Important Formulas for NSEA Stage 1

1. Angular Diameter Formula

The angular diameter of an object is given by:

$$ \\theta = \\frac{2R}{d} $$

Where:

• \( R \) = Radius of the object
• \( d \) = Distance to the object

2. Light Gathering Power of a Telescope

Light gathering power is proportional to the area of the objective lens or mirror:

$$ \\text{Light gathering power} \\propto D^2 $$

Where:

• \( D \) = Diameter of the telescope's objective lens or mirror

3. Pressure Calculation

The pressure at a point is given by:

$$ P = nkT $$

Where:

• \( n \) = Particle density
• \( k \) = Boltzmann constant
• \( T \) = Temperature

4. Distance from Parallax

Distance from parallax is calculated using:

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

Where:

• \( d \) = Distance in parsecs
• \( p \) = Parallax angle in arcseconds

5. Mass of Milky Way Galaxy (Keplerian Motion)

The mass enclosed within the Sun's orbit in the Milky Way is given by:

$$ M = \\frac{v^2 r}{G} $$

Where:

• \( v \) = Orbital velocity of the Sun
• \( r \) = Distance from the Sun to the center of the galaxy
• \( G \) = Gravitational constant

6. Doppler Shift Formula

The velocity of a galaxy moving away can be calculated using the Doppler shift formula:

$$ v = c \\frac{\\Delta \\lambda}{\\lambda_0} $$

Where:

• \( v \) = Velocity of the galaxy
• \( c \) = Speed of light
• \( \\Delta \\lambda \) = Change in wavelength
• \( \\lambda_0 \) = Rest wavelength

7. Orbital Radius of Geosynchronous Satellite

The orbital radius of a geosynchronous satellite is given by:

$$ R = \\left( \\frac{GMT^2}{4\\pi^2} \\right)^{\\frac{1}{3}} $$

Where:

• \( R \) = Orbital radius
• \( G \) = Gravitational constant
• \( M \) = Mass of the Earth
• \( T \) = Orbital period (equal to Earth's rotation period)