Blog: Sputnik I was launched into orbit around Earth in 1957. It had a perigee (the closest approach to Earth, measured from Earth’s center) of 6.81 x...

Thursday, June 25, 2015

Sputnik I was launched into orbit around Earth in 1957. It had a perigee (the closest approach to Earth, measured from Earth’s center) of 6.81 x...

Hello!

When a relatively light body rotates around much heavier body, its orbit is elliptical with the heavier object at one focus of this ellipse. And the speed of a smaller object is

$\displaystyle{v}=\sqrt{{{G}\cdot{M}\cdot{\left(\frac{{2}}{{r}}-\frac{{1}}{{a}}\right)}}},$

where $\displaystyle{G}$ is the constant of universe gravitation, $\displaystyle{M}$ is the mass of a heavier body, $\displaystyle{r}$ is the current distance between objects and $\displaystyle{a}$ is the major semi-axis of the elliptic orbit.

Denote the minimal distance as $\displaystyle{r}_{{p}}$ and the maximum as $\displaystyle{r}_{{a}}.$ Then $\displaystyle{r}_{{p}}+{r}_{{a}}={2}{a},$ or

$\displaystyle{a}=\frac{{{r}_{{p}}+{r}_{{a}}}}{{2}}.$

Also, at the perigee $\displaystyle{r}={r}_{{p}}.$ And we know $\displaystyle{G}$ and $\displaystyle{M}.$

The final formula is

$\displaystyle{v}_{{p}}=\sqrt{{{G}\cdot{M}\cdot{\left(\frac{{2}}{{r}_{{p}}}-\frac{{2}}{{{r}_{{p}}+{r}_{{a}}}}\right)}}}.$

In numbers,

$\displaystyle{v}_{{p}}=\sqrt{{{5.97}\cdot{{10}}^{{24}}\cdot{6.67}\cdot{{10}}^{{-{11}}}\cdot{2}\cdot{\left(\frac{{1}}{{{6.81}\cdot{{10}}^{{6}}}}+\frac{{1}}{{{6.81}\cdot{{10}}^{{6}}+{7.53}\cdot{{10}}^{{6}}}}\right)}}},$

which is approximately

$\displaystyle\sqrt{{{79.64}\cdot{{10}}^{{7}}\cdot{\left({0.147}+{0.070}\right)}}}\approx{13141}{\left(\frac{{m}}{{s}}\right)}.$

P.S. Sources give the completely different values for $\displaystyle{r}_{{p}}$ and $\displaystyle{r}_{{a}}.$

P.P.S. The main formula may be derived from the basic physical laws, see the first link.

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Thursday, June 25, 2015

Sputnik I was launched into orbit around Earth in 1957. It had a perigee (the closest approach to Earth, measured from Earth’s center) of 6.81 x...

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