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I have always been fascinated to see that with so little knowledge, scientists have succeeded in discovering the planets of the solar system, and have succeeded in defining them by their masses as well as by their orbits, temperatures, and so on.

I would like to try to trace this process myself but I do not know how to do.

My starting postulate is simple, I have at my disposal a simple telescope and DSLR camera.

My goal is to define the solar planets, ie to know their masses, their sizes, their orbits, their revolutions.

How to achieve it? Do you have any leads?

EDIT : Thank you for your awesome answers and sorry for my late reply.

I think that I have to separate this problem into 3 different points :

  • earth's variable : I can find it by reproducing the Greek calculus
  • Sun's variable : with the two-body problem
  • planets' variable : with the Gauss' method & observation

I keep you update about my advancements ;)

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    $\begingroup$ FYI: The #1 purpose of a telescope is to measure angles so that you can determine the celestial coordinates of sky objects at different points in time. Those angles are the starting points for the calculations mentioned in Mark Olson's answer. If your telescope and its mount can not give precise angular measurements, then it is only a toy. $\endgroup$ – Solomon Slow Jul 27 '18 at 3:20
  • $\begingroup$ #2 purpose is to make sky objects look bright. Bigger is better. You can use a relatively large telescope to find objects that are too faint to be seen with a smaller telescope or, with the naked eye. $\endgroup$ – Solomon Slow Jul 27 '18 at 3:28
  • $\begingroup$ Your edit is rather unclear. However, if you have a new question you would do better to post a new question rather than editing this one. $\endgroup$ – James K Aug 15 '18 at 12:52
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This is a fascinating goal, and very difficult. It would make a book on recreating the history of science.

It took many people spending lifetimes collecting data to get anywhere. However, you don't need to rediscover that the planets go around the Sun I would think.

There are some key milestones that need to be reached. For example, you can distances to planet orbits in terms of ratios of a reference distance, like the Earth Sun distance, WITHOUT knowing the actual Earth-Sun distance. You only know that Mars' orbit is x times bigger than Earth-Sun orbit.

The first calculation of Earth-Moon distance was done by the Greeks. Aristarchus or Eratosthenes perhaps. One of them found the diameter of the Earth using shadows of monuments and in wells at noon, and the other estimated the arc of the Earth by Its shadow on the Moon, and thus the size of the Moon.

I think then it goes something like Newton measure the mass of the Earth by measuring g for a dropped object. Then using estimates of the Moon's distance and its orbital period he could determine the constant G in his law of universal gravitation. Then you can calculate the Earth-Sun distance and the rest of the planets fall into place based on their periods.

Now the other challenge is collecting useful data for determining orbits of planets. There are a couple of ways to "determine an orbit from observations". And this is a particularly useful project. Getting a mental picture of what to do can be difficult. But basically, you record the time and the position of a planet in some Earth centered coordinate system, like Right Ascension and Declination. Or relative to the know position of nearby stars.

Now, the orbits are ellipses with the Sun at one focus, so the Earth based system is not much good. You convert all your observations to Sun centered coordinates in the form of lines or vectors that point to the planet. But they point to the Planet from where the Earth was when you took the measurements. These are a set of lines in space and they don't go through the Sun or anything like that. They are just from some point in space (where the Earth was) pointing to the Planet.

C.F. Gauss (I think Laplace improved on this - not sure) showed that you only need three observations to find an orbit because only one ellipse will have points on the three or more lines and have a focus at the Sun. Two points are enough if you have other information like is it getting closer or further and which way is it going (it might be in retrograde so you can't tell).

More measurements taken further apart will increase accuracy and refine the orbit.

Also, the distance to Jupiter or Saturn could be found by watching their moon's orbits and noticing how they stray from prediction with distance due to the speed of light at opposition versus being further away.

So, you can see that the steps are not too daunting if you don't need to wait for a lunar eclipse and take a few other things as known.

I think you could do the temperatures with a small reflecting telescope. It is by detecting the infra-red radiation from the planet. A reflector is used so as not to absorb the IR. The instruments used were a bolometer that responds to heat and a galvanometer. The galvanometer is an incredible precise and sensitive instrument with no electronics.

Search on "calculate orbits from observations" and there are bunch of PDFs from good sources.

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Some parts of this are easier than others!

By observing the positions of the planets among the stars from night to night over an extended period of time, and then applying a lot of not especially complicated -- but quite tedious -- calculation you can determine the planets' orbits around the Sun with good accuracy.

Secondly, with careful observation, you can measure the angular diameter of the planets as observed from Earth. Combining those angular measurements with the orbits you have determined lets you determine the planets' actual sizes. This will be quite accurate in some cases (e.g., Jupiter, where the error should be less than one percent), and much higher in others cases (e.g., Uranus). This is because Jupiter can be as big as 50 arc seconds, while Uranus is never bigger than just over two arc seconds. Since the practical resolution of an amateur telescope is around a half arc second when the seeing is very good, your estimated diameters will be pretty rough for the ice giants.

Measuring the masses of the planets is much harder. The only way I can think of the do it is to continue to measure the planets' positions as accurately as possibly for as long as possible. Applying a lot of tedious arithmetic (much better done on a computer) you can show that their orbits are affected by more than just the Sun's gravity and can make estimates of each planet's mass based on how their gravity perturbs the other planets. Close, massive planets are easier to measure this way than distant, light planets.

Rotation period is easy for planets like Mars and Jupiter which have large disks with easily visible markings and much harder for featureless planets like Venus and hard-to-observe ones like Mercury. (In fact, we really didn't know the rotation periods of either prior to sending probes. Some astronomers got it right, but they did not convince their colleagues.)

Temperature is hard. I don't think you can measure it given your specified instruments, but you can certainly make a rough estimate using the known distance from the Sun.

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    $\begingroup$ The "hard" part that is missing here is calculating the astronomical unit. To do that you'll need a watch, a friend, a ticket to Tahiti, and a time machine to 2117 $\endgroup$ – James K Jul 26 '18 at 23:05

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