How do astronomers find the "parallax angle" of a star?

[Karagoz]

https://www2.jpl.nasa.gov/teachers/attachments/parallax.html

It's written that they find the distance by calculating from the parallax angle.

But how do astronomers find the parallax angle?

 

[Hawksteinman]

It's how much the star appears to move in the sky per six weeks

 

[Hawksteinman]

Six months *

 

[Karagoz]

Six months *

But if the star appears to move 1 millimeters, then how can they find the angle and distance from that?If I see a tree. And the background is far away. There's a tall building that's far away.

E.g. when I run 50 meters, it looks like the tree is moving 3 cm from the building.

How can I find the distance to the tree?

 

[Hawksteinman]

But if the star appears to move 1 millimeters, then how can they find the angle and distance from that?

We can imagine a line drawn from the star to Earth, and as the star appears to move, the line would appear to rotate through an angle x

Once we know the angle and the distance between the Earth and the Sun, we can calculate the distance to the star

This might help: https://www.space.com/30417-parallax.html

 

[phyzguy]

But if the star appears to move 1 millimeters,

There are no distances on the sky. It makes no sense to say a star moves 1 millimeter. All measurements of the position of objects on the sky are angles. We are measuring the direction of the star, which is an angle relative to some reference direction, not a distance.

 

[Karagoz]

There are no distances on the sky. It makes no sense to say a star moves 1 millimeter. All measurements of the position of objects on the sky are angles. We are measuring the direction of the star, which is an angle relative to some reference direction, not a distance.

How is that angle measured? How do they know that angle?

 

[Hawksteinman]

How is that angle measured? How do they know that angle?

The entire world is divided into 360 degrees. These are then 'projected' onto the celestial sphere

https://en.m.wikipedia.org/wiki/Celestial_sphere?wprov=sfti1

 

[Karagoz]

How do they know that a parallax angle of one arcsecond is 1 parsec (1 parsec = ~206265 AU)?

And like in this picture:

When what we see is only a star changing its position, how is the parallax angle determined from that observation?

In other words:
How is the "apparent motion of stars" measured in angle degrees?

 

[newjerseyrunner]

How do they know that a parallax angle of one arcsecond is 1 parsec (1 parsec = ~206265 AU)?

Basic trig.

You have an isosceles triangle. The Earth is two endpoints and the star in question is the third. You also have measured the angle for the star. Create a right triangle by bisecting the isosceles (basically making the sun one of the endpoints of the triangle.) Then remember SOHCAHTOA? You have the opposite side (93 million miles,) and the angle, so use an inverse sine, and you'll get the distance to the star.

 

[Drakkith]

When what we see is only a star changing its position, how is the parallax angle determined from that observation?

When we look at two photographs and find that one of the stars has moved relative to the background stars we simply measure the distance from the nearest background star whose position hasn't noticeably moved. Note that the sky has been mapped out with a coordinate system for centuries and it is trivial to find the position of a star relative to this celestial coordinate system. Since our coordinate system is in angles, when you measure the distance to the nearest background star on an image you are also measuring the angular distance.

 

[Nik_2213]

The six-month result is a first approximation; you'll need longer series of observations to correct for 'proper motion', wobble due to unseen companion(s) etc. At least Doppler measurements may provide the latter, albeit subject to 'relative angle of orbit on sky'...

 

[Simon Peach]

It's found by a formula called the 'small angle approximation' I think!

 

[glappkaeft]

It's found by a formula called the 'small angle approximation' I think!

No, the small angle approximation is just a convenient way to calculate trig functions for small angles, it is not a specific formula for calculating distance from parallax.

 

[mfb]

@Drakkith: In the past it has been easy to find the position of a star with respect to these coordinates, but with increasing precision in star measuments it got more difficult. Gaia performs the most accurate measurements ever at the moment, and determining its current orientation is one of the most challenging aspects of the data analysis. The measurements are so precise that the vast majority of sources move within the measurement time and the accuracy of Gaia. It relies mainly on cross-references between its two telescopes. Their relative distance has to be monitored with a precision of the size of an atom.

 

[marksyncm]

I have a related question.

Wouldn't the movement of the star itself, in a direction similar to that of Earth, distort our calculations? In other words, how do we know that the calculated parallax angle is only a result of Earth's displacement during the 6 month period, and not the result of Earth's + the object's displacement?

 

[glappkaeft]

I have a related question.

Wouldn't the movement of the star itself, in a direction similar to that of Earth, distort our calculations? In other words, how do we know that the calculated parallax angle is only a result of Earth's displacement during the 6 month period, and not the result of Earth's + the object's displacement?

Yes, that was briefly explained in post #12.

 

[mfb]

You need at least three measurements, and every parallax measurement comes with a measurement of the motion in the sky because you have to consider that of course.
As comparison: Gaia does on average 70 measurements over several years.

 

[marksyncm]

Yes, that was briefly explained in post #12.

I see, thank you. It seems like a major problem. So if during those six months between the two observations, the object's and Earth's displacement (in the same direction) were, say, equal, we would conclude the star was too far for us to measure using parallax, when in fact it could be "only" 20 light years away?

Assuming the first measurement is taken at point A in orbit, and the other measurement six months later at point B, then does that mean we also take another measurement back when we are at point A? Seems this would help. By comparing the angle change while going from A to B with the change when going from B to A, it seems we could "subtract" the object's own motion. Is that more or less part of how it's done?

 

[mfb]

Did you see my post above? I posted while you were typing already I guess.

 

[marksyncm]

Did you see my post above? I posted while you were typing already I guess.

Ah yes, thank you.

Sounds like using parallax to measure the distance to binary stars might be a bit of a pain.

 

[glappkaeft]

Not really, there are a lot of thing much more complicated in science and it depends a lot on how accurate you want it to be. The basic concept is very simple but if you want ridiculously accurate measurements (and an equally ridiculously amount of them) you get something like the GAIA spacecraft .

 

[mfb]

It depends on the orbit, but some binary stars are certainly more complex. You can also see this in the Gaia data release plans. Data releases for multiple star systems are behind releases for single-star systems because it takes longer to measure and analyze them.

 

[marksyncm]

This is fascinating. Thank you.

 

[sophiecentaur]

How is that angle measured? How do they know that angle?

You can use any star database to find the angular separation between two known stars and, compare two photos, taken 6months apart. The stars that are far enough away to be regarded as 'fixed' are your backdrop and you measure the change in position of your star-under-test. You need to do some measuring of distances on the photos and use the known angular separations and some 'Ratios' to tell you the angle scale of your photo. (Use several known stars to help your accuracy).
To increase your accuracy, you may need to measure and correct for the field distortions in your lens (shift the image around and compare the distance when stars are on the edge and in the middle) and do your measurements near the axis of the lens. Of course, you need to choose stars that are separated by only a very few degrees so the error is small. (tanθ ≅ sinθ ≅ θ in radians as θ→ 0)

Wouldn't the movement of the star itself, in a direction similar to that of Earth, distort our calculations? In other words, how do we know that the calculated parallax angle is only a result of Earth's displacement during the 6 month period, and not the result of Earth's + the object's displacement?

To ameliorate the effect, you do it with the Earth moving one direction (right-left) in its orbit and then, later the other way (left-right). The 'correct' / better answer is the mean of the two apparent parallax values. A tedious and time consuming process but less so than when people had to sketch or use photographs. We have it relatively easy these days.

 

[stefan r]

How is that angle measured? How do they know that angle?

A mural instrument was the classic: