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**UPDATE:** The on-line simulation seems to be working beautifully now! I'd recommend anyone to go back and take another look to enjoy both the mathematical beauty of orbital mechanics, and the aesthetics of the solar system visualization!

New York Times article Visualizing the Cosmic Streams That Spew Meteor Showers links to a solar system viewer that allows one to visualize aspects of meteor-shower-inducing comet orbits by Ian Webb.

Now, astronomers and engineers have created an animation that lets you witness the entire journey. Using data from the Cameras for Allsky Meteor Surveillance, a network of about 60 cameras pointed at the sky above San Francisco Bay, researchers have recorded more than 300,000 meteoroid trajectories since 2010. They plan to use the data to confirm more than 300 potential meteor showers that scientists have observed, but not verified.

“Each dot that you see is a shooting star that was captured by one of our cameras,” said Peter Jenniskens, an astronomer at the SETI Institute and NASA Ames Research Center in Silicon Valley, Calif., who runs CAMS. His interactive transforms meteor showers like the Geminids and the Orionids into shimmering rivers of space rocks. Viewers can pinpoint the moment the Lyrids or Eta Aquarids light up the night by watching when their streams intersect with the Earth's orbit, shown in blue. There is even an option to see all the meteor showers at once, making it look like a meteor hurricane.

I need some help understanding what I'm looking at. When I first open it up I believe I see a number of objects following the same general hyperbolic (or very elliptical) orbit. What bothers me is that some seem to zip past very fast, some seem to crawl slowly, and some are moving at intermediate speed. If they are all associated with one primary comet, and follow similar paths in this large (roughly 10 AU) view, shouldn't they have at least roughly similar velocities?

**Edit:** I've snapped screenshots from`30-10-2017`

to`04-11-2017`

at a speed setting of`0.005`

and made two GIFs below The second one is annotated with a red, green, and blue arrow indicating three objects (slow, medium, and fast) following nearly the same orbit with at least a factor of 10 different speeds between them.

I have replayed viewing from different angles, they follow nearly the same orbit in 3D, all the way around the sun and back into space - it is not related to a particular view.

*I believe this to be completely unphysical!*

Red, Green Blue (slow to fast)

Looking at the simulation, and the gifs attached here, some things are clear. The dots cannot represent actual meteoroids in their orbits, as the 6 orbital elements fully determine the state vectors of a body. You can't be in the same orbit, at the same place, at the same time and have very different velocity. Thus the dots don't obey Kepler's third law.

It is not clear that they obey the second law either (that equal areas are swept out in equal times) They do seem to move faster at perihelion but perhaps not fast enough.

The graphic doesn't seem to show the "lumpiness" of some meteor streams. Notably the Leonids, which have a very intense storm, roughly every 33 years. Other streams are also lumpy which doesn't seem to be apparent in this graphic.

What it does show is the variation in orbit of the meteoroids that form a meteor shower. This is where (I guess) Nasa was involved. From the observation of a meteor (ideally several observations of the same meteor to triangulate its position) it should be possible to work out its orbit before it fell into Earth's gravitational field and hit the atmosphere. Observe enough meteors and you can get an idea of the distribution of orbits: the mean and standard deviation in the inclination, eccentricity, semimajor axis, ascending node and angle of perihelion. From these, you can get a distribution of elliptical orbits. There may be some modelling work do be done: we can only observe those meteorids that hit the Earth, but we can suppose the they are distributed all around the parent body. But from our sample of meteoroids that hit the Earth, we can model the distribution of those that never will.

The dots are drawn on these Keplerian orbits, but the speed at which the move on the orbit doesn't seem to represent the actual speed of meteoroids in space. Rather it is a way of illustrating the variation in elliptical orbits, without drawing a solid ring for each orbit.

Over time, small objects relatively close together at the start will be pulled into quite different orbits by the perturbations of larger objects (like Jupiter) and even Earth if they approach closely.

Relatively small changes at close approach can have quite large effects when the objects reach their furthest approach.

The orbits are approximately elliptical (some may be hyperbolic) and the orbital velocity will vary in such an orbit (unlike a circular orbit). So you'll see objects drifting along at the furthest reach of their orbit and speeding up as they "fall" in to the closest point of their orbit. They'll again slow down as they "rise up" away from the closest approach (exactly as a stone you throw into the air).

So the combination of these things means that the objects can separate significantly over time into quite distinct orbits that have quite significant differences in orbital velocities.

## A satellite moves around the earth in a circular orbit with constant speed.Explain is the motion uniform or accelerated ?

Uniform circular motion, but the satellite is accelerated towards the centre of the earth.

#### Explanation:

There are a few concepts with regards to a satellite orbiting the earth.

- The orbit is actually elliptical, but it is treated as circular for easier calculations.
- The satellite is orbiting with constant speed.
- The satellite's velocity is always changing.
- The satellite is accelerating, because there is a net force acting on it.
- The force of gravity on the satellite is the centripetal force.

Key concept:

Speed - is a scalar quantity, which only has *magnitude*.

Velocity - is a vector quantity, it possesses both *magnitude* and *direction*.

A geostationary satellite orbits the earth with a velocity of 3.07km/s.

So, the satellite orbits the earth with a constant **speed** of 3.07km/s because the magnitude of its speed is constant.

However, its direction is constantly changing, as seen in the diagram below.

At the west side of its orbit, the direction of the satellite is *upwards*. At the north of its orbit, the direction changes to *rightward*.

Therefore, the **velocity** of the satellite is always changing.

(The direction of velocity is tangential to its circular path.)

According to Newton's Second Law, the satellite is **accelerating** because it experiences a net force acting on it, and also because its velocity is changing.

The direction of the satellite's acceleration is not tangential to the circular motion, but rather perpendicular to its velocity/towards the centre of the earth.

This acceleration is a result of earth's gravitational force on the satellite. The acceleration is also known as **centripetal acceleration**.

It is also useful to know that the force of gravity provides the necessary centripetal force for a stable orbit.

Hence, the astronauts experience weightlessness because there are no reaction forces when they touch the objects around them.

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##### Figuring

## Hitchhiker’s Guide to the Solar System

Hitchhike with us for a ride amongst the planets in the Solar system, with a new review paper released by a collaboration of Australian planetary scientists - paving the way for what they expect Exoplanets will be like.

Whenever you learn or experience something new, it is common to compare it with something you are already familiar with. So, when scientists study exoplanets, it makes sense for them to compare these planets to the ones closest to home. The Solar system is a great source of information when it comes to trying to understand the composition and dynamics of planets that reside outside of our system.

Human understanding of the Solar system itself has changed throughout history, but especially since the development of telescope and photographic technology. Prior to this technological revolution which shifted paradigms, the Solar system was a place that sky gazers have filled with mythology and divine intervention since we first looked up at the sky in wonder.

To the ancients, and for millennia, the five bright wanderers were tied to omnipotent deities, dancing with enough regularity that their orbital values could be calculated - and even predicted. One such example of these intricate observations were the measurements made by some of the first Mesopotamian astronomers, when studying the orbit of Venus or cycles of the Moon. Another, the brightness, motion and connection to zodiacal light by Indigenous Australian astronomers.

For thousands of years, the five bright moving “stars” - Mercury, Venus, Mars, Jupiter and Saturn - ruled all knowledge of the Solar system only as bright points of light that differed from the fixed, eternal stars.

In the 17th Century, the human perspective of the Solar system changed. Optics and telescopes opened our eyes to the Jovian moons, the rings of Saturn, dark features on Mars, and craters on the lunar surface - all raising as much excitement as they did questions. At the height of this new era our exploration of the Solar system (in particular from 1684) a total of 16 bodies were considered part of the our planetary system - taking into account the Sun, Moon, and the known Jovian and Saturnian moons.

Through studying these newly discovered subjects, a range of astronomers - of the likes of Kepler, Huygens, Halley and Newton - started formulating mathematics around their observations, finding relations between the orbital dance of objects, and distance from the Sun. Following the discovery of Uranus by Sir William Herschel, in 1781, these relationships were used to great effect in the discovery of Neptune -- found in 1846 as a result of its gravitational pull nudging Uranus away from the path it would otherwise take across the sky.

The 1800s also saw the number of ‘planets’ in the Solar system rise dramatically, with the discovery of the first objects orbiting between Mars and Jupiter - objects we now know as the asteroids. Indeed, the largest of those asteroids, Ceres, was considered a planet in text books until at least the mid-1930s - around the time that Clyde Tombaugh stumbled across a new, odd, small planet moving on a highly elliptical orbit, even going so far as crossing that of the giant Neptune. With the asteroids demoted, and the new find (Pluto) considered a planet, the system capped out at 9 planets, from the 1930s until very recently.

For 75 years, following the discovery of Pluto, instruments advanced, telescopes got bigger, cameras got better, and rockets sent interplanetary spacecraft to make in-situ observations we learned so much about our own, unique planetary system. We expanded our knowledge so much that in 2006 we changed our definition of a planet and reclassified the largest of the Solar system’s smaller astronomical bodies (such as Pluto) as dwarf planets.

This brief summary of the evolution of how we consider the Solar system highlights the ever changing banks of knowledge that planetary astronomers contribute to over time.

Now, a new review paper about the Solar system, resulting from collaborations between a dozen scientists and led by an Australian team, creates a new bridge in planetary astronomy, extending to a new class of planets that reside beyond the Sun’s gravitational influence. The paper bridges the gap between our most up to date Solar system science and extends it into the science of Exoplanets, in order to benefit both fields.

To do this, first author Jonti Horner from the University of Southern Queensland (USQ) and his collaborators took on the behemoth task of describing the Solar system and its planets, their evolution (as it is currently understood), and then examining how knowledge of the Solar system informs our understanding of exoplanetary systems. In much the same way as the revolutions of the last four centuries (the telescope, photography and Newtonian mathematics) revolutionised our understanding of our Solar system - the last 400 years of learning are now helping us to expand our knowledge to systems beyond our own.

## Is there a site on SE that I could ask about whether I have found an error in a scientific paper?

Is there a site on SE that I could ask about whether I have found an error in a scientific paper?

"Living too long" by Guy C Brown has the following (it seems to me) self-contradictory paragraph:

Human life expectancy has been increasing at a rapid rate. Better health care and hygiene, healthier life styles, sufficient food and improved medical care and reduced child mortality mean that we can now expect to live much longer than our ancestors just a few generations ago. Life expectancy at birth in the EU was about 69 years in 1960 and about 80 years in 2010, which corresponds to a rate of increase in life expectancy of 2.2 years per decade. If this rate of increase remains unchanged, as it has for the last century, then someone born in the EU today would be expected to live about 100 years.

Edit: FWIW I now think the most likely cause of the error is that the author meant to say "in the EU in a hundred years time" or "in the EU in 2110" but either made a typo or his text got edited later by someone else to what seemed to make more sense.

## Components of the Acceleration Vector

We can combine some of the concepts discussed in Arc Length and Curvature with the acceleration vector to gain a deeper understanding of how this vector relates to motion in the plane and in space. Recall that the unit tangent vector (vecs T) and the unit normal vector (vecs N) form an osculating plane at any point (P) on the curve defined by a vector-valued function (vecs

Theorem (PageIndex<1>): The Plane of the Acceleration Vector

Here, (v(t) = |vecs v(t)|) is the speed of the object and (kappa) is the curvature of (C) traced out by (vecs

Now we differentiate this equation:

A formula for curvature is (kappa=dfrac<||vecs

The coefficients of (vecs**tangential component of acceleration** and the **normal component of acceleration**, respectively. We write (a_vecs

Theorem (PageIndex<2>): Tangential and Normal Components of Acceleration

These components are related by the formula

Here (vecs

The normal component of acceleration is also called the *centripetal component of acceleration* or sometimes the *radial component of acceleration*. To understand centripetal acceleration, suppose you are traveling in a car on a circular track at a constant speed. Then, as we saw earlier, the acceleration vector points toward the center of the track at all times. As a rider in the car, you feel a pull toward the *outside* of the track because you are constantly turning. This sensation acts in the opposite direction of centripetal acceleration. The same holds true for non-circular paths. The reason is that your body tends to travel in a straight line and resists the force resulting from acceleration that push it toward the side. Note that at point (B) in Figure (PageIndex<4>) the acceleration vector is pointing backward. This is because the car is decelerating as it goes into the curve.

Figure (PageIndex<4>): The tangential and normal components of acceleration can be used to describe the acceleration vector.

The tangential and normal unit vectors at any given point on the curve provide a frame of reference at that point. The tangential and normal components of acceleration are the projections of the acceleration vector onto (vecs T) and (vecs N), respectively.

Example (PageIndex<2>): Finding Components of Acceleration

A particle moves in a path defined by the vector-valued function (vecs

- Let&rsquos start deriving the velocityand acceleration functions:

An object moves in a path defined by the vector-valued function (vecs r(t)=4t,hat

Use Equations
ef

## Transcript

The planets orbit the Sun in a counterclockwise direction as viewed from above the Sun's north pole, and the planets' orbits all are aligned to what astronomers call the ecliptic plane.

The story of our greater understanding of planetary motion could not be told if it were not for the work of a German mathematician named Johannes Kepler. Kepler lived in Graz, Austria during the tumultuous early 17th century. Due to religious and political difficulties common during that era, Kepler was banished from Graz on August 2nd, 1600.

Fortunately, an opportunity to work as an assistant for the famous astronomer Tycho Brahe presented itself and the young Kepler moved his family from Graz 300 miles across the Danube River to Brahe's home in Prague. Tycho Brahe is credited with the most accurate astronomical observations of his time and was impressed with the studies of Kepler during an earlier meeting. However, Brahe mistrusted Kepler, fearing that his bright young intern might eclipse him as the premier astronomer of his day. He therefore led Kepler see only part of his voluminous planetary data.

He set Kepler, the task of understanding the orbit of the planet Mars, the movement of which fit problematically into the universe as described by Aristotle and Ptolemy. It is believed that part of the motivation for giving the Mars problem to Kepler was Brahe's hope that its difficulty would occupy Kepler while Brahe worked to perfect his own theory of the solar system, which was based on a geocentric model, where the earth is the center of the solar system. Based on this model, the planets Mercury, Venus, Mars, Jupiter, and Saturn all orbit the Sun, which in turn orbits the earth. As it turned out, Kepler, unlike Brahe, believed firmly in the Copernican model of the solar system known as heliocentric, which correctly placed the Sun at its center. But the reason Mars' orbit was problematic was because the Copernican system incorrectly assumed the orbits of the planets to be circular.

After much struggling, Kepler was forced to an eventual realization that the orbits of the planets are not circles, but were instead the elongated or flattened circles that geometers call ellipses, and the particular difficulties Brahe hand with the movement of Mars were due to the fact that its orbit was the most elliptical of the planets for which Brahe had extensive data. Thus, in a twist of irony, Brahe unwittingly gave Kepler the very part of his data that would enable Kepler to formulate the correct theory of the solar system, banishing Brahe's own theory.

Since the orbits of the planets are ellipses, let us review three basic properties of ellipses. The first property of an ellipse: an ellipse is defined by two points, each called a focus, and together called foci. The sum of the distances to the foci from any point on the ellipse is always a constant. The second property of an ellipse: the amount of flattening of the ellipse is called the eccentricity. The flatter the ellipse, the more eccentric it is. Each ellipse has an eccentricity with a value between zero, a circle, and one, essentially a flat line, technically called a parabola.

The third property of an ellipse: the longest axis of the ellipse is called the major axis, while the shortest axis is called the minor axis. Half of the major axis is termed a semi major axis. Knowing then that the orbits of the planets are elliptical, johannes Kepler formulated three laws of planetary motion, which accurately described the motion of comets as well.

**Kepler's First Law:** each planet's orbit about the Sun is an ellipse. The Sun's center is always located at one focus of the orbital ellipse. The Sun is at one focus. The planet follows the ellipse in its orbit, meaning that the planet to Sun distance is constantly changing as the planet goes around its orbit.

**Kepler's Second Law:** the imaginary line joining a planet and the sons sweeps equal areas of space during equal time intervals as the planet orbits. Basically, that planets do not move with constant speed along their orbits. Rather, their speed varies so that the line joining the centers of the Sun and the planet sweeps out equal parts of an area in equal times. The point of nearest approach of the planet to the Sun is termed perihelion. The point of greatest separation is aphelion, hence by Kepler's Second Law, a planet is moving fastest when it is at perihelion and slowest at aphelion.

**Kepler's Third Law:** the squares of the orbital periods of the planets are directly proportional to the cubes of the semi major axes of their orbits. Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun. The earth takes 365 days, while Saturn requires 10,759 days to do the same. Though Kepler hadn't known about gravitation when he came up with his three laws, they were instrumental in Isaac Newton deriving his theory of universal gravitation, which explains the unknown force behind Kepler's Third Law. Kepler and his theories were crucial in the better understanding of our solar system dynamics and as a springboard to newer theories that more accurately approximate our planetary orbits.

## Answers and Replies

Welcome to PF

I'm guessing you are thinking that energy is mass so a high kinetic energy would result in a higher "mass" which means extra gravity compared to if it is still.

I think the short answer is "yes and no".

The generator of the gravitational field is the stress-energy-momentum tensor ##T^

Anything with a non-zero T μν will feel gravity. In the non-relativistic limit co-momentum and co-stress vanish and energy reduces to mc 2 , which explains why masses appear in a non-relativistic description of gravity.

Kinetic energy contributes to gravity, mainly in the energy and co-momentum parts.

A quick calculation should show you how fast the Sun would have to be going to give it additional energy similar enough to it's mass-energy, and so show up as additional gravity that you'd notice. Even so - the earth moves with the Sun, so the Sun is not going all that fast wrt us. and it is *relative* speeds that count here.

You also seem to be asking if overall gravity increases due to motion - I think the best answer here is "no", after a circuit of the galaxy, the Solar system has the same gravity that it started with.

Introduction to general relativity.

http://preposterousuniverse.com/grnotes/grtinypdf.pdf [Broken]

Thanks for your response! It looks as though I have some more reading to do!

I guess what I'm wondering is whether or not the gravitational pull or stars on their respective orbiting planets is in some part due to the velocity at which they orbit the center of our galaxy.

Our sun for example orbits the center of the Milky way at about 139 MPS. Pretty darn fast! Let's say it wasn't orbiting anything at all, just standing "still" in space. Would its gravitational pull be less than it currently is moving at its impressive speed?

Along these same lines, would an object that rotates around the center of the Milky Way have less gravitational force than the same object rotating around the outer extents of the Milky Way (assuming their moving at the same RPM)? The object that is further out from the center would be going faster since it would have to be covering more space to maintain the same RPMs as the inner object.

## Circular Motion: Linear and Angular Speed

Radian measure and arc length can be applied to the study of *circular motion*. In physics the *average speed* of an object is defined as:

So suppose that an object moves along a circle of radius r, traveling a distance *s* over a period of time *t*, as in Figure 1. Then it makes sense to define the (average) **linear speed** *ν* of the object as:

Let *θ* be the angle swept out by the object in that period of time. Then we define the (average) **angular speed** *ω* of the object as:

Angular speed gives the rate at which the central angle swept out by the object changes as the object moves around the circle, and it is thus measured in radians per unit time. Linear speed is measured in distance units per unit time (e.g. feet per second). The word linear is used because straightening out the arc traveled by the object along the circle results in a line of the same length, so that the usual definition of speed as distance over time can be used. We will usually omit the word average when discussing linear and angular speed here.

Since the length *s* of the arc cut off by a central angle *θ* in a circle of radius *r* is *s* = *r θ*, we see that

so that we get the following relation between linear and angular speed:

An object sweeps out a central angle of (frac<π><3>) radians in 0.5 seconds as it moves along a circle of radius 3m. Find its linear and angular speed over that time period.

**Solution**: Here we have *t* = 0.5 sec, *r* = 3 m, and *θ* = (frac<π><3>) rad. So the angular speed *ω* is

and thus the linear speed *ν* is

$v = ω r = left(frac<2π> <3>rad/sec
ight) (3 m) ⇒ oxed

Note that the units for *ω* are rad/sec and the units of *ν* are m/sec. Recall that radians are actually unitless, which is why in the formula *ν* = *ωr* the radian units disappear.

An object travels a distance of 35 ft in 2.7 seconds as it moves along a circle of radius 2 ft. Find its linear and angular speed over that time period.

**Solution**: Here we have *t* = 2.7 sec, *r* = 2 ft, and *s* = 35 ft. So the linear speed *ν* is

and thus the angular speed *ω* is given by

$v = ω r = 12.96 ft/sec = ω(2 ft) ⇒ oxed<ω =="" 6.48="" rad/sec="">$

An object moves at a constant linear speed of 10 m/sec around a circle of radius 4 m. How large of a central angle does it sweep out in 3.1 seconds?

**Solution**: Here we have *t* = 3.1 sec, *ν* = 10 m/sec, and *r* = 4 m. Thus, the angle *θ* is given by

In many physical applications angular speed is given in *revolutions per minute*, abbreviated as *rpm*. To convert from rpm to, say, radians per second, notice that since there are 2*π* radians in one revolution and 60 seconds in one minute, we can convert *N* rpm to radians per second by “canceling the units” as follows:

This works because all we did was multiply by 1 twice. Converting to other units for angular speed works in a similar way. Going in the opposite direction, say, from rad/sec to rpm, gives:

A gear with an outer radius of *r*_{1} = 5 cm moves in the clockwise direction, causing an interlocking gear with an outer radius of *r*_{2} = 4 cm to move in the counterclockwise direction at an angular speed of *ω*_{2} = 25 rpm. What is the angular speed *ω*_{1} of the larger gear?

**Solution**: Imagine a particle on the outer radius of each gear. After the gears have rotated for a period of time *t* > 0, the circular displacement of each particle will be the same. In other words, *s*_{1} = *s*_{2}, where *s*_{1} and *s*_{2} are the distances traveled by the particles on the gears with radii *r*_{1} and *r*_{2}, respectively. But *s*_{1} = *ν*_{1} t and *s*_{2} = *ν*_{2} t, where ν_{1} and ν_{2} are the linear speeds of the gears with radii r_{1} and r_{2}, respectively. Thus,