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<H1>Elementary Orbital/Escape Velocity Calculations</H1>

<P>This program computes simple parameters for cirular orbits, escape
velocity and surface gravitation, where the mass of the secondary body
can be neglected in comparison with the primary, which is assumed to be
spherical.

<P>First, enter 2 of the following 3 parameters for the primary, in any order:
<UL>
<LI><SPAN CLASS="yellowbutton">2nd</SPAN> <SPAN CLASS="alttext">A´</SPAN> — Mass in kilograms
<LI><SPAN CLASS="yellowbutton">2nd</SPAN> <SPAN CLASS="alttext">B´</SPAN> — Density in kgm<SUP>-3</SUP>
<LI><SPAN CLASS="yellowbutton">2nd</SPAN> <SPAN CLASS="alttext">C´</SPAN> — Radius in metres
</UL>

<P>Then you can perform any of the following computations:
<UL>
<LI>Enter orbital radius (from the centre of the primary) in metres, and press
<SPAN CLASS="brownbutton">&nbsp;A&nbsp;</SPAN> to get the orbital velocity in ms<SUP>-1</SUP>.
<LI>Enter the orbital velocity in ms<SUP>-1</SUP>, and press
<SPAN CLASS="brownbutton">&nbsp;B&nbsp;</SPAN> to get the orbital radius in metres.
<LI>Enter orbital radius (from the centre of the primary) in metres, and press
<SPAN CLASS="brownbutton">&nbsp;C&nbsp;</SPAN> to get the orbital period in seconds.
<LI>Enter the orbital period in seconds, and press <SPAN CLASS="brownbutton">&nbsp;D&nbsp;</SPAN>
to get the orbital radius in metres.
<LI>Press <SPAN CLASS="yellowbutton">2nd</SPAN> <SPAN CLASS="alttext">E</SPAN> to compute
the escape velocity from the surface in ms<SUP>-1</SUP>.
<LI>Press <SPAN CLASS="yellowbutton">2nd</SPAN> <SPAN CLASS="alttext">E´</SPAN> to compute
the surface gravity in ms<SUP>-2</SUP>.
</UL>

<H2>Examples</H2>
<P>Mass of the Earth: 5.972×10<SUP>24</SUP> kg, mean radius 6371009 m:
<BLOCKQUOTE><P>5.972 EE 24 <SPAN CLASS="yellowbutton">2nd</SPAN> <SPAN CLASS="alttext">A´</SPAN><BR>
6371009 <SPAN CLASS="yellowbutton">2nd</SPAN> <SPAN CLASS="alttext">C´</SPAN>.</P></BLOCKQUOTE>
<P>Density of the Earth: <SPAN CLASS="brownbutton">RCL</SPAN> <B>11</B> shows the figure
5513.235374 kgm<SUP>-3</SUP>.
<P>Surface gravity: <SPAN CLASS="yellowbutton">2nd</SPAN> <SPAN CLASS="alttext">E´</SPAN>
shows 9.819916256 ms<SUP>-2</SUP>.
<P>Escape velocity: <SPAN CLASS="brownbutton">&nbsp;E&nbsp;</SPAN> shows
11185.95323 ms<SUP>-1</SUP>.
<P>What’s the height above the surface for a 90-minute orbit?
<SPAN CLASS="whitebutton">&nbsp;9&nbsp;</SPAN> <SPAN CLASS="whitebutton">&nbsp;0&nbsp;</SPAN>
<SPAN CLASS="yellowbutton">&nbsp;×&nbsp;</SPAN> <SPAN CLASS="whitebutton">&nbsp;6&nbsp;</SPAN>
<SPAN CLASS="whitebutton">&nbsp;0&nbsp;</SPAN> <SPAN CLASS="yellowbutton">&nbsp;=&nbsp;</SPAN>
(orbital period in seconds) <SPAN CLASS="brownbutton">&nbsp;D&nbsp;</SPAN> (orbital radius
from centre) shows 6652486.493 metres; <SPAN CLASS="yellowbutton">&nbsp;-&nbsp;</SPAN>
<SPAN CLASS="brownbutton">RCL</SPAN> <B>12</B> <SPAN CLASS="yellowbutton">&nbsp;=&nbsp;</SPAN> to
subtract planetary radius shows 281477.925 metres.

<P>What’s the radius of a geosynchronous orbit? That is, where the orbital period equals
the Earth’s rotation period. Note this is <I>not</I> exactly 24 hours; 24 hours is the
period between the Sun returning to the same angle in the sky. Because the length of the
year is 365.2422 days approximately, the rotation period is 24 × 365.2422 ÷ 366.2422 which
is closer to 23 hours, 56 minutes and 4 seconds, or 86164 seconds.
<P>So the radius is 86164 <SPAN CLASS="brownbutton">D</SPAN> which gives about 42164 km.

<H2>Formulas Used</H2>
<P>Volume of a sphere = 4π<I>R</I><SUP>3</SUP>/3 where <I>R</I> is the radius. Mass = density
× volume, or conversely density = mass ÷ volume.

<P>Orbital velocity at radius <I>r</I> = sqrt(GM/<I>r</I>) where G is the gravitational constant,
which is taken as 6.67428×10<SUP>11</SUP> m<SUP>3</SUP>kg<SUP>-1</SUP>s<SUP>-2</SUP>.

<P>Orbital period at radius <I>r</I> = orbital velocity at radius <I>r</I> × circumference
of orbit, which is 2π<I>r</I>.

<P>Escape velocity at surface = sqrt(2) × orbital velocity at surface radius.

<P>Surface gravity = GM/R<SUP>2</SUP>.

<P>Registers used: 10-14.

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