Estimating Displacement of Uranus Due to Neptune's Gravity

[jp40684]

I've been scratching my head over this for a while now. Seems pretty simple, but I just can't get anywhere!

So, this is a problem out of pater & Lissauer's Planetary Sciences book (prob 2.10). Starts off by describing how Neptune was discovered by observing anomalies in Uranus's orbit. Here's the question:

Estimate the maximum displacement in the position of Uranus caused by the gravitational effect of Neptune as Uranus catches up and passes this slowly moving planet. Quote answer in km along Uranus's orbital path (and in sec of arc against the sky as observed from Earth.

Assumptions:
U & N are in ciruclar, coplanear orbits (initially, unperturbed)
neglect effects of other planets and affect of Uranus on Neptune.
assume that the potentail energy released as Uranus gets closer to neptune increases URanus's semimajor axis and thus slows Uranus down.
assume that Uranus's average semimajor axis during the interval under consideration is halfway between its semimajor ais a the beginning and the end of the interval

Use the synodic (relative) period of the pair of planets rather than just Uranus's orbital period

This is a crude estimate (ie could integrate along U's path, but this can be solved using simple algebra and conservation laws).

So in the most basic statement I guess it would be: what is delta r for Uranus given a stationary Neptune (what is the change in distance between where it should be with no Neptune, and what it actually is with Neptune tugging on it and expanding it's orbit in the vicinity of Neptune).

ANY help, leads, hints etc would be greatly appreciated!

 

[willworkforfood]

The satellite is orbiting Earth in a fixed elliptical orbit. It must have 3D coordinates to account for whether a telescope on the Earth can see it (but not as part of these equations). The Earth is at/near one focus of this elliptical orbit.

So I'm pretty much looking for something of this form where the equations are in terms of time since 'perihelion' and constants about the ellipse such as a, b, eccentricity, and so on:

rho(t) = ...

theta(t) = ...

phi(t) = ...

Thanks for all your help.

 

[CometDude]

Hello I have a question about calculations of orbits. Mostly my question is about Comet orbits and eccentricity and period. To help myself learn more about orbits I'm witting a simple calculator for myself so I can better grasp it. I'm getting the orbital elements from the MPC website ( I can't post the url as I need 15 posts )

for a elliptical orbit e > 0 and e < 1
While Periodic Comets e (eccentricity)
name eccentricity
17P 0.432668
26P 0.633008
180P 0.357635
P/2006 HR30 0.842973

I see a lot of comets list that have a e > 0 and e < 1 and they are not listed as a periodic comet.
name eccentricity
C/2006 M4 0.999933
C/2007 E2 0.999235
C/2007 T5 0.913398

This is just a few, but shouldn't they be listed as a periodic comet also?

I understand to get the semi major axis a = q/(1 − e)
and then to get the period is $$T=\sqrt{a^3}$$

Comet C/2007 B2 Skiff
e 0.995869
q 2.974897

I get
a =720.13967562334
T = 19325.249433634 years

Does the IAU have a limit on the period time that they count a comet as periodic? I could not find it on their website.

thank you

Tony

 

[RERT]

The plane of the moon's orbit is currently inclined at 5 degrees to the plane of the ecliptic.

However, the plane of the ecliptic 'nods' relative to the invariable plane, according to wikipedia at least, with a period of about 100K years.

It seems reasonable to believe that the plane of the moon's orbit makes a varying angle to the plane of the ecliptic.

Does anyone out there know what the range of values of inclination of the moon's orbital plane to the ecliptic, and what the period of variation is?

Is there ever an epoch when the planes are close enough aligned to give an eclipse every month? If so, what percent change would that give to annual total solar insolation?

 

[poopcaboose]

I have been trying to map the location of the sun in the sky by using three dimensional vector analysis. I can find the rectangular equations of the elliptical orbit with the sun at one of the focal points, however I don't know the function of the angular acceleration of it's orbit. I know the maximum tangential velocity is 30.287E3m/s at perihelion and the minimum tangential velocity is 29.291E3m/s at aphelion but I don't know if the acceleration is a linear function. I would assume it is not linear, or not constant acceleration. I was wondering if anyone knew the angle as a function of time equation?

 

[charon]

I've been struggling with orbit calculations for Mars' moons and satellites. My Phobos and Deimos orbits look correct - orbital velocity and radius seem to be accurate. But satellites with pronounced eccentricity and/or inclination don't seem to work at all.

What I do during each frame is:
1) Calculate a new True Anomaly (v) and use that for orbit angle.
2) Calculate an new orbit radius using r = a(1 - e^2) / (1 + e cos v).
3) Calculate x = r * cos(i) * sin(v);
4) Calculate y = r * sin(i) * -cos(v);
5) Calculate z = r * cos(i) * cos(v);

Orbital elements are from Horizons and/or Celestia. It's distressing because the moons and some satellites look fine and other satellites are way off.

Thanks in advance.

charon

 

[sup3r_n00b]

Hi guys,

Can anyone tell where can I find reference materials about satellite flight dynamics for geostationary orbits? The codes from other satellite flight dynamics software would be also useful. I'm kinda new in this flight dynamics stuff and I need to learn the mathematics of the geostationary orbit specifically those listed below:

1. Orbit Determination
2. Maneuver Effects on the orbit (East-West and North-South Maneuvers)
3. Orbit Propagation and Prediction
4. Prediction of Orbital Events such as the movement of the Sun and Moon
5. Attitude Determination and Control

Like I said I'm new at this stuff so it would be very helpful to me if the mathematical equations were thoroughly explained and were kept as simple as possible.

Thanks and Regards,
sup3r_n00b

 

[nution]

Speaking in terms of fuel, what is a more efficient way of putting something into orbit? Should you go for low orbit and increasing speed to maintain the orbit? Or spending the fuel to get further out, then accelerate to the slower speed needed to orbit at that range with the same object?

Lets say you get to orbital velocity, will the object eventually fall back to Earth if it didnt get the occasional "push"? Or can it theoretically just sustain that orbit provided there is no interaction with anything else?

 

[sozkan]

HelloI am trying to develop a software for short term Orbit Propagation for GEO satellites and I wanted to consult you guys because I am scared that I might be missing some important points.

I am using NAIF SPICE MATLAB Toolkit to get Position of Moon and Sun w.r.t. Earth in inertial frame (SPICE calls it J2000 instead of ICRF). I wrote the relative acceleration of the satellite as described in D.A.Vallado(Fundament. of Astrodynamics) and my results seem to match with the Satellite Tool Kit simulation I ran. I did not use any aberration correction while getting the positions of sun and moon w.r.t. Earth since I was thinking that the time it takes for light to reach Earth does not matter as I thought gravitational effects were instantaneous. After reading some posts in this forum I see that gravity is also transferred with speed of light. Does this mean I need one way light time and stellar aberration correction?

Then I modeled the Earth Gravity Field using the EGM96 coefficients. For the level of accuracy I want I need to calculate up to terms around the order/degree 10. Here I am confused. I have the satellite position w.r.t. Earth written in Inertial Frame. How do I take into account the Precession, Nutation, Polar Motion of Earth? NAIF Supplies a High Accuracy Earth Rotation Model called ITRF93. If I transfer my pos&vel into this reference frame and then to geocentric lat/lon/alt, I believe I will automatically have included these effects?

Some of the Associated Lagrange Polynomials that MatLAB gives have different signs than the ones suggested in Vallado's book. I wrote my own m file for Associated Lagrange function looking at the recursive algorithm in the book. I am looking for methods of verifying my model of Earth Gravity Field. Any ideas what I can use for validation? I can not use LEO satellites since my model does not have drag as it is for GEO.

The gravity potential of Earth gives the direction of Earth gravity vector at any point but other than that, is there any effect acting on the satellites due to Earth's rotation around itself? Of course neglecting the atmosphere or its rotation. My guess is no, I hope I am not wrong :)

If you can elaborate on my four questions above I will be really happy. I will need to include Earth Tides, Ocean Tides, Solar Pressure as well but I am not sure if I can finish all in time. But at least the part I can finish should be reliable and accurate. I am using MatLAB ode45 with low tolerance(1e-11) values and I guess it is sufficient for my purposes.

 

[cph]

Would it seem reasonable that so-called ‘hot jupiters’ are in resonance i.e. have a stable orbit? If one utilized our stellar system as a simulation, with the addition of a hot jupiter, then what would the resonance be? Would it be calculated as non integral? Then if our cold gas giant were discarded, would this then seem to change the resonance; hence the possibility of a stable orbit? Thus might the data set of hot jupiters of approximately 70-100, all have systems with no cold gas giants? Would this then also be consistent with a 3-body scenario, with ejection (or effectively, since thrown into wide orbit?) of a cold gas giant and inward migration of what becomes a hot Jupiter in stable orbit? Hence would one have the prediction of no cold gas giants for any of the hot Jupiter systems?

 

[Adyaphede]

Hello again everyone.

I've got some questions hopefully someone can answer about planetary orbital periods, distances and the like in a fictitious solar system.

As I explained in my previous post I'm currently developing a Space-based game where I programmatically generate a galaxy, stars, solar systems, etc.

Right now I'm randomizing some basic values based on known ranges. This works, but things are disjointed and do not follow certain rules. For example, Kepler's Third Law for a solar system will generate different values for each planet when they should be similar.

Thus this post. I tend to be verbose but I'll try to be as concise as possible.

What I'd like to know is from where should I start so everything fits together. Here are the steps I use to make a solar system right now.

1 - Solar System's mass should be ~.15% more than its central star.
2 - Leftover mass is used to generate planets.
---
-- Repeat the following until we run out of mass (or mass is smaller than X). --
---
3 - Randomize type of planet from one of these: Minor, Rocky, Ice or Gas Giant.
4 - Get a mass for planet based on Min/Max of each type (as gathered from exoplanet library and other).
5 - Randomize a Rotation Period from 1 hour up to 60 days.
6 - Randomize a Radius between Min/Max for given planetary type.
7 - Finally, derive Volume, Density, Gravity, Escape Velocity, etc and last the planet's moons if any, from the generated values above.

Now, this works fine and end up with a variety of planets in a given solar system. The problem here is the whole. From what I've read and understood from Kepler's Third Law (and scaling), orbit distances (among other things) should be scaled (I guess in my case it would be based on the first generated planet since proportionality have to be similar for all planets in the same solar system. Right?)

Kepler's Third Law's equation, 4pi^2/MG, I don't know what to do with it. How can I use it? In what context should I use it? To calculate the next planet's orbital rotation, period, other values? I'm at a complete loss here.

The next problem is with the .15 percent mass of a central body, which is a lot, if the algorithm gives a lot of minor and rocky planets, I end up with sometimes more than 20 planets. I don't personally have a problem with that, but when you look at the last planet's semi-major axis in AU, they are so far out it's unlikely they would still be orbiting the star. Right now, orbit radius (circular for simplicity's sake) for a planet is derived from the previous planet's orbit radius * 1.7. This value, 1.7, was taken from wikipedia for our solar system.

So the question is: Is there a formula to get the extent of a star's gravitational pull which, once passed, would simply make it impossible for a planet to remain? I guess I'm wondering if something like Hill's Sphere exist for stars.

---
I imagine some people would scoff at those questions (I hope not), but I do want to make things as life-like as possible. I have to make concessions since the game is not a galaxy simulator, but if I can harmonize the basic ingredients to make my game as close as possible to real life I will try to do so.

Thank you for your time.

 

[fructivore]

Hi astrophysics people!

I am a programmer trying to teach myself the (very) basics of orbital mechanics for a simulation project. Thanks to previous discussions on the subject here, and some articles on Wikipedia, I think I have figured out most of the pieces.

There is still one thing left, though. Sometimes when I am collecting information about an orbit I have time of periapsis passage (call it Tp) and sometimes I have anomaly (call it m) at a given time (call it Tm).

(The reason why is that sometimes I have orbital data for well-known celestial bodies, like planets, but other times I only know that at Tm, when it was at anomaly m, the satellite entered the orbit in question for the first time.)

So the issue is that, as I understand it, the equations for predicting orbital position (v, r) and speed at a given future time depend on knowing Tp. That seems to mean that if I started with {m, Tm} that I need a way to figure out how long it takes to get from m to periapsis.

So my questions:

Is my understanding correct?

If so, how do you get Tp from {m, Tm}?

If not, what am I missing?Thanks to everyone on this forum -- you have already been incredibly helpful. And I apologize if this is really simple, I just have not been able to wrap my head around it.

 

[solarblast]

I'm looking at some computer code that purports to compute the correction for the Earth's rotation involving two Earth stations (A and B) that are collecting meteor data. The closest reference I can offer that talks about this is by Whipple and Jacchia, Reduction Methods for Photographic Meteor Trails, 1956. It's available on a pdf at <http://www.sil.si.edu/smithsoniancontributions/Astrophysics/pdf_hi/SCAS-0002.pdf>. I'm just trying to get some basic understanding of what's going on. The section involved is "Computation of orbital elements".

The corrected velocity involves V∞ and Vc, magnitudes. and equations (81) and (82). (82) is somewhat familiar from a post I made several days on "Position and velocity vector of point referred to the equinox". V∞ in understandable as explained by the use of (46) and the lower portion of the left column on the following page, 194. I'm not sure what's going on with Vc. Is it derived somehow from (81)?

The second part of a correction is for the zenith attraction, Vcc, which continues after (81) on page 201. (85) Vcc, and the ΔZ correction is shown in (87). I suppose some of what baffles me about Vc and Vcc is that another source calls Vc the observed speed, and Vcc the vector sum of its and the Earth's heliocentric velocities. Meanings do not match up with the Whipple article. This source provides the same equation as (88).

 

[aerrowknows]

Im trying to calculate the orbit of a planet rotating a star after "x" amount of mass has been transferred form the planet to the star. I took our own solar system for example and assumed Earth was the only planet orbiting the sun.

I used the orbit equilibrium equation :

(GM1M2)/ R = M2 (V)2

where m1 is the mass of the sun and m2 is mass of the earth, v is Earth's orbital velocity and r is its orbital radius.

then i added the value x to m1 and subtracted it from m2 ( mass added to the sun and stolen from earth), getting :

(G(M1+x)(M2-x))/ R = (M2-x) (V)2

but V or orbital velocity is simply:

V= [G(M1+x)/R]^1/2

Substituting that back into our equilibrium equation, we get:

(G(M1+x)(M2-x))/ R = (M2-x) ([G(M1+x)/R]^1/2)2

which is simplified to :

G(M1+x)(M2-x)/ R = (M2-x)G(M1+x)/R

As one can see , the terms "R" and "G" can be canceled out from both sides , giving:

(M1+x)(M2-x)= (M2-x)(M1+x)

Which implies that no matter how much mass is transferred from an orbiting body to the body being orbited , the orbital radius WILL NOT change. ONLY the orbital speed would change.

Is this the right conclusion or did I go wrong somewhere ?

 

[ecastro]

Given:
According to my reference (An Introduction to Modern Astrophysics), I should start with this equation:

$r = \frac{a\left(1 - e^2\right)}{1+ e \cos\left(\theta\right)}$

And by using Kepler's Second Law, I should be able to derive the expression for $\mathbf{v}_r$ and $\mathbf{v}_\theta$.

Attempt:

Since $\mathbf{v}_r = \frac{dr}{dt}\mathbf{\hat{r}}$, then,

$\frac{dr}{d\theta}\frac{d\theta}{dt} = \frac{d}{d\theta} \left[\frac{a\left(1 - e^2\right)}{1+ e \cos\left(\theta\right)}\right]\frac{d\theta}{dt} \mathbf{\hat{r}}$, and

$\frac{d}{d\theta} \left[\frac{a\left(1 - e^2\right)}{1+ e \cos\left(\theta\right)}\right] = \frac{ae\sin\left(\theta\right)\left(1 - e^2\right)}{\left[1+ e \cos\left(\theta\right)\right]^{2}}$.

According to Kepler's Second Law,

$\frac{dA}{dt} = \frac{1}{2}r^2\frac{d\theta}{dt} \\ \frac{1}{2}\frac{L}{\mu} = \frac{1}{2}r^2\frac{d\theta}{dt} \\ \frac{L}{\mu r^2} = \frac{d\theta}{dt} \\ \frac{d\theta}{dt} = \frac{L}{\mu r}\frac{1}{r} = \frac{v}{r} = \frac{2\pi}{P}\\$

Thus,

$\mathbf{v}_r = \frac{2\pi ae\sin\left(\theta\right)\left(1 - e^2\right)}{P\left[1+ e \cos\left(\theta\right)\right]^{2}} \mathbf{\hat{r}}.$

The same goes for $\mathbf{v}_\theta$,

$\mathbf{v}_\theta = r \frac{d\theta}{dt} \mathbf{\hat{\theta}} = \frac{2\pi a\left(1 - e^2\right)}{P\left[1+ e \cos\left(\theta\right)\right]} \mathbf{\hat{\theta}}.$

My final answers should only be functions of $P, a, e,$ and $\theta$ only. Please check if my calculations are wrong.

Now, I need to square both and add them up to be able to acquire this,

$v = G\left(m_1 + m_2\right)\left(\frac{2}{r} - \frac{1}{a}\right)$

Squaring and adding the two velocities,

$v_r^2 = \frac{4\pi^2 a^2e^2\sin^2\left(\theta\right)\left(1 - e^2\right)^2}{P^2\left[1+ e \cos\left(\theta\right)\right]^{4}}$

$v_\theta^2 = \frac{4\pi^2 a^2\left(1 - e^2\right)^2}{P^2\left[1+ e \cos\left(\theta\right)\right]^2}$

Their sum should be something like this,

$v_r^2 + v_\theta^2 = \frac{4 \pi^2 a^2 \left(1 - e^2\right)^2}{P^2\left[1+ e \cos\left(\theta\right)\right]^2}\left\{\frac{e^2\sin^2\left(\theta\right)}{\left[1+ e \cos\left(\theta\right)\right]^2} + 1\right\}$

If I multiply $\frac{a}{a}$, I will be able to bring out the gravitational constant $G$,

$\frac{G\left(m_1 + m_2\right)\left(1 - e^2\right)^2}{a\left[1+ e \cos\left(\theta\right)\right]^2} \left\{\frac{e^2\sin^2\left(\theta\right)}{\left[1+ e \cos\left(\theta\right)\right]^2} + 1\right\}$.

This is where I get stuck. I tried combining the two latter terms and having a trigonometric identity of $e^2\sin^2\left(\theta\right) + e^2\cos^2\left(\theta\right) = e^2$, but it seems I don't get anywhere doing it. Could you help me or give me tips on what should I do?

Thank you in advance.

 

[NRotunda]

Is there a formula or other easy method or algorithm for finding stable, periodic orbits in the Circular Restricted Three Body Problem (CR3BP), besides trial and error?
Example: In a two-dimensional x-y plane, the two primaries lie along the x-axis and are not in motion. The third body is in periodic motion around the smaller primary. If I know the initial position of the third body, i.e. 100 km to the left of the smaller primary, and I want to apply an initial velocity in the positive y-direction; what initial velocity should I apply in order for the resulting orbits around the smaller primary to periodic, stable and close together (meaning each successive orbit is the same distance from the primary as the one before). Is there a formula or other method for determining this initial velocity? Currently, I only know to use trial and error, incrementally changing the initial velocity until I find something that works. Thank you!

 

[cptolemy]

Good evening,

This could be one of the questions: if I have a set of orbital elements (mainly inclination, ascending node and perihelion) referred to the equinox and mean ecliptic of J2000, how do I translate them to the current Julian date epoch and ecliptic reference plan so I don't have to make reductions to the coordinates?

My question comes from the fact that I'm using a set of orbital elements referred to J2000, and I get allways a difference of about 11 seconds towards the real results. I assume that after calculating the heliocentric positions from the keplerian elements, I must find the geocentrics ones, and then reduce them to the equinox and ecliptic of date.

After that, I calculate the light time correction, add the nutation and I was suposed to be done.

Any ideas - or experience with orbital elem. calculations?

Kind regards,

C. Ptolemy