The parallax of a heavenly body measured from two points diametrically opposite on equator of earth is 1.0 minute. If the radius of the earth is 6400 km, find the distance of the heavenly body from the centre of the earth in AU. Given 1 AU=1.5 x 10¹¹ m.

The correct option is 0.293 AU.

Step-by-step Explanation:

1. Understand the Parallax Method:
The parallax angle ($\theta$) is defined as the angle subtended by the basis (distance between the two observation points, $b$) at the heavenly body. The formula relating the distance ($D$) of the body, the basis ($b$), and the parallax angle ($\theta$ in radians) is given by:
$D=\frac{b}{\theta }$
where $D$ is the distance of the heavenly body from the center of the Earth.

2. Calculate the Basis ($b$):
The observation points are diametrically opposite on the Earth's equator. Therefore, the basis is equal to the diameter of the Earth.
Given the radius of the Earth, $R=6400\text{ km}=6.4\times {10}^6\text{ m}$.
$b=2R=2\times 6400\text{ km}=12800\text{ km}=1.28\times {10}^7\text{ m}$

3. Convert the Angle ($\theta$) to Radians:
The parallax angle is given as $1.0\text{ minute}\left(1^{\prime }\right)$. We must convert this angle into radians.
Since $1^{\prime }={\left(\frac{1}{60}\right)}^{ˆ}$ and $1^{ˆ}=\frac{\pi }{180}\text{ radians}$:
$\theta =\frac{1}{60}\times \frac{\pi }{180}\text{ rad}$
Using $\pi \approx 3.1416$:
$\theta =\frac{3.1416}{10800}\approx 2.91\times {10}^{-4}\text{ rad}$

4. Calculate the Distance ($D$) in Meters:
Substitute the values of $b$ and $\theta$ into the distance formula:
$D=\frac{1.28\times {10}^7}{2.91\times {10}^{-4}}$
$D\approx 4.40\times {10}^{10}\text{ m}$

5. Convert the Distance to Astronomical Units (AU):
Given that $1\text{ AU}=1.5\times {10}^{11}\text{ m}$:
$D=\frac{4.40\times {10}^{10}\text{ m}}{1.5\times {10}^{11}\text{ m/AU}}$
$D\approx 0.293\text{ AU}$
Thus, the distance of the heavenly body from the center of the Earth is approximately 0.293 AU.