How would it be possible to calculate the orbital parameters of a planet from that planet's moon?

Question

My story takes place on a moon in orbit around a Gas giant. The moon has 0 tilt relative to the sun but the gas giant it orbits does have an eccentric orbit around its parent star. (it has an orbit of about 1 year)

So my question is simply, by using astronomical observation would it be possible to calculate the orbital parameters of the gas giant from the moon? There is also a catch in that my characters want to be able to determine this within a relatively short timespan (the situation is pretty dire). So any plan that involves waiting for a for a long period of time won't work.

The whole point of this would be to determine when winter will end and how severe it will be, (they also want to do the same for summer). Thanks in advance for answering I know it's probably a tough question.

Answer 1

As you orbit the gas giant, it will appear to change it’s phase, waxing and waning, over one lunar orbit (let’s call it a “month”), just like Earth’s moon appears to.

Let’s imagine your first observation is of a waxing half-planet. During the observation you noticed that the ANGLE of the shadow-edge (terminator) across the moon, matched exactly the vertical line of the noonday sun.

After one month however, the angle of the sun, relative to that of the planet will have changed, due to the planet’s solar orbit inclination. Since you told us that on the moon, the noonday sun angle will not change throughout the year (a very peculiar, unnatural arrangement- that proves extremely useful), we can use it as a reference to measure the angle of the NEXT waxing half-planet terminator.

Here is an animation of earth doing exactly this over the course of 12 months: https://commons.wikimedia.org/wiki/File:XEphem-sunset-animation.gif#/media/File:XEphem-sunset-animation.gif

If this were a circular orbit, the detected change in angle will allow one to compute the orbital inclination of the planet, assuming one knew how many “months” are in a “year”(planetary orbit).

Computing the eccentricity of the orbital ellipse requires the above information PLUS soe additional observations. The AMOUNT of angular change detected each month in the planet’s terminator will CHANGE, depending on the eccentricity of the orbit. As the planet reaches it’s smallest orbital radius, the angular change will happen faster and faster, as the planet’s orbital velocity increases. Then it will slow after that point, until it reaches it’s furthest orbital radius, where is orbital velocity will be slowest. Kepler's laws would be a good place to get a sense of this part. https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion

Note that after a full planetary orbit, (one year), the planets terminator angle, will return to the original measurement, the same as excatly one year ago.

I have not actually tried figuring out the formula for all this; but I suspect this basic system of deduction would provide the eccentricity information you need (where they are in the elliptical orbit, how many months of winter to expect, etc..) by simply noting the changes in the angle of the planet’s terminator, each month.

This would NOT provide everything about the orbit, like the orbital radius. Nor does it account for things like, how long it takes for the moon's weather to warm and cool.

Answer 2

It's a matter of measurement, and depends on what equipment you have for making measurements. If you're stuck with telescopes and sextants, you might have to watch for a very long time. We had to wait until a transit of Venus to get an accurate measure of our distance to the sun in 1769.

With more advanced technology, you can get you answers much faster. We measured range to terrestrial planets in our solar system with radar. It's conceivable that a large radar could map out the system and establish orbital parameters in a few days, maybe weeks.

With spectrum analysis, you could measure the red/blue shift of the parent star to get your motion toward or away from it.

Also, remember error! All measurements have error. Sometimes you can figure out the error bars on your measurements. The longer you watch and the more measurements you take the more confidence you can have in your conclusions. You might be taking more measurements every day and refining your models.

I don't think being on a moon makes it more difficult. It adds an extra step or two, but it doesn't change the technology or math required.

Answer 3

"The whole point of this would be to determine when winter will end and how severe it will be."

You just need to count days and see when winter comes for several consecutive years. In our history it was done by Ancient Egyptians, thousands of years before word orbit was coined.