Measurement of Distances: Parallax Method and Precision Instruments

Astronomers estimate the distance of nearby objects in space by using a method called stellar parallax, or trigonometric parallax. Simply put, they measure a star's apparent movement against the background of more distant stars as Earth revolves around the sun

Given,

$\theta =0.03316\,\,rad$

Also $b = \mathrm { AB } =$ diameter of earth $= 1.276 \times 10 ^ { 7 } \mathrm { m }$

Now d $=\dfrac{b}{\theta }=\dfrac{1.276\times {{10}^{7}}}{0.03316\,}=3.84\times {{10}^{8}}\text{m}$

Hence, distance between earth and moon is $3.84\times {{10}^{8}}\text{m}$

Spherometer is an instrument which is used to measure the curvature of spherical surfaces.

Parallax method is used for the measurement of planet distance and also used to measure distance of starts and planets

Optical microscopes are enabled with objective fitted with lenses with different magnifications. This helps to magnify a small objects and so necessary measurements.

In parallax method, an approximation is made that distance between a point on the earth and the planet is very large as compared to the distance between two points on the earth's surface.

$Answer:-$ D

Astronomers use an effect called parallax to measure distances to nearby stars. Parallax is the apparent displacement of an object because of a change in the observer's point of view.

$Answer:-$ B

$Answer:-$ A

Astronomers use an effect called parallax to measure distances to nearby stars. Parallax is the apparent displacement of an object because of a change in the observer's point of view.

Relationship between a star's distance and its parallax angle:

$d=\dfrac{1}{p}$

The distance $d$ is measured in parsecs and the parallax angle $p$ is measured in arcseconds.

Hence, $d=\dfrac{1}{0.723}=1.38 parsecs$

$Answer:-$ C

Astronomers use an effect called parallax to measure distances to nearby stars. Parallax is the apparent displacement of an object because of a change in the observer's point of view.

Distance measurement by parallax is a special case of the principle of triangulation, which states that one can solve for all the sides and angles in a network of triangles if, in addition to all the angles in the network, the length of at least one side has been measured. Thus, the careful measurement of the length of one baseline can fix the scale of an entire triangulation network. In parallax, the triangle is extremely long and narrow, and by measuring both its shortest side (the motion of the observer) and the small top angle (always less than 1 arcsecond leaving the other two close to 90 degrees), the length of the long sides can be determined.

Parallax effect depends upon the path of light that travels from object to the observer. For distant objects observed from Earth, the light that reaches Earth has to go through a number of layers in Earth's atmosphere, during which it refracts and disperses and hence decreases the accuracy of the method. Resultantly parallax angles less than $0.01/arcsec$ are very difficult to measure from Earth.

Least count of the spherometer is $LC=$ pitch/number of circular divisions $=\dfrac{1/10}{50}=0.002 cm$
The total reading for height is $h=MSR+(CSR\times LC)=(2\times 0.1)+(40\times 0.002)=0.28 cm$
The formula for radius of curvature is $R=\dfrac{l^2}{6h}+\dfrac{h}{2}=\dfrac{4^2}{6(0.28)}+\dfrac{0.28}{2}=9.66 cm$

The least count of the spherometer, $LC=$pitch/ total number of circular divisions $=\dfrac{p}{n}$
Here, pitch $p=1/20=0.1\ cm$ and $n=50$
Thus, $LC=\dfrac{0.1}{50}=0.002\ cm$

Given :    $1$ AU $ = 1.5\times 10^11$ m                $d = 5.2\times 10^{15}$ m

One light year $=$ speed of light $\times$ one year