Why Does Acceleration Point Towards the Center in Elliptical Orbits?

[schaefera]

If you parameterize an ellipse such that x=acos(t) and y=bsin(t), then you quite easily get the relations:

r={acost, bsint}
v={-asint, bcost}
a={-acost, -bsint}

But my issue is that now, if I think of the equations as representing the motion of a planet about its sun, the acceleration vector listed above always points toward the center of the ellipse and not toward the ellipse's focus. (Take, for example, t=pi/2... with this, the position is along the y-axis at a distance b, and acceleration points toward the origin, not the ellipse's focus).

That is, the acceleration is always directed normal to velocity, which should only happen in a circle... so what is wrong with my math?

 

[lavinia]

If you parameterize an ellipse such that x=acos(t) and y=bsin(t), then you quite easily get the relations:

r={acost, bsint}
v={-asint, bcost}
a={-acost, -bsint}

But my issue is that now, if I think of the equations as representing the motion of a planet about its sun, the acceleration vector listed above always points toward the center of the ellipse and not toward the ellipse's focus. (Take, for example, t=pi/2... with this, the position is along the y-axis at a distance b, and acceleration points toward the origin, not the ellipse's focus).

That is, the acceleration is always directed normal to velocity, which should only happen in a circle... so what is wrong with my math?

The acceleration is not normal to the velocity except when a = b, the case of the circle.

In the motion of a planet, the parameterization is not the same as the one you have given.
Try solving for the parameters starting with Newton's law of gravitation

 

[schaefera]

In the parameterization above, isn't velocity normal to acceleration?

 

[lavinia]

In the parameterization above, isn't velocity normal to acceleration?

no. the inner product is

a^2sintcost - b^2bsintcost

In planetary motion, the acceleration is not normal to the ellipse except at extreme points.

Generally there is a component of acceleration that is tangent to the ellipse in addition to the normal centripetal component. I think of the direction centripetal component as being rotated by the tangent component so that the total acceleration points towards the focus of the ellipse.

 

[D H]

If you parameterize an ellipse such that x=acos(t) and y=bsin(t), then you quite easily get the relations:

r={acost, bsint}
v={-asint, bcost}
a={-acost, -bsint}

But my issue is that now, if I think of the equations as representing the motion of a planet about its sun, the acceleration vector listed above always points toward the center of the ellipse and not toward the ellipse's focus.

Yours is but one of many (an infinite number) of parameterizations of an ellipse with the center at the origin. It is not *the* parameterization to use to describe an orbiting body. For one thing, your parameterization describes an ellipse with the center at the origin of the reference frame. The Sun is at one of the foci of an ellipse.