- #1
Mike2
- 1,313
- 0
The gravitational potential, U, can be calculated at any point, [tex]\[
{\rm{\vec r}}
\][/tex], for a mass density distribution, [tex]\[
{\rm{\rho (r)}}
\][/tex], using the formula:
[tex]\[
{\rm{U = - }}\int_{{\rm{all space}}} {G\frac{{{\rm{\rho }}({\rm{\vec r')}}}}{{\left| {{\rm{ \vec r - \vec r' }}} \right|}}} \,\,\,d^3 {\rm{r'}}
\][/tex].
See:
http://scienceworld.wolfram.com/physics/GravitationalPotential.html
My question is how is this calculated for points inside the distribution. For points outside the distribution, [tex]\[
{\rm{\rho (r)}}
\][/tex] is zero, and there is no problem. But inside the distribution where [tex]\[
{\rm{\rho (r)}}
\][/tex] is not zero, there will be points where [tex]\[
{{\rm{\vec r - \vec r' }}}
\]
[/tex] does become zero and send the integrand to infinity. Wouldn't this cause a problem? And how would you work around it? Thanks.
Could it be as simple as adding a small constant in the denominator that goes to zero after the integration, such as:
[tex]\[
{\rm{U(\vec r) = }}\mathop {\lim }\limits_{\varepsilon \to 0} {\rm{( - }}\int_{{\rm{all space}}} {G\frac{{{\rm{\rho }}({\rm{\vec r')}}}}{{\left| {{\rm{ \vec r - \vec r' }}} \right| + \varepsilon }}} \,\,\,d^3 {\rm{r')}}
\][/tex]
{\rm{\vec r}}
\][/tex], for a mass density distribution, [tex]\[
{\rm{\rho (r)}}
\][/tex], using the formula:
[tex]\[
{\rm{U = - }}\int_{{\rm{all space}}} {G\frac{{{\rm{\rho }}({\rm{\vec r')}}}}{{\left| {{\rm{ \vec r - \vec r' }}} \right|}}} \,\,\,d^3 {\rm{r'}}
\][/tex].
See:
http://scienceworld.wolfram.com/physics/GravitationalPotential.html
My question is how is this calculated for points inside the distribution. For points outside the distribution, [tex]\[
{\rm{\rho (r)}}
\][/tex] is zero, and there is no problem. But inside the distribution where [tex]\[
{\rm{\rho (r)}}
\][/tex] is not zero, there will be points where [tex]\[
{{\rm{\vec r - \vec r' }}}
\]
[/tex] does become zero and send the integrand to infinity. Wouldn't this cause a problem? And how would you work around it? Thanks.
Could it be as simple as adding a small constant in the denominator that goes to zero after the integration, such as:
[tex]\[
{\rm{U(\vec r) = }}\mathop {\lim }\limits_{\varepsilon \to 0} {\rm{( - }}\int_{{\rm{all space}}} {G\frac{{{\rm{\rho }}({\rm{\vec r')}}}}{{\left| {{\rm{ \vec r - \vec r' }}} \right| + \varepsilon }}} \,\,\,d^3 {\rm{r')}}
\][/tex]
Last edited: