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alexb
Charter Member
1592 posts 
Jul1305, 11:27 AM (EST) 

"Cassini's determination of distance to Sun"

In 1672 a chap named Cassini found the distance from the earth to the sun. He used a scheme which required that he send a friend to South America while he stayed in Paris, and both of them looked at Mars on the same day just as it was rising (as I understand it). The details are given at https://www.astronomyforbeginners.com/astronomy/howknow.php#distsun Each observed the star that was in the background behind Mars. Then they got together again, and by knowing the identity of the two stars, were able to figure the angle L (in the attached diagram). And from L and the radius of the Earth, they could find the distance to Mars (from R/D=sin(L/2)). Then from Kepler's third law they could find the distance to the sun. The trouble is, I am unable to see how to compute L from the information (their angular displacement in the sky, presumably) on the two stars. Can you see how to do it? I suspect it's obvious, and it's just my blindsightnedness that keeps me from understanding. Monty


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alexb
Charter Member
1592 posts 
Jul1405, 07:02 PM (EST) 

1. "RE: Cassini's determination of distance to Sun"
In response to message #0

The essential thing to remember is that at the night Cassini took measurements and determined the distance to Mars, Sun, Earth and Mars were collinear and Mars was farther from the Sun, so that, at that time, dist(Sun, Mars) = dist(Sun, Earth) + dist(Earth, Mars) 1.542 AU = 1 AU + d, where d is the measured dist(Earth, Mars). Thus AU = d/.542 How did they find d? They assumed to make measurements on the opposite locations on earth. The two points of measurements and Mars formed an isosceles triangle whose base was known diameter of Earth. What they used their measurements for was to determined the angle at the apex (Mars). (The formula actually uses tan() and sin()). How did the find the angle? As a guess, the star in question may have been assumed to be infinitely far, so that both lines from the points of observation to the star may be thought as being perpendicular to the diameter of Earth between the points of measurement. (Far as I understand, their result was of by 7%. Perhaps, because of an assumption like that.) The two fellows measured the angles between the lines to Mars and to the star. If you draw a diagram you'll see that the angle we are after is the sum of the two measured angles. 
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mr_homm
Member since May2205

Jul1505, 11:08 AM (EST) 

2. "RE: Cassini's determination of distance to Sun"
In response to message #1

>The essential thing to remember is that at the night Cassini >took measurements and determined the distance to Mars, Sun, >Earth and Mars were collinear and Mars was farther from the >Sun, so that, at that time, > >dist(Sun, Mars) = dist(Sun, Earth) + dist(Earth, Mars) > >1.542 AU = 1 AU + d, > >where d is the measured dist(Earth, Mars). Thus > >AU = d/.542 I hope it will not be amiss to bring in some nonmathematical cosiderations here. Certainly the calculation is easiest in the case where Mars is in opposition to the sun, but since the orbits of the planets had been accurately known since Kepler (except for the allimportant overall scale, which Cassini found from this experiment), it would have been an easy exercise for any astronomer of the day to calculate the distance to the sun even if the observation was taken at some other point in Mars's orbit. The real reason for making the measurement at opposition is that Mars is closest to Earth then, and parallax measurements are more accurate for nearer objects. As the object is further away, the angle between lines of sight becoms smaller, and is more difficult to measure. Additionally, since Mars is then on the opposite side of the sky from the sun, the viewing is particularly good (Mars is in the part of the sky with the least scattered sunlight). > >How did they find d? > >They assumed to make measurements on the opposite locations >on earth. The two points of measurements and Mars formed an >isosceles triangle whose base was known diameter of Earth. >What they used their measurements for was to determined the >angle at the apex (Mars). (The formula actually uses tan() >and sin()). > The article at Astronomy for Beginners has somewhat oversimplified the situation. First, the point opposite Paris on the Earth's surface is a section of ocean just southeast of New Zealand, which is farther west than South America by the entire width of the Pacific Ocean. Of course the parallax calculation can still be done, one must simply use the actual distance from Paris to the other observation point somewhere in South America, rather than the diameter of the Earth. This does not change the essential nature of the parallax calculation, merely one of the details. A related interesting question is, why could Cassini NOT have used two exactly antipodal points for his observations? After all, that would have provided the longest baseline for parallax. The answer is simple geometry: At each point on the Earth's surface, the portion of space that is visible is everything above a tangent plane to the earth at that point. On antipodal points, these regions do not overlap, since the two planes are parallel. Therefore, the two observers could not see Mars at the same time. Time is important here, since Mars moves against the background stars as it orbits the sun (and also because our line of sight changes as Earth orbits the sun). If the observations are not made at exactly the same time, this motion will carry Mars to a slightly different position in the sky, changing the observed angle, and hence ruining the estimate of Mars's distance. Most points on the Earth's surface would work. If I try to find the size of the spot where you can definitely not see Mars when Cassini in Paris can see it, I get a circle of radius about 3.27km. Except for this very small spot, the observations work. I found the distance by considering the two horizon planes to intersect at Mars. In fact, measuring the radius of this "blind spot" would be another way of getting the parallax angle and hence the distance to Mars. It would not be practical of course, since the observer would have to move around very quickly trying to find the edges of the blind spot simultaneously, or we would have to use many observers. >How did the find the angle? > >As a guess, the star in question may have been assumed to be >infinitely far, so that both lines from the points of >observation to the star may be thought as being >perpendicular to the diameter of Earth between the points of >measurement. (Far as I understand, their result was of by >7%. Perhaps, because of an assumption like that.) The two >fellows measured the angles between the lines to Mars and to >the star. If you draw a diagram you'll see that the angle we >are after is the sum of the two measured angles. Yes, it is a typical assumption in planetary astronomy that the stars are infinitely distant. Since the nearest star is roughly 500000 times as far as Mars, this assumption would probably still give about 5 or 6 decimal accuracy in the estimate of Mars's distance, so it cannot account for the 7% error. The three most likely sources of error would be inaccuracies in Cassini's knowledge of the radius of the Earth, or in the longitude of the second observer, or in the timekeeping. The method of finding the radius of the earth has been the same since Eratosthenes, and the difficult part has always been the actual measurement of the reference distance along the ground. This is a problem of cumulative errors in measurement that arise while surveying the distance between the two points where the observations were to take place. The proper way to handle these errors was introduced by C. F. Gauss (who did much practical engineering and scientific work, in addition to being a towering figure in pure mathematics), but in Cassini's day, this method was almost 150 years in the future. A few percent error in the radius of the Earth is therefore certainly possible. Longitude and time were also troublesome in the 17th century. Good clocks were unavailable, and longitude was usually determined by comparing local noon with clock noon. (This gave navigators no end of trouble until better more stable clocks were invented.) However Cassini was an astronomer after all, and could easily have made measurements that would have been unavailable to ordinary navigators. For instance, send the second observer to South America with the best clock available, then just before the observation of Mars, have both observers synchronize their clocks using some celestial event they could both observe with a good telescope. The transits of the moons of Jupiter would be good, or better yet, the occultation of a star by the moon  which they would observe at DIFFERENT times because of the Moon's parallax, but as the Moon's distance was already known, they could correct for that. Then using the synchronized clocks, measure the longitude of the second observer by the usual local solar noon method, and finally, perform the actual observation of Mars. There are plenty of places here where the 7% error could creep in, I think. This has got rather far from the original question, so I'll stop here, only mentioning that this is a common situation in experimental science: the original simplicity of the mathematical idea is often necessarily destroyed by practical considerations, without which an accurate result cannot be obtained. Stuart Anderson 

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Monty Phister
guest

Jul1505, 01:25 PM (EST) 

3. "RE: Cassini's determination of distance to Sun"
In response to message #2

Regarding "how did they find d", and "how did they find the angle": The two cities from which the observations were taken were in fact not on a common diameter, so there is no "diameter of Earth between the points of measurement." The observations were taken AT THE SAME TIME, so there wouldn't be an isosceles triangle whose base is the diameter of the earth. After the results of the observations were joined, Cassini could measure the angle between the two stars, from a common point. But how is that angle related to d? 

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alexb
Charter Member
1592 posts 
Jul1505, 01:28 PM (EST) 

4. "RE: Cassini's determination of distance to Sun"
In response to message #3

> The two cities from which the observations were taken >were in fact not on a common diameter, so there is no >"diameter of Earth between the points of measurement." The >observations were taken AT THE SAME TIME, so there wouldn't >be an isosceles triangle whose base is the diameter of the >earth. mr_homm explained this point: so there were two points with a known between them. The points were not diametrically opposite. >After the results of the observations were joined, >Cassini could measure the angle between the two stars, from >a common point. They did not measure the angle between two stars but between a star and Mars. >But how is that angle related to d? Please see the previous message. The angle related to d was the sum of two they measured.


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mr_homm
Member since May2205

Jul1505, 02:25 PM (EST) 

5. "RE: Cassini's determination of distance to Sun"
In response to message #3

>Regarding "how did they find d", and "how did they find the >angle": > The two cities from which the observations were taken >were in fact not on a common diameter, so there is no >"diameter of Earth between the points of measurement." The >observations were taken AT THE SAME TIME, so there wouldn't >be an isosceles triangle whose base is the diameter of the >earth. This is correct. Further, if the triangle is NOT isoceles, there is no direct relationship between the angle and the distance d to Mars. For the case of an isoceles triangle, you only need to know the base (distance between measurement points on Earth) and the angle at the apex (Mars), because for an isoceles triangle, knowing one angle will determine the other two for you. If you do not have an isoceles triangle the situation is more complicated, but can still be handled. It is easier to use rectangular coordinates to do the math, so that's how I will proceed: Let's call Paris "P",the other observation point "Q" and Mars "M". The known latitude and longitude of P and the radius of the earth describe the position of P relative to the center of the Earth in spherical coordinates r = earth radius, theta = 90 degrees  latitude, phi = longitude. Convert these to x, y, z coordinates by z=r*cos(theta), x = r*sin(theta)*cos(phi), y = r*sin(theta)*sin(phi). Do the same for Q. Now you have the coordinates of both points, so subtracting one set from the other gives the vector PQ. The length of PQ is the straight line distance between the two observation points. Now divide the vector PQ by this length to make it into a vector with length = 1. The observation of M in the sky from P gives the celestial coordinates of M, which are declination ( same as latitude) and right ascension (same as longitude, expressed in units of hours, where 24 hours = 360 degrees). We of course don't know the distance coordinate "r" for M yet, as that's what we're trying to find. Assume r=1 for M as seen from P, and convert to rectantular coordinates as before. Now you have a vectors of length 1 from P to Q and from P to M. Take the dot product of these two vectors (product of x components + product of y comoponents + product of z components). This is the cosine of the angle between the two vectors, so applying inverse cosine gives the angle at P in the triangle QPM. Since Cassini can measure the angle at M by looking at the two reference stars, he now has two of the three angles. Since the angle sum up to 180 degrees, this gives him the third angle. Knowing all three angles and the straight line distance PQ, he can use the law of sines to find the distances QM and QP. Notice that this doesn't give the distance from Mars to the center of the Earth, but instead from Mars to Paris. Of course, thid difference is negligible in comparison to the scale of the solar system. I think this is what you were looking for, right? I have deliberately given a lot of detail about the calculation in order to show how it would actually be carried out. The short answer to your question would have been: Knowing the positions of P and Q, Cassini knew the length and orientation of the line joining them. From this and the known directions of the sight lines to Mars, he could compute the angles at the two nearby corners of the triangle, and his observations of the background stars gave him the angle at Mars. Then he had enough information to apply the law of sines. > After the results of the observations were joined, >Cassini could measure the angle between the two stars, from >a common point. But how is that angle related to d? Now that you have the answer to this, I should say that the situation you find yourself in is a common one. People who are trying to explain something (for example astronomy) at an introductory level usually try to hide the messy details so that the central idea can be clearly seen. However, the details are often necessary, and so the clarity is only apparant. Inevitably someone (you in this case) comes along and takes a deeper look at the presentation and asks a question that can't be answered from the oversimplified explanation, and all these messy details come rushing right back into the discussion. This is a good thing, in my opinion, because actual understanding often means resisting the oversimplified presentation and demanding the details. Stuart Anderson


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Monty
guest

Jul2005, 06:10 PM (EST) 

6. "RE: Cassini's determination of distance to Sun"
In response to message #5

Right. That's EXACTLY what I was looking for, and I think I finally understand. I have a math newsletter https://www.gnarlymath.com/news/gnews1_1.html and am working on an "astronomy" issue. I was hoping the math would be simple enough that I could include it in the newsletter, but it won't do. So I'll be "leaving the messy details" (or at least many of them) out. I hope to do a little better than was done on the page referenced in my first note, though. While I was puzzling over this, another idea came to mind. Would Cassini have been able to predict the time when Mars was in such a position that a line from Mars to the earth's center was perpendicular to the earth's diameter? Putting it another way: consider a plane passing through P, the center of the Earth, and Mars. The plane will rotate as Mars and the Earth move. At some point, the line from Mars will be perpendicular. If he then measured the angle from Paris to Mars, could he compute the EarthMars distance from that angle and the latitude/longitude of Paris? In other words, could he have found the Mars distance without sending his buddy off to the ends of the Earth? One needs two positions to locate Mars. But could one of them be the Earth's center instead of Cayenne? 

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mr_homm
Member since May2205

Jul2105, 09:53 AM (EST) 

8. "RE: Cassini's determination of distance to Sun"
In response to message #6

>Right. That's EXACTLY what I was looking for, and I think I >finally understand. Glad to help. It was an interesting question you raised, and I had never really thought it all the way through myself until you asked. >I have a math newsletter >https://www.gnarlymath.com/news/gnews1_1.html and am working >on an "astronomy" issue. I was hoping the math would be >simple enough that I could include it in the newsletter, but >it won't do. So I'll be "leaving the messy details" (or at >least many of them) out. I hope to do a little better than >was done on the page referenced in my first note, though. Yes, perhaps a little bit to messy for a newsletter. You could if you wanted substitute the parallax measurement of the distance to a nearby star based on the width of the Earth's orbit. No matter where a star is in the sky, there are always points on the Earth's orbit that make an isoceles triangle with the base being the full diameter of the Earth's orbit. So in this case, the math really is just as simple as it appears. You need to have a much more distant star in the nearby background to compare against, but this is very common. > >While I was puzzling over this, another idea came to mind. >Would Cassini have been able to predict the time when Mars >was in such a position that a line from Mars to the earth's >center was perpendicular >to the earth's diameter? Putting it another way: consider a >plane passing through P, the center of the Earth, and Mars. >The plane will rotate as Mars and the Earth move. At some >point, the line from Mars will be perpendicular. If he then >measured the angle from Paris to Mars, could he compute the >EarthMars distance from that angle and the >latitude/longitude of Paris? In other words, could he have >found the Mars distance without sending his buddy off to the >ends of the Earth? One needs two positions to locate Mars. >But could one of them be the Earth's center instead of >Cayenne? There are two problems with this approach, one algebraic and one geometric. Algebraically, we need the same number of measurements as unknowns, in this case the radius of the earth and the angles describing the lines of sight from two points on Earth to Mars. Since we cannot actually perform an observation from the center of the Earth, we have to find some other way to determine exactly when the diameter that passes through Paris is exactly perpendicular to the line from Earth to Mars. Inferring the direction of some unreachable observer's line of sight is another parallax calculation, so we need to know the angle at Paris, and either two sides or one other angle. But both other angles are inaccessible (one at Mars and one at the center of the Earth), and for any two sides we pick, one of them leads to Mars, so we would have to already know that distance, which is of course what we are trying to find out. This shows that the method can't work. Geometrically, when the diameter through Paris is perpendicular to the line from the center of the Earth to Mars, then Mars lies in the plane perpendicular to that diameter, of course. But the horizon at Paris is also perpendicular to that same diameter, so a plane tangent to the earth at Paris is parallel to the plane containing Mars. As I mentioned in my earliest post to this thread, that means Mars cannot be seen from Paris at the required time, because it is (just barely) below the horizon at Paris. This shows that the method won't work. Either of these is a showstopper for the method you describe, of course. You could theoretically make repeated observations from Paris by taking advantage of the fact that Paris is not standing still. You could observe Mars just after it rises, when it reaches its zenith, and again just before it'sets, on two consecutive days. At Marsrise, Paris is ahead of the center of the Earth in its orbit around the sun, and at Marsset, Paris is behind the center of the Earth. Therefore, you would get a slight wobble in the position of Mars, superimposed on the motion it has due to its and Earth's orbital motions around the sun. The trouble with this method is that Earth moves about 1.8 million miles per day in its orbit, but is only about 8000 miles wide, so this wobble is about half a percent of the total motion of Mars across the sky during the observations. That means that you need to observe very accurately, because what you want are the tiny changes in already tiny angles. This is certainly feasible with modern equipment, but in Cassini's day, it would have given such a crude estimate as to be worthless, something like "Mars is 70,000,000 plus or minus 70,000,000 miles away. Stuart Anderson 

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AFB
guest

Jul2005, 06:36 PM (EST) 

7. "RE: Cassini's determination of distance to Sun"
In response to message #5

Hi, I have been reading this thread with interest, as Astronomy for Beginners is my site. Thanks for the clarification Stuart (and for the derivation), I never really thought about it that much detail and I agree my article was oversimplified. I changed the text in the main article and removed the offending equations, and put a link to a new page (https://www.astronomyforbeginners.com/astronomy/mars.php) with the mathematics in more detail, so if people want to know more detail its available. Thanks Gavin 

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mr_homm
Member since May2205

Jul2105, 10:12 AM (EST) 

9. "RE: Cassini's determination of distance to Sun"
In response to message #7

You're welcome, Gavin. I'm glad you found the material useful. By the way, sorry if it'seemed I was criticising your site with the word "oversimplified." I really just meant that it is necessary to simplify the initial explanation to make it clear, but later if you want to go deeper, that explanation won't do any more. So from that point of view, the explanation was oversimplified, but this kind of oversimplification really is often the best way to introduce a concept. I just took another look at your website, and I really like what you've done with the explanation I gave. Keeping it as simple as possible on the front page, then having a link to more details is (I think) the best and clearest way to present it. In fact, it is clearer now than the way I said it, because you have explained things that a general audience might not know, such as unit vectors, law of sines, and using the vector "dot" product to find the cosine of an angle. I just assumed those things in my posting, because I knew I was talking to people who would know them. It's good to see a website whose owner really thinks about his audience and what they might need to know. Very nicely done! Stuart Anderson 

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