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Thread: Orbit Calculation

  1. #1
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    Orbit Calculation

    Yesterday, I came across this problem and managed to solve it all, which stated as follows:

    - Consider a planet lying at an incredible distance of 710 AU from its giant host star, which has a mass 8 times the Sun’s mass. Assuming a circular orbit, calculate each of the following:
    a) The orbital period of the planet in years. (Earth)
    b) The orbital velocity.
    c) The time required for the light of the host star to reach the planet in days. (Earth)
    And I got the results:

    a) T = approx. 6,900 years, b) V = 3 km/second, and c) 4 days
    My question deals with item b – the orbital speed.

    Suppose this same planet was located even further from its parent star. Since the further the planet, the slower the orbital speed, would the planet end up reaching a point where the star’s gravitational force had zero influence on it? (i.e. a point where the planet’s orbital speed reached zero)

    Here’s what I think: Even at its awesome original distance (710 AU), the planet still orbits at 3 km/sec. But this also indicates that at a slightly greater distance the planet would lose all gravitational attraction from the star; it would then be considered out of the entire star system, just a deep-space object.

    Is my layman conclusion correct? Or am I misunderstanding something, as usual? Also, while we’re at it, is this same phenomena, by any chance, possible? I mean, deep-space objects. Or, in other words, planets that have somehow totally escaped their parent stars’ gravitational pull and spent the rest of their lives standing alone and still in interstellar space. And would such a planet still spin on its axis? Why or why not?

    That’s all. Sorry if it’s too much; I know I could have summarized everything in just a couple of lines, if not less, but I thought a few further details should work better. Even if my conclusion is perfectly correct, I would love to see some more elaboration from you. But please try to put things simply and use easy words for me to understand better.

    Thanks in advance.

    Fiery Phoenix,
    Last edited by Fiery Phoenix; 2009-Jan-31 at 06:56 AM.
    “Of all the sciences cultivated by mankind, Astronomy is acknowledged to be, and undoubtedly is, the most sublime, the most interesting, and the most useful. For, by knowledge derived from this science, not only the bulk of the Earth is discovered, but our very faculties are enlarged with the grandeur of the ideas it conveys, our minds exalted above their low contracted prejudices.” - James Ferguson

  2. #2
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    Quote Originally Posted by Fiery Phoenix View Post
    Is my layman conclusion correct? Or am I misunderstanding something, as usual?
    Impressive thinking about it.
    But; my take is that it doesn't matter how far away the object is, but how close other objects are to disturb the orbit.

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    Quote Originally Posted by NEOWatcher View Post
    Impressive thinking about it.
    But; my take is that it doesn't matter how far away the object is, but how close other objects are to disturb the orbit.
    Could you please explain that further?

    Thank you.
    “Of all the sciences cultivated by mankind, Astronomy is acknowledged to be, and undoubtedly is, the most sublime, the most interesting, and the most useful. For, by knowledge derived from this science, not only the bulk of the Earth is discovered, but our very faculties are enlarged with the grandeur of the ideas it conveys, our minds exalted above their low contracted prejudices.” - James Ferguson

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    Suppose this same planet was located even further from its parent star. Since the further the planet, the slower the orbital speed, would the planet end up reaching a point where the star’s gravitational force had zero influence on it? (i.e. a point where the planet’s orbital speed reached zero)
    This might not be exactly what your after, but I was looking about for some answers and came across this wiki article, http://en.wikipedia.org/wiki/Oort_cloud, which mentions:

    The point at which the Sun's gravity concedes its influence to the galactic tide is called the tidal truncation radius. It lies at a radius of 100,000 to 200,000 AU, and marks the outer boundary of the Oort cloud.

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    Quote Originally Posted by NEOWatcher View Post
    But; my take is that it doesn't matter how far away the object is, but how close other objects are to disturb the orbit.
    Yes, the galactic tide that Tzarkoth mentioned would always be there to disrupt the orbit at large enough distances from the star, but even inside that distance, passing stars that are not normally there (so not part of the galactic tide, which is a combined effect from very distant masses) would still occasionally arrive to disrupt the orbit. That kind of interaction should mean that a planet at that distance should not be able to maintain a circular orbit-- it should have to either escape the solar system or fall into an elliptical orbit that spends most of its time much closer to the star. I think the latter is what tends to happen, and certainly the comets in the Oort cloud tend to have noncircular orbits. (Note that the OP problem needed to specify a circular orbit to be doable, no doubt that was the intention.)

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    Quote Originally Posted by Fiery Phoenix View Post
    Yesterday, I came across this problem and managed to solve it all, which stated as follows:



    And I got the results:



    My question deals with item b – the orbital speed. Lying at such an inconceivable distance from its star, the planet travels around the star at 0.03 m/second; which, in fact, is equivalent to a mere 3 centimeters per second.
    I calculated the planet's orbital speed as about 3 kilometers per second. How did yours come out a factor of 100,000 less?

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    Quote Originally Posted by Fiery Phoenix View Post
    Could you please explain that further?
    Hopefully, the latter posts did. They are more eloquant than my post.

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    Wink

    Fiery Phoenix, here are the answers I get to three significant digits when considering the planet to have the same mass as Earth and with a circular orbit.

    a) 6690 sidereal years
    b) 3.16 km/sec
    c) 4.10 days

    Note that my (and Hornblower's) answer to “b” is 100,000 times greater than yours. The circumference of the orbit is 710 AU x 2π = 4460 AU. 4460 AU / 6690 yrs = 0.667 AU/yr = 3.16 km/sec.

    If this were a two-body problem, then the star’s gravitational influence would be proportional to the inverse square of the distance. Thus the planet’s orbital velocity would slow down with greater distance but never reach zero. However, as others note, in reality at great distances the effect of the star’s gravity would be washed out by that of other large bodies.
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    NEOWatcher said:
    Impressive thinking about it. But; my take is that it doesn't matter how far away the object is, but how close other objects are to disturb the orbit.
    Fiery Phoenix said:
    Could you please explain that further? Thank you.
    There's no limit on gravity. As you walk around on the Earth, the dominant gravitational influence on you is obviously the Earth. The Sun and Moon also have small but noticeable influences on you. But so do the other planets in the Solar System - they're just a lot smaller.

    I assume you had formulas to calculate the answers you got (even though others seem to think they're wrong). Why not plug some larger distances into those formulas, and see what the result is. Can you plug in a distance between the sun and planet that is so large that the orbital velocity is ever zero? Or does it just keep getting smaller? Now imagine a distance between the star and planet which is, say, 10 light years. How fast will the planet be orbiting now? And what do you think might happen if there was another star situated between the two, five light years from each.

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    Quote Originally Posted by Fiery Phoenix View Post
    Now to the question.... would the planet end up reaching a point where the star’s gravitational force had zero influence on it?
    Of course not. As others have noted, gravity never goes to zero. Theoretically, the 'test body' could be moving arbitrarily slowly if its orbital radius was large enough. But as others have also implied, if there is a single body of mass anywhere else in the universe, the orbiting body wouldn't be able to go arbitrarily slowly and maintain its orbit.
    Everyone is entitled to his own opinion, but not his own facts.

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    (Renamed thread to be a bit more specific, and useful to future searchers.)
    Measure once, cut twice. Practice makes perfect.

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    Quote Originally Posted by Fiery Phoenix View Post
    Here’s what I think: Even at its awesome original distance (710 AU), the planet still orbits at 3 centimeters/sec. But this also indicates that at a slightly greater distance the planet would lose all gravitational attraction from the star; it would then be considered out of the entire star system, just a deep-space object.
    The orbital velocity approaches zero only as the distance from it's star approaches infinity.

    On a more practical note, with a bit more distance, the gravitation effects of other stars are enough to knock it out of it's orbit completely.

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    Oops! Guys, I'm so sorry for the wrong calculation. I just recalculated the orbital speed and found it to be about 3 km/sec -- just like stated above. I probably forgot to convert the orbital distance to meters at first.

    Thanks for the correction. I'll be back for another reply later.
    “Of all the sciences cultivated by mankind, Astronomy is acknowledged to be, and undoubtedly is, the most sublime, the most interesting, and the most useful. For, by knowledge derived from this science, not only the bulk of the Earth is discovered, but our very faculties are enlarged with the grandeur of the ideas it conveys, our minds exalted above their low contracted prejudices.” - James Ferguson

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    Quote Originally Posted by Ken G View Post
    Yes, the galactic tide that Tzarkoth mentioned would always be there to disrupt the orbit at large enough distances from the star, but even inside that distance, passing stars that are not normally there (so not part of the galactic tide, which is a combined effect from very distant masses) would still occasionally arrive to disrupt the orbit. That kind of interaction should mean that a planet at that distance should not be able to maintain a circular orbit-- it should have to either escape the solar system or fall into an elliptical orbit that spends most of its time much closer to the star. I think the latter is what tends to happen, and certainly the comets in the Oort cloud tend to have noncircular orbits. (Note that the OP problem needed to specify a circular orbit to be doable, no doubt that was the intention.)
    The problem actually mentions "assume a circular orbit". I just forgot to write it down -- original post edited.

    So, basically, other stars/massive objects would keep disturbing the orbit of the planet, right? But why wouldn't the planet's orbital speed reach zero at some point? Is it also due to the gravity of the other stars here and there?

    I think this implies that deep-space objects don't exist, or not? Let's assume a planet that's somehow escaped from its star system, how would the gravity of the stars around it affect it, considering that the planet was no longer a "child" of its original star?
    “Of all the sciences cultivated by mankind, Astronomy is acknowledged to be, and undoubtedly is, the most sublime, the most interesting, and the most useful. For, by knowledge derived from this science, not only the bulk of the Earth is discovered, but our very faculties are enlarged with the grandeur of the ideas it conveys, our minds exalted above their low contracted prejudices.” - James Ferguson

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    Quote Originally Posted by Fiery Phoenix View Post

    But why wouldn't the planet's orbital speed reach zero at some point?
    All objects in the universe with mass gravitationally attract all other objects in the universe with mass. The force decreases with the inverse square of the distance. In other words if the Earth were to be moved a thousand times further from the Sun (1000 AU), the gravitational force between it and the Sun would become a millionth (1 ∕ 1000˛) as much. No matter what finite distance you square for the denominator, the force would never be zero and the theoretical circular orbital speed in a two-body system would never be zero.

    As we have been stating, the case is simple with a two-body system. But in the real universe there are more than two bodies. When the original two bodies become widely separated, the gravitational effects from other bodies overwhelm the original relationship. So even though the two bodies still attract each other, the gravitation from the other bodies can force the original pair to separate.
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    3167 m/s

    Quote Originally Posted by Fiery Phoenix View Post
    ...

    Suppose this same planet was located even further from its parent star. Since the further the planet, the slower the orbital speed, would the planet end up reaching a point where the star’s gravitational force had zero influence on it? (i.e. a point where the planet’s orbital speed reached zero)

    Here’s what I think: Even at its awesome original distance (710 AU), the planet still orbits at 3 km/sec. But this also indicates that at a slightly greater distance the planet would lose all gravitational attraction from the star; it would then be considered out of the entire star system, just a deep-space object.

    Is my layman conclusion correct? Or am I misunderstanding something, as usual? Also, while we’re at it, is this same phenomena, by any chance, possible? I mean, deep-space objects. Or, in other words, planets that have somehow totally escaped their parent stars’ gravitational pull and spent the rest of their lives standing alone and still in interstellar space. And would such a planet still spin on its axis? Why or why not?
    ...

    Fiery Phoenix,

    Hi Fiery Phoenix,

    your result appears to be correct. I get 3167 m/sec as a result. However, such a large mass has alot of attraction hence it has to be "balanced" by centrifugal force (I know - I know there isn't such thing as centrifugal force but it makes such calculations easier!).
    In theory nothing ever leaves a gravitational field but an object can be so far away from a mass that it's influence is nil compared to the next passing dust peck. Hence, by all practical definitions, this mass is in a gravity free zone.
    And yes, there might be "rogue" planets wandering between the stars until, by chance, they're captured.
    This would make for an interesting Science Fiction novel "When worlds collide" by John Campbell (I believe) comes to my mind.

    Extracelestial

  17. #17
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    Quote Originally Posted by Fiery Phoenix View Post
    So, basically, other stars/massive objects would keep disturbing the orbit of the planet, right?
    Yes. We can fairly accurately calculate the orbit of the Earth around the Sun, but the other planets do perturb it. However, the influence of, say, Pluto would be almost undetectably small. By contrast, Mars's orbit is quite noticeably affected by Jupiter, and NASA engineers need to take Jupiter's influence into account when calculating a spacecraft's trajectory from the Earth to Mars.

    But why wouldn't the planet's orbital speed reach zero at some point? Is it also due to the gravity of the other stars here and there?
    No it isn't. At least not directly. Here you have to separate out the pure answer (assuming the star and planet are the only two objects in the universe) and reality (the fact that there are other objects in the universe which affect matters). When looking at the pure answer, have a look at the formula you used to calculate the speed of the orbiting planet. No matter what distance from the star you plug in, you'll never get an answer of zero. But when looking at the influence of other stars and planets, their effects will usually be small enough that the orbiting planet won't ever achieve a velocity of zero. Of course, if we postulate a star diving into this solar system, then its gravitational effect will be so large that all bets are off.

    I think this implies that deep-space objects don't exist, or not?
    It depends on how far out you mean when you say "deep-space". Remember, the nearest star to our Sun is about 284 000 AU away, so an orbital radius of 10 000 AU wouldn't be out of the question. Much beyond that and the collective influence of nearby stars might make things a bit tricky for a permanent circular orbit.

    Let's assume a planet that's somehow escaped from its star system, how would the gravity of the stars around it affect it, considering that the planet was no longer a "child" of its original star?
    I imagine it'll have a fairly chaotic orbit, though this would be slow motion chaos. I suspect it might be likely to spend a lot of its time wandering a long way from stars, simply becase they're small targets compared to the vastness of space. An interesting comparison might be the third stage of the Apollo 12 Saturn V rocket, which has alternated between a solar orbit and a tenuous Earth orbit a couple of times since 1969.

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