Why Does a(tau) Not Equal af in This Orbital Migration Equation?

[tony873004]

**** Edit, appearently, Latex is not working right now. Here's my best effort to duplicate this post without Latex: ***

In the following journal paper:
http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1995AJ...110..420M&data_type=PDF_HIGH&type=PRINTER&filetype=.pdf

the top of page 424 contains this formula
a(t)=a_f-delta a exp(-t/tau)

which if I'm not mistaken is the same as
a(t)=a_f-delta a e^(-t/tau)

It gives an object's semi-major axis at time t for an object that is migrating from the orbit where it formed, to where it is in the current epoch.

tau is the timescale of the migration
and a_i can be computed by
a_f-delta a

It would make sense to me that a(0) should equal a_i, and a(tau) should equal a_f.

My first assumption is correct. a(0) does equal a_i since e⁰=1.

But my second assumption can only work if e^-1 = 0, which it does not.

Does anyone care to guess what I'm doing wrong?

*** The Latex version for when TEX starts working again ***
In the following journal paper:
http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1995AJ...110..420M&data_type=PDF_HIGH&type=PRINTER&filetype=.pdf

the top of page 424 contains this formula
a(t)=a_{f}-\delta a exp(-t/\tau)

which if I'm not mistaken is the same as
a(t)=a_{f}-\delta a e^{(-t/\tau)}

It gives an object's semi-major axis at time t for an object that is migrating from the orbit where it formed, to where it is in the current epoch.

\tau is the timescale of the migration
and a_i can be computed by
a_{f}-\delta a

It would make sense to me that a(0) should equal a_i, and a(\tau) should equal a_f.

My first assumption is correct. a(0) does equal a_i since e⁰=1.

But my second assumption can only work if e^-1 = 0, which it does not.

Does anyone care to guess what I'm doing wrong?

 

[tony873004]

I figured it out. As long as I posted the question here, I might as well post the answer too incase anybody was curious. I e-mailed the author of the paper and she responded:

af is the value of a(t) at t-->infinity (i.e., very long time... many times
tau). a(tau) is just the value of 'a' at one e-folding time.