# How did Kepler know that the time of Mars' orbit was 687 days ?

How did Kepler knew that the time of Mars' orbit was 687 days ?

It doesn't seem to me a simple task to determine that timing because after 687 days, Earth is at a different place in the solar system and mars seems to be also in a different place in the solar system.

How was that 687 days orbit determined/calculated ? And who the first precisely calculated that period ?

Regards,

Jonathan

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phyzguy

Kepler had access to the observations made by Tycho Brahe. Tycho had observed the positions of the planets over a period of years, to an unprecedented accuracy of about 1 arcminute, which is about the resolution iimit of the human eye (this being before the invention of the telescope).

These observations were more than sufficient to determine the orbital periods of the planets, and allowed Kepler to test various theories and to determine his three laws of planetary motion. Note that to accurately reproduce the positions of the planets, you have to know not just the orbital period, but the fact that the orbits are ellipses (Mars orbit especially is quite far from circular), as well as how fast the planet is moving at different parts of the orbit. This is all spelled out in Kepler's three laws of planetary motion. In my mind, Kepler's deduction of his three laws from the raw observations was an amazing accomplishment.

To answer your specific question, accurately determining Mars orbital period had to wait until we had a sun-centered model of the solar system, which of course was due to Copernicus. Trying to understand the positions of the planets was a problem that occupied the best minds since antiquity. Aristotle's epicyclic model was relatively successful at reproducing the positions of the planets, but not accurate enough to explain Tycho's more accurate observations. Tycho favored an Earth-centered model where the sun revolved around the Earth, and all of the other planets revolved around the sun. Observationally, this is indistinguishable from the Copernican sun-centered model.

Kepler guessed his third law from a great deal of data about planetary motions that had been taken by astronomer Tycho Brahe. Brahe mostly measured angles and relative distances in the sky without the aid of a telescope. Nonetheless, he was the first to collect precise and reliable information about the motion of planets in the solar system. However, he didn't believe in the heliocentric model, so I believe that he wasn't the first to calculate Mars period of revolution. On the other hand, Kepler was on Copernicus side. He knew the parallax of Mars and so its distance from earth, he knew the distance of the earth from the sun (this was first imprecisely measured by the greeks), he knew the angular velocity of earth relative to the sun and of mars relative to the earth, so he could get mars orbital velocity and finally, its period. I am not an expert on this, so take all with some precautions. Hope it helps

Edit: I must correct myself. Kepler third law dates back to 1619, while mars parallax was first measured by Cassini in 1673. Its rotation period instead was measured by Huygens in 1659 who observed the motion of a great dark region on its surface called Syrtis major, or 'great swamp'.

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I read that the orbital period of mars (687 days) was known before Kepler. Kepler seems to have used this information for his own purposes but he is not the one that calculated it. (Maybe I'm mistaken)

But that still leaves me with the question of how these people of that time knew about the 687 days orbital period.

The difficulty seems to be that seen from the earth, Mars at day 0 and Mars at day 687 is not at the same place in the sky. It just seems uneasy, since Mars is at a totally different place in the sky, to conclude that it has just finished its orbit around the sun.

I'll read into Huygens biography to see if I can get that info.

Personally, when I look at mars in the sky, I see a dot but in order to figure out when that dot has exactly done one rotation around the sun seems impossible to my brain. These people were very intelligent. There must be some kind of theory behind that orbital calculation of the period of Mars orbit.

Janus
Staff Emeritus
Gold Member

I read that the orbital period of mars (687 days) was known before Kepler. Kepler seems to have used this information for his own purposes but he is not the one that calculated it. (Maybe I'm mistaken)

But that still leaves me with the question of how these people of that time knew about the 687 days orbital period.

The difficulty seems to be that seen from the earth, Mars at day 0 and Mars at day 687 is not at the same place in the sky. It just seems uneasy, since Mars is at a totally different place in the sky, to conclude that it has just finished its orbit around the sun.

I'll read into Huygens biography to see if I can get that info.

Personally, when I look at mars in the sky, I see a dot but in order to figure out when that dot has exactly done one rotation around the sun seems impossible to my brain. These people were very intelligent. There must be some kind of theory behind that orbital calculation of the period of Mars orbit.
If you track Mars' position against the stars you will see something like this:

The little "loop" occurs when the Earth catches up to and passes Mars as they orbit. If you measure the time that occurs between loops you get what is called the "synodic period" which is the time it takes for Earth and Mars to go from lining up with the Sun to lining up with the Sun again.

There is a relationship between the synodic period and the periods of Earth's and Mars' orbits. It is:

$$\frac{1}{P_{syn}} = \frac{1}{P_E}- \frac{1}{P_M}$$

We know that Earth's orbital period is 1 year and we can measure the Earth-Mars synodic period (779.94 days), so its just a simple matter of solving the above equation for the period of Mars.

phyzguy

These people were in fact very intelligent, and you must remember that it took the best minds of the human race millenia to figure all of this out. You might try reading this:

http://en.wikipedia.org/wiki/Synodic_period

It's a simple matter to measure that the synodic period of Mars is about 780 days. This is the time between successive oppositions of Mars, meaning that Mars is due South at midnight. Given the heliocentric model, it is easy to see, as shown in the Wikipedia article, that 1/Synodic Period = 1/Earth's orbital period - 1/Mars orbital period. So, once th heliocentric model was understood, it all becomes easy.

Thanks, Janus, very informative. I will try to derive the equation above. Calling S the synodic period, t_E the period of earth and t_M the period of mars, clearly t_E+t=S, where t is the interval of time that is needed after a complete revolution of earth to catch up again with mars and theta is corresponding angle on the two orbits (I'll assume that the orbits are just regular circles and the motion is uniform, which were also the same assumptions Copernicus could make). The following geometric relations hold:

t_E/(2pi)=t/theta and t_M/(2pi)=(T_E+t)/theta

We can solve the system for S and we get S= t_M t_E/(t_M-t_E)

PS I still don't get how to write formulas in latex. Can someone please tell me? The usual  seems not to work...

Hi Janus,

I see on your drawing ( http://upload.wikimedia.org/wikipedi..._path_2003.png [Broken] ) that Mars makes a loop. Do you know how long this loop last ? Because it is said that the synodic period of Mars is exactly 779.94 days, this is very very precise. If the loop lasts a few days, how can they come up with such precise measurement? What exactly are they measuring ? The start of the loop, the end of it ?

Also, as for the equation for the period of Mars in regards to the synodic period, I had to convince myself with a drawing because the equation 1/Psyn=1/PE−1/PM (even though it works fine) doesn't help me connect the dots. But after 4 hours in Visio I am now convince ).

This 687 days is a geometric calculation based on the fact that planets orbit around perfect circles which is not the case. I'm wondering how this 687 days is really exact in reality?

Regards,

Jonathan

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Janus
Staff Emeritus
Gold Member

Hi Janus,

I see on your drawing ( http://upload.wikimedia.org/wikipedi..._path_2003.png [Broken] ) that Mars makes a loop. Do you know how long this loop last ? Because it is said that the synodic period of Mars is exactly 779.94 days, this is very very precise. If the loop lasts a few days, how can they come up with such precise measurement? What exactly are they measuring ? The start of the loop, the end of it ?
The easiest point is from where Mars' path crosses itself. Also, the value of 779.94 is the average synodic period. Since, as you note below, neither Mar's or Earth's orbits are perfectly circular, how fast Mar's and the Earth are moving with respect to each other at slightly different speeds from pass to pass, so the actual synodic period varies slighty from this mean value. By measuring the actual synodic period for many circuits, you can arrive at the mean value.
Also, as for the equation for the period of Mars in regards to the synodic period, I had to convince myself with a drawing because the equation 1/Psyn=1/PE−1/PM (even though it works fine) doesn't help me connect the dots. But after 4 hours in Visio I am now convince ).

This 687 days is a geometric calculation based on the fact that planets orbit around perfect circles which is not the case. I'm wondering how this 687 days is really exact in reality?

Regards,

Jonathan
It doesn't matter whether the orbit is perfectly circular or not. The period is based on the average distance from the Sun. IOW, a planet that is elliptical and at its nearest to the Sun is 190,000,000 km distant and at it furthest 210,000,000 km distant will have exactly the same period as a planet with in a perfectly circular orbit of 200,000,000 km.

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I have the same question on this point.

Dear Sir,
I collected the documents from NASA's horizon's system to find that the synodic period of the Earth and Mars during past 10 years is very uneven, which is respectively 806 days, 802 days, 777days, and 767days. If one uses fomula 1/Ps=1/Pe-1/Pm to calculate the orbital period of the Mars (Ps is synodic period, Pe and Pm are assumed to be the orbital period of the Earth and Mars), and will find that the orbital period of the Mars is respectively 668 days, 671days, 689days, and 697 days. This means if the observations last very long, one will obtain a seriers of different synodic period of the Earth and the Mars. So, it is unclear how they use observed synodic period and the orbital period of the Earth to deduce the orbital period of the Mars to be 687days.
Best regards
roufeng

The "687 days", in Kepler's time, was probably an average of hundreds of years of observations. Mars is naked-eye visible, and has been observed since ancient times.
Calculating an exact orbit, from one (or less) orbit's worth of position data, is non-trivial, but there are mathematical methods.

That means different people uses different observations (within different time length) to obtain different orbital period for Mars. So, how do we now accept that the orbital period of Mars is just 687 days. And I even think the real orbital period for Mars is still unknown.
Best wishes!

AlephZero
Homework Helper
IIRC, Kepler had a lot of bother with the data for Mars, compared with the other planets, until he finally guessed that the orbits were ellipses. The orbit of Mars has a significantly higher eccentricity than the other naked-eye planets.

And I even think the real orbital period for Mars is still unknown.
That is nonsense. How do you think we send spacecraft to land on Mars, if we don't know exactly where it is?

The orbits of the four inner planets are known well enough to place them (that is, their center point) within a sphere 1-2 km in radius. The outer planets, not as good.

IIRC, Kepler had a lot of bother with the data for Mars, compared with the other planets, until he finally guessed that the orbits were ellipses. The orbit of Mars has a significantly higher eccentricity than the other naked-eye planets.

That is nonsense. How do you think we send spacecraft to land on Mars, if we don't know exactly where it is?

This is a different thing. We see, the 687 days for Mars is from an average of a series of observations, and based on 687 days, if one adds enough mathematical skill, along with the information of other bodies (Sun, Earth, and so on), he could predict position for Mars in the future. So, send spacefract to Mars by this predicted position does not mean that the real orbtial period of the Mars is just 687 days.
Kind

AlephZero
Homework Helper
Do you understand the difference between the synodic period of an orbit and the sidereal period?

Of course the synodic period changes from one orbit to the next for every planet, because the orbits are elliptical. Your simplified formula 1/Ps=1/Pe-1/Pm is only accurate for circular orbits.

This discussion has gone far from the topic of this thread. Now, a conclusion is reached that the 687 days for Mars is from the average of a series of observations that last very long time. That means that if we select different observation time length, the average could be different. And then, the real orbital period for Mars could be 687 days, or not 687 days, or more or less.

Well of course, if you average something over hundreds of years, it almost never will be that exact value at any particular time. It is like the average air temperature at a certain city on a certain day of the year - it is almost meaningless for practical use, because it's just an average. There will seldom be a March 5th on which the air temperature matches the average. Likewise for orbital elements of the planets - because of perturbations, they will virtually never be that average value.
Is this a new concept to some of you?
1+2+3+4+6+7+8+9=40
average=40/8=5
In our sample size of 8, the value "5" occurred exactly zero times.

TumblingDice
Gold Member
This discussion has gone far from the topic of this thread. Now, a conclusion is reached that the 687 days for Mars is from the average of a series of observations that last very long time. That means that if we select different observation time length, the average could be different. And then, the real orbital period for Mars could be 687 days, or not 687 days, or more or less.
And you think we land spacecraft on Mars by calculating only when we need to know? That might explain a lot...

What surprises me the most is your generalized statements based on observations made hundreds of years ago. "Who cares?" how many years of naked-eye information astronomers had available to them way back when. We have much more accurate measurements today:

Mars orbital period:
686.971 days
1.8808 Julian years
668.5991 sols (Mars days)

D H
Staff Emeritus
This discussion has gone far from the topic of this thread.
No, it hasn't. You first need to understand the difference between synodic and sidereal period. Then you need to understand how an elliptical orbit makes the synodic period vary but not the sidereal period.

"Who cares?" how many years of naked-eye information astronomers had available to them way back when.
Astronomers care. If you use data from a short time frame, and then try to extrapolate it out for thousands of years, your result may be worthless. Yes, observations in ancient times were not very precise because there were no instruments. However certain of them can put some good limits on where, for instance, a planet or Moon was at that time - eclipses, rise/set times from a known location, etc. If your extrapolation does not show an eclipse which was known to have been observed at a particular place on a particular day, then it is no good.

This is important, because our very precise data from spacecraft ranging and radar observations of the planets is of very limited duration. Because of unknowns, for instance perturbations by asteroids, the very precise predictions break down pretty quickly, and any limits we can set by the above methods allow us to make more accurate long-term predictions.

Also, a simple average over a very large sample size is more accurate than over a small sample size, and the observations need not be as precise, which makes ancient observations even more useful.

1 person
I used a different model (hierarchical two-body) to simulate the angle of Sun and Mars (S-O-T) from 9/10/2000 to 8/10/2010, time span is 1 day, 3652 position points are totally selected, and make comparison with NASA's horizon system to find an average of the deviation to be 2.98 degrees, and a maximum of the deviation is 10 degrees. Beyond Kepler's elliptical orbital laws, could you please tell me whether there is other model/why in the past to reach such an approach.
Best regards