Question

best answer will Mark as brainlist​

Answer 1

✭ Parallax Angle ✭

• Parallax Angle refers to the angle formed between the two apparent positions of an object when observing from two different positions.

• The Earth revolves around the Sun with an orbital period of 12 months (1 year). The mean distance between the Earth and the Sun is 1 AU(Astronomical Unit).

• At six months apart, the Earth will be at diametrically opposite positions in its orbit. So, the distance between two positions of the Earth becomes 2 AU.

• Now, consider a distant star. When we observe the Star from Earth six months apart, we find a difference in the direction we see the star in. The two positions form a specific angle. This is the Parallax Angle, which we want to find out in this question.

✭ See the diagram attached ✭

• We will consider the standard formula, as seen in the second image attached.

$\huge\boxed{\theta = \dfrac{l}{r}}$

• The $l$ in the image, and in reality, is a curve. However, objects like Stars are really far away and the angle $\theta$ is very very small. So, the curve represented by length $l$ can be safely approximated by a straight line.

• Here, it is represented as the distance between positions of Earth 6 months apart.

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• We have a couple of standard astronomy constants: Parsec (pc) and Astronomical Unit (AU).

• $\sf 1 \: \textsf{pc} = 3.086 \times 10^{16} \: m\\ \\ 1 \: \textsf{AU} = 1.496 \times 10^{11} \: m$

Let's now look at our data:-

$l = 2 \: AU = 2\times 1.496\times 10^{11} \: m = 2.992 \times 10^{11} \: m \\ \\ r = 1.45 \: pc = 1.45 \times 3.086 \times 10^{16} \: m = 4.4747\times 10^{16} \: m$

We need the Parallax Angle. Let us call it $\theta$.

We can obtain it by our simple formula:-

$\displaystyle \theta = \frac{l}{r} \\\\\\ \implies \theta = \frac{2.992 \times 10^{11}}{4.4747\times 10^{16}} \approx 6.686 \times 10^{-6} \: radians \\\\\\ \implies \huge \boxed{\sf\theta \approx 6.686 \times 10^{-6} \: \textsf{rad}}$

• So, we obtained the answer in radians. However, since the notion of degrees, minutes and seconds is so much more comprehensible than a radian, let's change the units.

$\displaystyle\pi \: rad = 180^{\circ} \\\\\\ \left[1^{\circ} =60\,' \text{ and } 1\,' = 60\,'' \right] \\ \\ \\ \implies \pi \: rad = (180 \times 60 \times 60)\, '' \\\\\\ \implies 1\: rad = \left(\dfrac{180\times 60\times 60}{\pi}\right)'' \\\\\\ \implies \theta = 6.686\times 10^{-6} \: rad = \left(\frac{6.686\times 10^{-6}\times 180\times 3600}{\pi}\right)'' \\\\\\ \implies \huge \boxed{\theta\approx1.379\,''}$

• As we see, the Parallax Angle is indeed extremely small!! It is hardly just 1 second!

• This just shows us the scale of the Universe! Stars are so far apart. The angles subtended get so small.

Finally, we have our answer:-

$\Large \boxed{\boxed{\sf \theta \approx 6.686\times 10^{-6} \textsf{ rad} \approx 1.379\,''}}$

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EXTRA INFO:-

The answer has finished. This is the Extra Info Section. Read on if you wish to know more!

1) Astronomical Unit (AU)

The Astronomical Unit is a unit of distance.

• The mean distance between the Sun and Earth is defined as 1 AU.

• The Earth's orbit is not perfectly circular. It is slightly elliptical. So, at some points in its orbit, the Earth is closer to the Sun than other points.

• The point in the Earth's orbit when it is closest to the Sun is called the Perihelion. At this point, the Earth is about 147 million kilometres from the Sun.

• The point in the Earth's orbit when it is farthest from the Sun is called the Aphelion. At this point, the Earth is about 153 million kilometres from the Sun.

• The mean distance is around 150 million kilometres, and this is defined as 1 AU.

[See the third image attached]

Precisely, it is 149.6 million kilometres.

$\sf 1 \textsf{ AU} = 1.496 \times 10^{11} \textsf{ m}$

2) Parsec (pc)

• Parsec is another unit of length.

• A parsec is defined as the distance at which an arc length of 1 AU subtends an angle of 1 second.

[See the fourth image attached]

• Here, as seen, we have an arc length of 1 AU, and the angle subtended as 1''. On calculating from:

$\sf\tan 1'' = \dfrac{\textsf{1 AU}}{\textsf{1 pc}}$

• Since 1'' is an extremely small angle, we can directly approximate it as:

$\sf\displaystyle 1'' = \frac{\textsf{1 AU}}{\textsf{1 pc}} \\\\\\ \implies \frac{1}{3600} \times \frac{\pi}{180} \: rad =\frac{\textsf{1 AU}}{\textsf{1 pc}} \\\\\\ \implies 1\ pc = \frac{1.496\times 10^{11}\times 3600\times 180}{\pi} \: m \\\\\\ \implies 1\ pc \approx 3.086 \times 10^{16}\ m$

• 1 parsec is huge! The order of $10^{16}$ shows just this. It is around $10^5$ times the Astronomical Unit.

• For comparison, the diameter of Solar System is estimated to be 122 AU! Much less than even a parsec!

• The Universe is huge! And this is where we live :)

Answer 2

Explanation:

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refer to the attachment