CarlB
Sep14-06, 01:43 AM
It includes differential equations for Schwarzschild and Painleve coordinates:
Effects of general relativity in the motion of minor planets and comets
G. Sitarski,
Acta Astronomica (ISSN 0001-5237), vol. 33, no. 2, 1983, p. 295-304
http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1983AcA....33..295S&db_key=AST&d ata_type=HTML&format=&high=446dbc38dc06742
The Schwarzschild differential equation is:
\ddot{\vec{r}} = -\frac{\vec{r}}{r^3} +\vec{r}\left(
\frac{2}{r^4}-\frac{2\dot{vec{r}}\cdot\dot{\vec{r}}}{r^3}
+\frac{3(\vec{r}\cdot\dot{\vec{r}}}{r^5}\right)
+\dot{\vec{r}}\frac{2\vec{r}\cdot\dot{\vec{r}}}{r^ 3}.
For Painleve, the paper gives:
\ddot{\vec{r}} = -k^2\frac{\vec{r}}{r^3}\left(1 + 3\vec{r}\cdot\vec{r} - 6\dot{r}^2\right)
Of course I see now that they took a first order approximation of the true Schwarzschild metric. There should be factors of (r-2) in the denominators.
Carl
Effects of general relativity in the motion of minor planets and comets
G. Sitarski,
Acta Astronomica (ISSN 0001-5237), vol. 33, no. 2, 1983, p. 295-304
http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1983AcA....33..295S&db_key=AST&d ata_type=HTML&format=&high=446dbc38dc06742
The Schwarzschild differential equation is:
\ddot{\vec{r}} = -\frac{\vec{r}}{r^3} +\vec{r}\left(
\frac{2}{r^4}-\frac{2\dot{vec{r}}\cdot\dot{\vec{r}}}{r^3}
+\frac{3(\vec{r}\cdot\dot{\vec{r}}}{r^5}\right)
+\dot{\vec{r}}\frac{2\vec{r}\cdot\dot{\vec{r}}}{r^ 3}.
For Painleve, the paper gives:
\ddot{\vec{r}} = -k^2\frac{\vec{r}}{r^3}\left(1 + 3\vec{r}\cdot\vec{r} - 6\dot{r}^2\right)
Of course I see now that they took a first order approximation of the true Schwarzschild metric. There should be factors of (r-2) in the denominators.
Carl