Editing Quintic function (section)

==Application to celestial mechanics==
Solving for the locations of the [[Lagrangian point]]s of an astronomical orbit in which the masses of both objects are non-negligible involves solving a quintic.

More precisely, the locations of ''L''<sub>2</sub> and ''L''<sub>1</sub> are the solutions to the following equations, where the gravitational forces of two masses on a third (for example, Sun and Earth on satellites such as [[Gaia probe|Gaia]] and the [[James Webb Space Telescope]] at ''L''<sub>2</sub> and [[Solar and Heliospheric Observatory|SOHO]] at ''L''<sub>1</sub>) provide the satellite's centripetal force necessary to be in a synchronous orbit with Earth around the Sun:

: <math>\frac{G m M_S}{(R \pm r)^2} \pm \frac{G m M_E}{r^2} = m \omega^2 (R \pm r)</math>

The ± sign corresponds to ''L''<sub>2</sub> and ''L''<sub>1</sub>, respectively; ''G'' is the [[gravitational constant]], ''ω'' the [[angular velocity]], ''r'' the distance of the satellite to Earth, ''R'' the distance Sun to Earth (that is, the [[semi-major axis]] of Earth's orbit), and ''m'', ''M<sub>E</sub>'', and ''M<sub>S</sub>'' are the respective masses of satellite, [[Earth]], and [[Sun]].

Using Kepler's Third Law <math>\omega^2=\frac{4 \pi^2}{P^2}=\frac{G (M_S+M_E)}{R^3}</math> and rearranging all terms yields the quintic

: <math>a r^5 + b r^4 + c r^3 + d r^2 + e r + f = 0</math>

with:
:<math> \begin{align}
&a = \pm (M_S + M_E),\\
&b = + (M_S + M_E) 3 R,\\
&c = \pm (M_S + M_E) 3 R^2,\\
&d = + (M_E \mp M_E) R^3\ (\text{thus } d = 0\text{ for } L_2),\\
&e = \pm M_E 2 R^4,\\
&f = \mp M_E R^5.
\end{align}</math>

Solving these two quintics yields {{math|1=''r'' = 1.501 × 10<sup>9</sup> ''m''}} for ''L''<sub>2</sub> and {{math|1=''r'' = 1.491 × 10<sup>9</sup> ''m''}} for ''L''<sub>1</sub>. The [[List of objects at Lagrangian points|Sun–Earth Lagrangian points]] ''L''<sub>2</sub> and ''L''<sub>1</sub> are usually given as 1.5 million km from Earth.

If the mass of the smaller object (''M''<sub>E</sub>) is much smaller than the mass of the larger object (''M''<sub>S</sub>), then the quintic equation can be greatly reduced and L<sub>1</sub> and L<sub>2</sub> are at approximately the radius of the [[Hill sphere]], given by:

: <math>r \approx R \sqrt[3]{\frac{M_E}{3 M_S}}</math>

That also yields {{math|1=''r'' = 1.5 × 10<sup>9</sup> ''m''}} for satellites at L<sub>1</sub> and L<sub>2</sub> in the Sun-Earth system.