GRAVITY IN EQUATOR V/S POLE OF ALPHA ARAE

GRAVITY IN EQUATOR V/S POLE OF ALPHA ARAE

The Alpha Arae is a distant star in our Solar System with a distance of 270 light years from the Earth. Gravity on the surface of Alpha Arae varies due to its fast rotation. It rotates as fast as 375km/s so it looks like an ellipsoid. Since Alpha Arae rotates, all points on its surface perform circular motion except the North and South Pole. The points at equator perform circular motion of maximum radii as compared to other points. The polar points do not perform circular motion but rotate about their own axis. 

Consider two points, equator and North Pole on Alpha Arae’s surface. We’ll compare acceleration due to gravity (g) between these two points.

ASSUMPTIONS

The equator and pole locations are not terrain but plains

Density of Alpha Arae is constant throughout

Angular velocity is constant throughout

PHYSICAL CHARACTERISTICS

Mean Radius R= 3,130,650 km

Mass M = 1.92*10³¹ kg

Rotational velocity = 375 km/s

CALCULATION

Acceleration due to gravity on pole:

g = GM/R²

G – Universal Gravitation constant = 6.67*10^-11 Nm²/kg²

g = [6.67*10^-11 * 1.92*10³¹]/ [3,130,650*1000]²

g = 130.6646 m/s²

g (φ) = 130.665 m/s²

φ – Latitude = 90⁰ for poles

Average Angular velocity:

ω = v / r (rad/s)

v- Tangential velocity (m/s)

r- Radius (m)

ω = 375*1000 / [4.5*695700*1000] (rad/s) 

ω = 375/3130650

ω = 1.1978*10^-4 rad/s

Acceleration due to gravity on equator:

We can find acceleration due to gravity on equator by using the formula,

g’(φ) = g(φ) – Rω²cos²φ

φ - Latitude = 0⁰ for equator

g’(φ) = 130.665 – 4.5*695,700,000*(1.1978*10^-4)²

g’(φ) = 130.665 – 44.9162

g’(φ) = 85.7484 m/s²

CONCLUSION

On comparing g (φ) and g’ (φ) we observe that gravity at pole is greater than that at the equator. This is because the Alpha Arae rotates so fast that the angular velocity at the equator decreases the effective pull due to gravity.

The difference is g (φ) - g’ (φ) = 130.665 – 85.7484 = 44.9162 m/s²