Trajectory of Sun's motion

[lidocorc]

In Version 1.5.0 the mass centre of the solar system is the origin of the ecliptical frame. This means the Sun is moving around this centre on a very interesting trajectory. It would be nice to make it visible.

I know, the trajectory is not easy to draw, because it is not closed like an elliptic orbit. Therefore it might be an idea to draw only a few rotations of it. But doesn't the same problem exist for other trajectories, too? Mercury's trajectory for example, shows a perihel rotation which completes in 225'000 years. Up to now Celestia doesn't show this. Both problems are of the same kind, but on different time scales.

[selden]

Part of the Sun's trajectory was drawn by some of the prereleases, corresonding to about a full orbit around the SSB.

Chris was persuaded to turn it off, apparently because it is "confusing" that it is not a closed orbit. Personally, I agree with you: the Sun's orbital path should be visible, perhaps with a selectable number of orbital periods.

[chris]

I want to introduce an orbit plotting feature that will let the user select a time range and reference frame for plotting any orbit.

For trajectories that are well-approximated by ellipses, I'd like to switch to drawing ellipses using the osculating elements. Right now, Celestia chooses a time window starting at the current time and extending for one orbital period. Since the orbits of planets aren't perfectly elliptical, this results in discontinuities at the ends of the time window.

--Chris

[Cham]

Why not draw the orbit with some fade-in and fade-out effect, at the extremities, instead of that discontinuity jump ?

[t00fri]

I want to introduce an orbit plotting feature that will let the user select a time range and reference frame for plotting any orbit.

Good! . Orbits are frame dependent. As we all know e.g. in the rest frame of Earth, the other planetary orbits also look quite "odd" and apparently nonclosed. Think of the strange retrograding movements of some planets... They represented a mystery for centuries and amazingly baroque "recipies" have been invented in the "olden days" to explain these complex movements at the skyglobe. Anyway as we discussed in detail, we should allow for a general framework of plotting orbits in different frames of reference. This is particularly relevant also for multiple star orbits.

F.

[mjoubert]

Great feature!

It will also be usefull to see spacecrafts' trajectories from a ground station.

Mathieu

[Epimetheus]

I want to introduce an orbit plotting feature that will let the user select a time range and reference frame for plotting any orbit.

Good! . Orbits are frame dependent. As we all know e.g. in the rest frame of Earth, the other planetary orbits also look quite "odd" and apparently nonclosed. Think of the strange retrograding movements of some planets... They represented a mystery for centuries and amazingly baroque "recipies" have been invented in the "olden days" to explain these complex movements at the skyglobe. Anyway as we discussed in detail, we should allow for a general framework of plotting orbits in different frames of reference. This is particularly relevant also for multiple star orbits.

F.

Are you referring to the way, say, Mercury appears to go backwards in its orbit as it races past the earth and begins to turn on its leg around the sun due to our frame of reference? That would be a very cool phenomenon to model! Some interesting math involved, eh?

[t00fri]

I want to introduce an orbit plotting feature that will let the user select a time range and reference frame for plotting any orbit.

Good! . Orbits are frame dependent. As we all know e.g. in the rest frame of Earth, the other planetary orbits also look quite "odd" and apparently nonclosed. Think of the strange retrograding movements of some planets... They represented a mystery for centuries and amazingly baroque "recipies" have been invented in the "olden days" to explain these complex movements at the skyglobe. Anyway as we discussed in detail, we should allow for a general framework of plotting orbits in different frames of reference. This is particularly relevant also for multiple star orbits.

F.

Are you referring to the way, say, Mercury appears to go backwards in its orbit as it races past the earth and begins to turn on its leg around the sun due to our frame of reference? That would be a very cool phenomenon to model! Some interesting math involved, eh?

Certainly, the retrograde movement of certain planets in the rest system of Earth is one familiar illustration of the frame dependence of orbits. But all this is of course much more general, as I have pointed out repeatedly in the past.

One way of expressing it, is to note that an orbit is the locus of a certain position vector as time proceeds.

Vectors are NOT invariant under reference frame transformations.! It's effectively as simple as that.

A relativistic extension of this are the familiar "world lines" in 4d-space (t, x,y,z), the locus of relativistically covariant 4-vectors (as function of proper time, i.e. time measured by a single clock between events that occur at the same place as the clock). ...

for the less initiated, here is a Wiki page reference

http://en.wikipedia.org/wiki/World_line
http://en.wikipedia.org/wiki/Lorentz_scalar

Along with the world line, one typically defines the 4-velocity vector, as the tangent vector to the world line (derivative wrto proper time = curve parameter!). The main reason for doing this in 4d rather than 3d in relativistic physics is that 4-vectors are transforming covariantly under relativistic frame transformations, while 3-vectors don't. The length of 4-vectors is frame invariant! World lines of particles/objects at constant speed are called geodesics.

So instead of Chris' 7-dimensional txyz+3d-velocity sampled trajectories (dev list) one would consider in relativistic movements preferrably the pair of 4d-positions and their 4d tangent vector (4-velocity) that satisfy certain simple (relativistic) constraints wrto frame invariance ... So Chris' 7d sample vectors may be very useful for practical non-relativistic space flight problems, for a theoretical physicist they are pretty horribly looking constructs .

..and so on.

F.

[selden]

General relativity does have to be taken into account when they calculate the trajectory of the Messenger probe heading for Mercury. I don't know if that is relevant to Celestia, if one uses JPL's precalculated trajectory files. See http://www.planetary.org/blog/article/00001329/

[t00fri]

General relativity does have to be taken into account when they calculate the trajectory of the Messenger probe heading for Mercury. I don't know if that is relevant to Celestia, if one uses JPL's precalculated trajectory files. See http://www.planetary.org/blog/article/00001329/

In case your post referred to mine,

1) I was not talking about general relativity.
2) the special relativity example was only to illustrate once more the all-important concepts of implementing frame invariant variables into a parametrization of orbits in a general context... I used special relativity to illustrate this, since there I know the results by heart. For non-relativistic movements one needs to invoke analogously variables that are invariant under frame transformations effected by the Galilei group. See my previous discussion at

viewtopic.php?f=9&t=12070&start=3

The question about relativistic corrections for Messenger orbit calculation, addressed in your Planetary Society link, is obviously of academic interest only.

So you misunderstood completely, it seems.

F.

[selden]

All I was trying to point out was that relativity is relevant to spacecraft trajectories. For a targeted flyby, being wrong by 13 seconds and thus 65km along the track (the difference between taking GR into account and using simple Newtonian mechanics for the recent Mercury flyby) isn't entirely academic, since it could cause cameras to be looking in the wrong direction.

[t00fri]

All I was trying to point out was that relativity is relevant to spacecraft trajectories. For a targeted flyby, being wrong by 13 seconds and thus 65km along the track (the difference between taking GR into account and using simple Newtonian mechanics for the recent Mercury flyby) isn't entirely academic, since it could cause cameras to be looking in the wrong direction.

Indeed you are right

"The concluding summary, of course, is that general relativity is definitely required for accurate navigation of MESSENGER."

Well I didn't read to the end, since spacecraft flight dynamics doesn't exite me all that much... Yet admittedly, this sort of evidence for GR in precision flight path calculations may be taken as another fascinating experimental support for GR a la Einstein.

Still GR is hardly relevant for Celestia in it's present context, I suppose.

F.

[selden]

Still GR is hardly relevant for Celestia in it's present context, I suppose.

Not directly, just in trajectory ephemeris files.
Maybe someday, though.
One can hope...

[chris]

General relativity does have to be taken into account when they calculate the trajectory of the Messenger probe heading for Mercury. I don't know if that is relevant to Celestia, if one uses JPL's precalculated trajectory files. See http://www.planetary.org/blog/article/00001329/

It's not relevant, for exactly the reason you specified: the relativistic effects are already accounted for in the trajectory from HORIZONS. If Celestia was doing trajectory propagation itself, then we'd have to take relativity into account. That Planetary Society article about the actual magnitudes of relativistic effects in spaceflight was a good read.

As for Epimetheus' comment:

Are you referring to the way, say, Mercury appears to go backwards in its orbit as it races past the earth and begins to turn on its leg around the sun due to our frame of reference? That would be a very cool phenomenon to model! Some interesting math involved, eh?

The apparent retrograde motion of Mercury is already modeled in Celestia--it's sort of an automatic result of modeling the Solar System in three dimensions.

--Chris

[t00fri]

Are you referring to the way, say, Mercury appears to go backwards in its orbit as it races past the earth and begins to turn on its leg around the sun due to our frame of reference? That would be a very cool phenomenon to model! Some interesting math involved, eh?

The apparent retrograde motion of Mercury is already modeled in Celestia--it's sort of an automatic result of modeling the Solar System in three dimensions.

--Chris

Wait a second:

we are talking about plotting the retrograde ORBIT in the Earth's rest system, NOT the modelling of the movement of Mercury itself. If one presently looks at Mercury's orbit from Earth during it's retrograde movement it really looks funny... The Mercury orbit in the sky is drawn around the sun and does NOT show these little spiraling loops that are so characteristic for retrograde motion.

Incidentally, we should have an option to advance time in units of 1 day!
(or any other sensible time diffeence)

F.

[Epimetheus]

Are you referring to the way, say, Mercury appears to go backwards in its orbit as it races past the earth and begins to turn on its leg around the sun due to our frame of reference? That would be a very cool phenomenon to model! Some interesting math involved, eh?

The apparent retrograde motion of Mercury is already modeled in Celestia--it's sort of an automatic result of modeling the Solar System in three dimensions.

--Chris

Wait a second:

we are talking about plotting the retrograde ORBIT in the Earth's rest system, NOT the modelling of the movement of Mercury itself. If one presently looks at Mercury's orbit from Earth during it's retrograde movement it really looks funny... The Mercury orbit in the sky is drawn around the sun and does NOT show these little spiraling loops that are so characteristic for retrograde motion.

Incidentally, we should have an option to advance time in units of 1 day!
(or any other sensible time diffeence)

F.

Huh oh!!! I'm being quoted by scientists! Is that a good thing?!?

BTW, Fridger, the article regarding world lines was fascinating. Thanks for the link. I haven't yet read about the Lorentz effect -- still trying to wrap my head around the math in the world lines article! So from what I understand the extra element is time element, letting t represent time, so, in 4D vectors we have (x,y,z, t). Does this apply to both positional and velocity vectors?, or is there some type of inverse of variant relationship between position and velocity vectors in 4D space that does not occur in 3D space? You may have already answered this question. My apologies if you have.

For those of you who are trying to wrap your head around this subject, as I am, you might want to follow this link and take a look at a 4D hyper-dimensional cube animation. To quote the artist,

In 3-dimensional rotation, rotation usually occurs around a linear axis but 4-dimensional rotation occurs around a planar axis.

http://dovsherman.deviantart.com/art/Hypercube-Rotations-25609383

You really get a visual feel for the reference frame transformations that Fridger mentions.

[Epimetheus]

More interesting examples of tesseracts (4-dimensional hypercubes).

Great animation
http://chaos-syndrome.deviantart.com/art/Tesseract-13963126

Flash animation of a tesseract. Allows manual spinning of the cube on its planar axis. The z-w plane axis is particularly interesting.
http://dovsherman.deviantart.com/art/Hypercube-Manual-Rotations-25706685

[t00fri]

Epimetheus,

While the rotating hypercube even exists as a well-known screensaver (under Linux => xscreensaver), the 4d space of relativistic physics is different in an essential aspect:

It's the metric! The 4d space is called a Minkowski space while that hypercube lives in a so-called 4d Euclidean space.

Without getting too mathematical, the metric determines e.g. how the length^2 of a vector is calculated in terms of it's components.

Let X=(x0,x1,x2,x3)

be a general 4-vector, then in Euclidean space we calculate the length^2 as

X.X = X^2 = x0^2 + x1^2 + x2^2 + x3^2,

i.e as the straightforward generalization of what we know and love in 3 dimensions. For the Minkowski space of relativity, there are some crucial '-' signs involved, such that the "length^2" of a vector is NOT necessarily positive!!!

X.X = X^2 = x0^2 - x1^2 - x2^2 - x3^2

(Note: sometimes you see another convention, which is /physically/ equivalent: X.X = - x0^2 + x1^2 + x2^2 + x3^2)

So from what I understand the extra element is time element, letting t represent time, so, in 4D vectors we have (x,y,z, t).

Almost...Time and space don't have the same physical dimensions, hence cannot be the components of one vector . What is lacking is c, the speed of light: c*t = x0 is the first, extra component (remember [velocity] = [length/time]), and x1=x, x2=y, x3=z.

So a 4d positional vector has the components

X= (c*t, x,y,z)

The amazing fact is, that with the Minkowski metric, the length^2 of any such vector is the SAME in EVERY relativistic reference frame, it's an invariant under Lorentz frame transformations as we say.

So we can evaluate the length^2 of X in ANY frame that is CONVENIENT, without changing the underlying physics!!

Consider the components of the vector X in 2 such frames, this means

X.X=X^2 = (c*t)^2 - x^2 -y^2 -z^2 = (c*t')^2 - x'^2 -y'^2 - z'^2

Solving that equation properly, will immediately give you the famous length contraction and time dilatation of special relativity!

The proper time tau that parametrizes the world lines is nothing but the time in a special frame: t'=tau, x'=y'=z'=0, such that

X^2 = (c*t)^2 - x^2 - y^2 -z^2 = (c*tau)^2

We can immediately parametrize the components of the vector X in some frame as function of the proper time such that this equation is automatically satisfied. Would you know how this is done??
(Hint use the hyperbolic sinus/cosinus, sinh(tau), cosh(tau), satisfying sinh^2 - cosh^2 =1 )

The task is the complete analog of the well-known parametrization of a 3d unit vector in terms of cos and sin with cos^2 + sin^2 =1, namely

Code: Select all

X_3d = (x1,x2,x3) = (sin(tau), sin(phi)*cos(tau), cos(phi)*cos(tau))
then

X_3d^2 = sin^2(tau) + (cos^2(phi) + sin^2(phi))* cos^2(tau) = sin^2(tau) + cos^2(tau) = 1


[You see, while a general 3d-vector depends on 3 components x1,x2,x3, the constraint X_3d^2=1 eliminates one. The above trigonometric parametrization makes this manifest in terms of only tau and phi!]

Since a speed vector is generally the derivative of the position vector, we now define a 4d-velocity vector, the /tangent vector/, by forming the derivative wrto the /proper time tau/

V = dX/dtau = (c*dt/dtau, dx/dtau,dy/dtau,dz/dtau)

Since the 4d velocity is once more a proper 4-vector, again, it's length^2

V.V = V^2

is frame independent!

Moreover one easily finds that the vector product

V.X

is also an invariant under Lorentz frame transformations.

+++++++++++++++++++
Hence, after identifying all invariants under Lorentz frame transformations, the world line orbits can be parametrized easily in terms of invariants and covariants such that one can read off what part of the orbit is frame independent and what is frame dependent. The straight analog of this proven procedure is what I am proposing since long for making the frame dependence of the non-relativistic orbits of Celestia manifest!
+++++++++++++++++++

F.

[ajtribick]

Great animation
http://chaos-syndrome.deviantart.com/art/Tesseract-13963126

Thank you

[Epimetheus]

Epimetheus,

Consider the components of the vector X in 2 such frames, this means

X.X=X^2 = (c*t)^2 - x^2 -y^2 -z^2 = (c*t')^2 - x'^2 -y'^2 - z'^2

Forgive my budding mathematical knowledge ( intermediate algebra and intro to discrete math), Dr. Shrempp, but in the above equation are we using ' as logical not operator? The right-hand side is the inverse?

Solving that equation properly, will immediately give you the famous length contraction and time dilatation of special relativity!

The proper time tau that parametrizes the world lines is nothing but the time in a special frame: t'=tau, x'=y'=z'=0, such that

X^2 = (c*t)^2 - x^2 - y^2 -z^2 = (c*tau)^2

When you speak of proper time tau, does the tau relate to the Uncertainty Principle for Proper Time and Mass?

We can immediately parametrize the components of the vector X in some frame as function of the proper time such that this equation is automatically satisfied. Would you know how this is done??

LOL! Yeah, maybe after sevens of study for my PhD! But I'll shut and listen good sir.

(Hint use the hyperbolic sinus/cosinus, sinh(tau), cosh(tau), satisfying sinh^2 - cosh^2 =1 )

The task is the complete analog of the well-known parametrization of a 3d unit vector in terms of cos and sin with cos^2 + sin^2 =1, namely

Code: Select all

X_3d = (x1,x2,x3) = (sin(tau), sin(phi)*cos(tau), cos(phi)*cos(tau))
then

X_3d^2 = sin^2(tau) + (cos^2(phi) + sin^2(phi))* cos^2(tau) = sin^2(tau) + cos^2(tau) = 1


[You see, while a general 3d-vector depends on 3 components x1,x2,x3, the constraint X_3d^2=1 eliminates one. The above trigonometric parametrization makes this manifest in terms of only tau and phi!]

Since a speed vector is generally the derivative of the position vector, we now define a 4d-velocity vector, the /tangent vector/, by forming the derivative wrto the /proper time tau/

V = dX/dtau = (c*dt/dtau, dx/dtau,dy/dtau,dz/dtau)
F.

Hmmm, I'm not sure I understand X_3d^2 and the use of the underscore. Is X_3 a constant, or is it X_3d? This is fascinating stuff. I can't wait to get to Calculus level in math so I can converse with you as an equal. Amazing stuff, Fridger!

I was reading some of your publications, or trying to!, and was also amazed by the work and research you are doing. I tried to understand what an Instantaton was but my head exploded! Is your research focusing on better understanding the strong interaction and the behavior of hadrons, quarks and gluons? Do I understand correctly that free quark searches have failed so far because the quarks were not where they were expected to be? Is this whole business with worldlines and Lorentz transformations a way to explain where these little quarks are within their confinement by the strong interaction? Conversely, I read that quarks are experiencing asymptotic freedom. Does this imply that two quarks would move closer toward unity as one tends toward infinity? Which brings me to something I've long about. What do mathematicians call this problem. I'm sure it's an elementary problem for PhDs.

When you take a real number like N to the negative n, I think that's right, where .99 -> .999 -> .9999 -> .99999 -> etc.
We get closer and closer to one but we never ever get there. The closer we get the farther away we are. What is this principle called in mathematics?

Thank you, Doctor, for your educational replies!


Best Regards,

Epimetheus