with eccentricity equal to 0. Kepler orbit with the eccentricity greater than 0 and less than 1 . Wikipedia

Trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. Orbit has the shape of a hyperbola. Wikipedia

Angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit. One of three angular parameters that define a position along an orbit, the other two being the true anomaly and the mean anomaly. Wikipedia

Dimensionless vector with direction pointing from apoapsis to periapsis and with magnitude equal to the orbit's scalar eccentricity. Constant of motion. Wikipedia

Angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body. The concept applies equally well to a small body revolving about a large, massive primary body or to two relatively same-sized bodies revolving about a common center of mass. Wikipedia

Orbit equation defines the path of orbiting body m_2,! around central body m_1,! Wikipedia

Orbit is the gravitationally curved trajectory of an object, such as the trajectory of a planet around a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. Wikipedia

Constant sum of their mutual potential energy (epsilon_p) and their total kinetic energy (epsilon_k), divided by the reduced mass. According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time: Wikipedia

One of the equations that model the motion of orbiting bodies. Direct result of the principle of conservation of mechanical energy which applies when the only force acting on an object is its own weight. Wikipedia

Motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. Wikipedia

Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line. Wikipedia

Angular difference between the actual position of a body in its elliptical orbit and the position it would occupy if its motion were uniform, in a circular orbit of the same period. Defined as the difference true anomaly, Wikipedia

Effect of the tidal forces between an orbiting body and the primary object that it orbits, whereby the eccentricity of the orbit is reduced over time so that the orbit becomes less and less elliptical. Tidal locking Wikipedia

Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. Wikipedia

List of types of orbits: Less commonly used Wikipedia

Orbit for an artificial satellite in which natural drifting due to the central body's shape has been minimized by careful selection of the orbital parameters. Orbit in which, over a long period of time, the satellite's altitude remains constant at the same point in each orbit. Wikipedia

Angular parameter that defines the position of a body moving along a Kepler orbit. Angle between the ascending node and the body. Wikipedia

Object's orbit around a celestial body. Expressed as the angle between a reference plane and the orbital plane or axis of direction of the orbiting object. Wikipedia

Time a given astronomical object takes to complete one orbit around another object, and applies in astronomy usually to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars. Often referred to as the sidereal period, determined by a 360° revolution of one celestial body around another, e.g. the Earth orbiting the Sun. Wikipedia

One of the orbital elements used to specify the orbit of an object in space. Angle from a specified reference direction, called the origin of longitude, to the direction of the ascending node, as measured in a specified reference plane. Wikipedia

One of the orbital elements of an orbiting body. Angle from the body's ascending node to its periapsis, measured in the direction of motion. Wikipedia

Imaginary line defined by an orbit's eccentricity vector. Strictly defined for elliptic, parabolic, and hyperbolic orbits. Wikipedia

Fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical two-body problem. Angular distance from the pericenter which a fictitious body would have if it moved in a circular orbit, with constant speed, in the same orbital period as the actual body in its elliptical orbit. Wikipedia


Affiliations

TEDA Applied Physics Institute and School of Physics, Nankai University, Tianjin, 300457, China

Juanying Zhao, Yi Hu, Daohong Song & Zhigang Chen

Science and Technology on Solid-State Laser Laboratory, North China Institute of Electronics Optics, Beijing, 100015, China

Department of Physics and Astronomy, San Francisco State University, San Francisco, CA, 94132, USA

Juanying Zhao, Peng Zhang & Zhigang Chen

Max Planck Institute for the Science of Light, 91058, Erlangen, Germany

Academy of Opto-Electronics, Chinese Academy of Sciences, Beijing, 100094, China

Department of Mathematics and Applied Mathematics, University of Crete, 70013, Heraklion, Crete, Greece



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