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# Parallax

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Modern astronomers use parallax to determine the distance to a star. After the observer on the earth propagates the distance of the earth's orbital diameter in a six month time interval (fig 33), the change in the angular position of the star is used to determine the distance to the star but the distance to a 4.22 light year star (4 x 1016 meters) is more than 10times larger than the earth's orbital diameter (2.99 x 1011 m). The earth's orbital diameter is too short of a distance to produce a change in the angular position that can be used to measure the distance to a 4.22 ly star. The resolution required to determine the distance to a 4.22 ly star is calculated using,

A/B = cos θ.........................................................................................................................................78

when A/B  0, equation 78 becomes,

A/B = θ.................................................................................................................................................79

Using A as the earth's orbital diameter, B is the distance to a 4.22 ly star, the resolution θ required to determine the distance to a 4.22 ly (4 x 1016 meters) star is calculated,

θ = A/B = (2.99 x 1011 m) / (4 x 1016 meters) = 7.475 x 10-6 degrees or 0.027 arcsec....................80

To measure the distance of a 4.22 ly star using the earth's orbital diameter as the parallax reference distance requires a telescopic resolution of 0.027 arcsec (equ 80) which is 3.7 times more power than the Hubble (.1 arcsec). The Hipparcos telescope is described with a resolution of .001 arcsec but the Hubble was launched after the Hipparcos and the Hubble's mirror diameter is 7.9 feet which is eight times larger than the Hipparacos mirror diameter (11 inches) yet the Hipparcos is 100 times more powerful than the Hubble which violates logic. Using A/B = θ when A/B  0, the maximum distance to a star calculated using the Hubble is,

B = A/θ = (2.99 x 1011 m) (3600) / (.1 arcsec) = 1.0764 x 1015 m = 0.114 light years.........................81

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If the maximum distance that can be determine is less than one light year is modern astronomy a fabrication similar to the photos of the Milky Way galaxy and the LIGO.

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Have a look at this.

Parralax can be measured to close stars with amateur telescopes.

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On 11/04/2019 at 23:59, alright1234 said:

θ = A/B = (2.99 x 1011 m) / (4 x 1016 meters) = 7.475 x 10-6 degrees or 0.027 arcsec....................80

Ah, you've missed the conversion between Radians (which is what A/B gives - technically it's sin-1 (A/B), and degrees.

Multiply by 180/pi and you get the right value of just over 1.5 arc seconds for 4.22 ly.

ETA

parallax itself is defined as the half angle, where 'A' is the earth's orbital radius rather than diameter

Edited by Gfamily

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I did a talk to my local AS about how Bessel measured the parallax of 61 Cygni - which was done using a telescope where the 6" objective was split, with half on a micrometer thread to allow the distance between the 'nearby' star and a distant star to be measured.

Of course, he measured the separation between 61 Cyg and two other stars over a year,

and from this (despite what looks like a dodgy measurement in Jan 1838), you can see the annual change in parallax

Edited by Gfamily
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On 11/04/2019 at 23:59, alright1234 said:

To measure the distance of a 4.22 ly star using the earth's orbital diameter as the parallax reference distance requires a telescopic resolution of 0.027 arcsec

No it doesn't.  As well as your schoolboy error mixing radians and degrees, You can measure the position of the star ie the centroid of the star image to much greater precision than the resolution

Edited by robin_astro

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On 12/04/2019 at 08:29, alright1234 said:

Modern astronomers use parallax to determine the distance to a star. After the observer on the earth propagates the distance of the earth's orbital diameter in a six month time interval (fig 33), the change in the angular position of the star is used to determine the distance to the star but the distance to a 4.22 light year star (4 x 1016 meters) is more than 10times larger than the earth's orbital diameter (2.99 x 1011 m). The earth's orbital diameter is too short of a distance to produce a change in the angular position that can be used to measure the distance to a 4.22 ly star. The resolution required to determine the distance to a 4.22 ly star is calculated using,

A/B = cos θ.........................................................................................................................................78

when A/B  0, equation 78 becomes,

A/B = θ.................................................................................................................................................79

Using A as the earth's orbital diameter, B is the distance to a 4.22 ly star, the resolution θ required to determine the distance to a 4.22 ly (4 x 1016 meters) star is calculated,

θ = A/B = (2.99 x 1011 m) / (4 x 1016 meters) = 7.475 x 10-6 degrees or 0.027 arcsec....................80

To measure the distance of a 4.22 ly star using the earth's orbital diameter as the parallax reference distance requires a telescopic resolution of 0.027 arcsec (equ 80) which is 3.7 times more power than the Hubble (.1 arcsec). The Hipparcos telescope is described with a resolution of .001 arcsec but the Hubble was launched after the Hipparcos and the Hubble's mirror diameter is 7.9 feet which is eight times larger than the Hipparacos mirror diameter (11 inches) yet the Hipparcos is 100 times more powerful than the Hubble which violates logic. Using A/B = θ when A/B  0, the maximum distance to a star calculated using the Hubble is,

B = A/θ = (2.99 x 1011 m) (3600) / (.1 arcsec) = 1.0764 x 1015 m = 0.114 light years.........................81

Two points:-

Point 1.

OK with equation 78 when A and B are the sides of a right angle triangle. B is the hypotenuse.

If A/B --> 0 then the angle theta tends to 90 degrees. Cos(90) = 0 and Sin(90)  = 1  (angle in degrees)

Seems to be a mix-up here, perhaps what was meant:-

Lim ((sin(a)) ) / a  --> 1

a --> 0

It is written like this because as the angle a=0, sin(a) = 0 and the division 0/0 is meaningless.

When the angle is tends to 0 radians then a = sin(a).

If A/B = sin(a) when a --> 0 then we can say the angle a in radians is

a = A/B

Point 2.

One does not need to go through the above reasoning if one uses the definition that on a circle (radius r) the length of the arc (S) is

S = r * a    where a is the subtended angle at the centre of the circle.

For very small angles the arc and chord may be taken to be of the same length.

Distance to the star  may be calculated from:

r = S/a.

Angle a may be measured by the large land based telescopes.

I see no problems here.

Re: You can measure the position of the star ie the centroid of the star image to much greater precision than the resolution.

Yes of course. Good point.

Jeremy

Edited by JRWASTRO
fix up a sentence

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Just for completeness - look up parsec as unit of measure, it is about 3.26 light years and it represents distance that produces PARallax of one arc SECond.

Amateurs can determine stellar position down to about 0.05" (and in some cases even less) which means that even with small amateur scopes, parallax distance measurements up to ~20pc or about 65ly are possible.

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