Kepler's second law and Wikipedia article

[PainterGuy]

Hi,

I'm sorry but I'm not sure if I should post it here or in homework section. It's not homework for sure.

This Wikipedia article, https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion, on Kepler's laws says the following under History section in the last para:

Newton was credited with understanding that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law; while the other laws do depend on the inverse square form of the attraction.

Does it simply mean that the law is not a consequence of inverse square law and even if the gravitational law was a linear relation instead of inverse square relation, the law would still work the same. Could you please elaborate on it? Thanks.

 

[Janus]

Yes that is pretty much what it means. Newton's theorem of areas showed that as long as the force acts radially, it doesn't matter how it changes with distance; an object acted on by that force will sweep out equal areas in equal time periods.

 

[jbriggs444]

Equal areas in equal time can be understood as conservation of angular momentum. Any central force conserves angular momentum about that center.

 

[PainterGuy]

Hi again,

I'm facing a problem which I'm trying to understand more qualitatively than quantitatively.

Please have a look on the attachment, or for hi-resolution have a look here: https://imageshack.com/a/img924/6138/hNTkVC.jpg

Equal areas are swept in equal times as Kepler's 2nd law says. In Figure 1 from the attachment, is there a fixed proportion between the time elapsed from X→Y and from Y→Z, i.e. XY/YZ=constant?

From looking at the figures, the relation I could see is around XY/YZ=4/2.

I understand that my question is too general and lacks any quantitative treatment so if you could possibly comment on it, I'd really appreciate. Thank you.

 

[gneill]

No. Consider the case as the eccentricity of the ellipse approaches zero (the ellipse becoming a circle). By symmetry the two times must then be equal. Clearly then the ratio cannot be the same for all (closed) orbits.

 

[PainterGuy]

Thank you.

Let me put it differently. Please have a look on the attachment. My confusion is arising from the understanding that time taken by a counterclockwise orbiting planet from G to H is T, from H to I is also T, and so on. In other words, the letters along the curve divide the curve into equal time segments. This way all the elliptical orbits with same eccentricity but with different lengths for major axis will have same number of 'T' time segments along the curve. I'm not sure if I'm correct.

To me, your reply suggests that as the eccentricity changes, the number of 'T' segments will change but my possibly flawed understanding assumes that the number of 'T' segments only remain the same as long as eccentricity is the same for the given orbit. Please guide me where I have it wrong. Thanks a lot.

 

[gneill]

Thank you.

Let me put it differently. Please have a look on the attachment. My confusion is arising from the understanding that time taken by a counterclockwise orbiting planet from G to H is T, from H to I is also T, and so on. In other words, the letters along the curve divide the curve into equal time segments. This way all the elliptical orbits with same eccentricity but with different lengths for major axis will have same number of 'T' time segments along the curve. I'm not sure if I'm correct.

That's fine, but keep in mind that the area of the ellipse can be carved into an arbitrary number of equal-area segments. So saying that different ellipses have the same number of 'T time segments' is not useful.

If, for a different ellipse with the same eccentricity you divide the area up into the same number of segments in the same way, then clearly the number of segments will be the same. However, for differently sized ellipses the actual value of T will be different.

To me, your reply suggests that as the eccentricity changes, the number of 'T' segments will change but my possibly flawed understanding assumes that the number of 'T' segments only remain the same as long as eccentricity is the same for the given orbit. Please guide me where I have it wrong. Thanks a lot.

Again, the number of segments shown is arbitrary, chosen by the illustrator to best convey the principle. You can divide any ellipse in the same way, so in that sense the number of segments is the same for all ellipses. But for any two ellipses, even those with the same eccentricity, the actual time associated with the segments will be different.

 

[PainterGuy]

Thank you.

Could you please help me to clarify another point from my fist post?

This Wikipedia article, https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion, on Kepler's laws says the following under History section in the last para:

Newton was credited with understanding that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law; while the other laws do depend on the inverse square form of the attraction.

So, if the gravitational law was of any other nature other than the inverse square relation, elliptical orbits (or, perhaps, parabolic and hyperbolic) won't possibly have come into existence. Is this what the underlined statement roughly imply? Thanks a lot.

 

[gneill]

So, if the gravitational law was of any other nature other than the inverse square relation, elliptical orbits (or, perhaps, parabolic and hyperbolic) won't possibly have come into existence. Is this what the underlined statement roughly imply?

That may well be an hypotheses that you could draw from the statement, but you would have to prove it separately to show that no other central force can produce those types of orbits. I doubt that the author meant to directly imply that though, as he was just commenting on what Newton's understanding was regarding Kepler's laws and their relation to his theory of gravity.