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Stick with me on this one, This is purely fictional, and curiosity.

Say we travel to another solar system we know nothing about. We land on another habitable planet. We don't know the mass of any planet around us or the one we landed on. We have no idea the mass of the sun. We also don't know the radius of any spherical body. We also don't know the distance between the planet we landed and the sun or any other planet for that matter.

In summary, we don't know anything.

How do we find the mass of the planet we are on?

How do we find the gravity of the planet?

How do we find the distance between the planet and the sun?

Is time our only friend here? How do we know when a full revolution around the sun has taken place? what if it takes 100 years for a full revolution?

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    $\begingroup$ Physics and math are your friend. Most of these questions are answerable from secondary school physics, and definitely from first-year university physics. Not surprising, since we here on Earth had to figure it all out... $\endgroup$ – Jon Custer Sep 29 '16 at 14:53
  • $\begingroup$ At least for mass and gravity of the planet, I can think of a quick solution. Suppose you have an exact copy of the original 1kg weight, and a spring scale. On Earth, 1kg is attracted by Earth's gravity with a force of roughly 9.81 N. Measuring this force on this uncharted planet may give you, say, 14 N. You can plug this into Newton's gravitational law F=G*m_1*m_2/r^2 with a known distance (say, 1 m) to calculate the mass of the celestial body (up to errors bc of uneven mass distributions). $\endgroup$ – John W. Sep 29 '16 at 14:54
  • $\begingroup$ The only problem you might run into is if the human lifetime were extremely short compared to all the relevant astronomica $\endgroup$ – Lewis Miller Sep 29 '16 at 15:02
  • $\begingroup$ Periods. The everything would look like it was unchanging and all the visual clues we had on earth might be absent. $\endgroup$ – Lewis Miller Sep 29 '16 at 15:05
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    $\begingroup$ This isn't a fictional scenario: our ancestors found themselves on Earth, ignorant of the answers to all these questions, and figured them out. $\endgroup$ – rob Sep 29 '16 at 15:44
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How do we find the mass of the planet we are on?

Did you arrive on a spaceship? If you were in any kind of orbit around the planet, then Kepler's third law enables you to estimate it's mass.

How do we find the gravity of the planet?

If you know the mass of the planet and have an estimate of it's radius, then the gravity can be calculated. If you are on the surface and have a spring balance and a 1 kg mass, then the measured weight is telling you what the gravitational acceleration is.

How do we find the distance between the planet and the sun?

Assuming you have no moon and no other planets in the system, then you could do it by parallax. Measuring the apparent angular position of the sun with respect to the (distant) background stars simultaneously from two points with known separation on the planet.

Is time our only friend here? How do we know when a full revolution around the sun has taken place? what if it takes 100 years for a full revolution?

You can measure the relative position of the sun with respect to the much more distant stars. This would tell you what fraction of an orbit you had completed.

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