Earth's Elliptical Path: Acceleration Direction Explained

[alexmahone]

Homework Statement

The Earth travels in an elliptical path with the sun at one of its foci. Is the acceleration of the Earth directed towards the sun or towards the centre of the ellipse?

Homework Equations

The Earth's elliptical path can be parametrized as:
$x=acos (pt)$
$y=bsin (pt)$

The Attempt at a Solution

Using the above parametrizations and differentiating twice, I get that the Earth's acceleration is directed towards the centre of the ellipse.

However the force on the Earth is due to the sun, and so I feel the Earth's acceleration must be directed towards the sun.

Please help.

 

[Redbelly98]

However the force on the Earth is due to the sun, and so I feel the Earth's acceleration must be directed towards the sun.

That's correct.

EDIT: you're paramaterization is incorrect if "t" is really the time. It implies the speed is the same at aphelion and perihelion, but in reality it is faster at perihelion.

 

[alexmahone]

That's correct.

EDIT: you're paramaterization is incorrect if "t" is really the time. It implies the speed is the same at aphelion and perihelion, but in reality it is faster at perihelion.

Thanks, Redbelly98! :)

Forget the Earth - sun example.

What if the x and y coordinates of a particle were given by the parametrization where t is time? Where would the acceleration of the particle be directed towards?

 

[Redbelly98]

That's a reasonable question. Well, you can take the 2nd derivative of both x and y with respect to t, and see what direction the vector
a = (d²x/dt², d²y/dt²)
points in.