Trivial questions regarding parallax and measurement of distances.

[JC2000]

While trying to understand parallax and its use in measuring distances here, I had a few questions.

(1) Parallax is defined as the apparent movement of an object with reference to another object in the background when one views it from different angles. As in the movement of a finger relative to the background when viewed through either eye. Also, the source states that the illusion of seeing two points of the background when focusing on the point closer to the observer. This is attributed to the crossing/uncrossing of the eyeballs.

(a) Is the second scenario also an instance of the parallax effect?
(b) If so, how? (The first scenario involves movement of the foreground with respect to a background when the observers vantage points change (distance between eyeballs). On the other hand, the second scenario involves 'duplication' of the point in the background simply due to crossing over of lines of vision of each eyeball. The only common aspect between the two is that the shift is greater when the distance is smaller.).
(c) Have I misunderstood something in the second scenario?

(2) On the following page, the parallax method is used to find the distance of the moon from the Earth using a star in the background and by changing vantage points between Athens and Selsey.

(d) Here it is assumed that the tree points (Athens, Selsey and the Moon) form an isosceles triangle. I am unclear about the justification behind this.
(e) How is the angle of apparent movement of the star calculated?
(f) Is the angle of apparent movement of the star the same as the angle of apparent movement of the Moon because both the scenarios that were described in (1) are the same?
(g) None of this seems to correspond to the given image with the moon as the vertex of the triangle.

Sorry for the huge wall of text and asinine questions, I seem to be having a tremendous mental lapse.

 

[sophiecentaur]

Binocular vision doesn't rely on parallax because you can judge some distances without background objects.
We can use parallax too and it works over much larger distances (buildings against distant mountains) than binocular vision will function. The baseline is what limits the range - just as the baseline for a ('binocular') military rangefinder limits the measurement of distances to around 10km (when conditions are good).
Measuring the distance to the Moon (one of the many steps in establishing the value of the Astronomical Unit or AU) required a long baseline and the (handy) distance from Syene and Alexandria had already been measured by Eratosthenes. Everything is on the move so some method was needed to make sure the angles are measured at the same time. Hipparchus used a Solar Eclipse to synchronise the measurement times (maximum 'shadow' as the Moon passed over the Sun). The two angles between the edge of the Moon and the vertical plus the previous measurement of the Earth's radius gave the triangle needed to find the lunar distance.
For the kinds of distances involved in this and other measurements, the approximation that
Θ ≈ sin(Θ) ≈tan(Θ) is often good enough and I don't think you need worry about the triangle being isosceles because, in practice, this was almost certainly taken care of. It's probably assumed for the purpose of a simple demonstration.
The use of parallax for measuring star distances uses the AU and involves very small parallax angles - even for the nearest stars so the method doesn't suit measuring the Moon's distance - for a start it moves too fast and synchronising observing times is too demanding. The parallax angles are measured by comparing against larger angular distances between 'fixed' stars. You may find this link of interest; it's results from an amateur astronomer.
A large can of worms here - you need to dig deeply to get all the answers you are after.

 

[JC2000]

Thanks for going through all the links and providing such a detailed answer.

Do you think that the sources that I used are unnecessarily confusing? I feel so since I am now reasonably clear about my questions under (1) since I understand the distinction between the parallax effect and binocular vision, thanks to your answer. Regarding (2) I feel that digging further into the specifics of the experiment will make things clearer. I stumbled onto that source just to understand the basics of parallax but it seemed to confuse me further, hence the long list of questions.

I don't think you need worry about the triangle being isosceles because, in practice, this was almost certainly taken care of.

I still do not understand how they ensure that the triangle is isosceles?

After reading your answer I had a number of follow on questions but I believe I can find the answers by digging further into the specifics. Nonetheless I listed them below, please ignore them if you think that these answers should be easy to find.

Binocular vision doesn't rely on parallax because you can judge some distances without background objects.
We can use parallax too and it works over much larger distances (buildings against distant mountains) than binocular vision will function. The baseline is what limits the range - just as the baseline for a ('binocular') military rangefinder limits the measurement of distances to around 10km (when conditions are good).

I see so binocular vision (vision using two eyes allowing overlapping fields of view) is what has been described in the second scenario. What you are saying is that normal vision allows us to perceive smaller distances while the parallax effect (magnitude of movement of distant objects relative to one another) helps perceive larger distances.

(h) If so does the link I mentioned (1) simply overcomplicate the parallax effect by also introducing binocular vision even though it is unrelated?
(i) I understand that the parallax effect combined with trigonometry allows us to measure distances fairly precisely. How does one do so with binocular vision?
(j) I got a sense of what you mean by baseline but since I am unaware of the trigonometry used to find distances through binocular vision, where can I find a simple yet fairly rigorous definition?

The two angles between the edge of the Moon and the vertical plus the previous measurement of the Earth's radius gave the triangle needed to find the lunar distance.

(k) Can I find a diagram somewhere?

 

[sophiecentaur]

I would not say that the method you are quoting is actually a Parallax method. It is a simple Rangefinder / Triangulation method - like binocular vision but our eyes don't actually have angle scales on them. Afaik, the Parallax method of Astronomers always produces an answer in AU and uses very distant stars.

Can I find a diagram somewhere

The link has everything in it about historical Moon distance measurement afaik.
I suggest you look up Binocular Vision or Rangefinder. Use the Images Button and you will find dozens of diagrams and chose something you fancy.

I still do not understand how they ensure that the triangle is isosceles?

You can assume sin(2Θ) ≈2sin(Θ) so it doesn't matter whether you chose it as an isosceles triangle with the legs on either side of the direct line , a single right angled triangle or an asymmetrical triangle. The error is very small. Afaics, we are only discussing the sort of approximations that apply when very small angles are involved.

 

[sophiecentaur]

I re-read that link of yours about the modern day moon measurement. Firstly, they make a point of choosing a situation where they are sure of an isosceles triangle. They can do that by getting the Moon directly overhead at the mid point and doing the measurement at that time. (This is only possible for certain locations, of course)
Also, the time of the measurements needs to be synchronised, which was not easy for the old Greeks. They had to use a one-off eclipse as the timing point.
The method used is interesting but I think it is needless complicated as a way into the problem. It confused you and I'm not too surprised. I read recently that, until the 1950's, the measurements of Moon Distance were done with 'rangefinder' methods which obtain the angle in that triangle by reference to the vertical, rather from parallax. That is much easier to understand, even if the parallax method is more accurate.
It struck me that the angular accuracy of their measurement method could be compromised if they are using the 'edge' of the Moon, which is actually bumpy. It could be better if they used a small individual feature (crater?) But we all look at things from our own perspective and there are always several ways of killing a cat.

 

[jbriggs444]

You can assume sin(2Θ) ≈2sin(Θ) so it doesn't matter whether you chose it as an isosceles triangle with the legs on either side of the direct line , a single right angled triangle or an asymmetrical triangle. The error is very small. Afaics, we are only discussing the sort of approximations that apply when very small angles are involved.

One could do a parallax measurement where the baseline is not at approximately right angles to the distance being surveyed. In that case one would have to apply a ##\cos \theta## correction to the angled distance between the base points to obtain the corresponding perpendicular distance.

For the typical situation with the baseline at approximately right angles, ##\cos \theta## is approximately 1 and no correction is needed.

 

[sophiecentaur]

One could do a parallax measurement where the baseline is not at approximately right angles to the distance being surveyed. In that case one would have to apply a cosθcos⁡θ\cos \theta correction to the angled distance between the base points to obtain the corresponding perpendicular distance.

Yes, there are many extensions to the basic methods and the OP's link is one of them. When you jump into a subject that you come across from browsing or from a Google search and you try to make sense of just one link, it is really hard to differentiate between the basics and the extra bells and whistles that you can find.
One should always look at a number of alternative links before trying to battle through just one, which may 1. Be wrong or 2. Off on a tangent to the 'real' problem.
This is why I always tend to recommend a Text Book as source for information because you get the bare bones and not someone's enjoyable diversion. Text Books are not favourite with many Internet Users because they can be hard work and because it will probably involve Spending Money.
The Hyperphysics Site is an excellent source of nearly all the Physics you need - it's free (+1) and usually correct (+2).

 

[LURCH]

Didn’t actually read the link, but regarding this:

I still do not understand how they ensure that the triangle is isosceles?

It was mentioned earlier that Eratosthenes had already measured the distance from Alexandria to Syene. That measurement was in response to the fact that on the longest day of the year, the Sun is directly overhead in Syene (and not in Alexandria). This may be their basis for considering the Moon to be directly over that city during an eclipse. Again, just guessing, as I have not read the link.