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Apologies, I don't know how to do the math graphics so I'll try it in words

You could just solve your first equation for $T$....

Simplistically:

For a circular orbit orbital velocity is constant at $$\sqrt{(G/r)(M+m)}$$

So for orbital period: you know $r$ (also constant), so calculate the length of the orbit (circumference, assuming it's a circle) and period is just length / velocity

Notice for a given primary the only thing that really matters is $r$.

Mass of the satellite (assuming it's artificial) is kinda negligible ($M+m$) where $M$ is the primary.

You are right about the geosynchronous orbit...but the period is determined by the radius...

You could just solve your first equation for $T$....

Simplistically:

For a circular orbit orbital velocity is constant at $$\sqrt{(G/r)(M+m)}$$

So for orbital period: you know $r$ (also constant), so calculate the length of the orbit (circumference, assuming it's a circle) and period is just length / velocity

Notice for a given primary the only thing that really matters is $r$.

Mass of the satellite (assuming it's artificial) is kinda negligible ($M+m$) where $M$ is the primary.

You are right about the geosynchronous orbit...but the period is determined by the radius...

props to HDE 22686 for adding the math graphics.

Apologies, I don't know how to do the math graphics so I'll try it in words

You could just solve your first equation for T....

Simplistically:

For a circular orbit orbital velocity is constant at sqrt((G/r)(M+m))

so for orbital period: you know r (also constant), so calculate the length of the orbit (circumference, assuming it's a circle) and period is just length / velocity

Notice for a given primary the only thing that really matters is r

Mass of the satellite (assuming it's artificial) is kinda negligible (M+m) where M is the primary.

You are right about the geosyn orbit...but the period is determined by the radius...

Apologies, I don't know how to do the math graphics so I'll try it in words

You could just solve your first equation for $T$....

Simplistically:

For a circular orbit orbital velocity is constant at $$\sqrt{(G/r)(M+m)}$$

So for orbital period: you know $r$ (also constant), so calculate the length of the orbit (circumference, assuming it's a circle) and period is just length / velocity

Notice for a given primary the only thing that really matters is $r$.

Mass of the satellite (assuming it's artificial) is kinda negligible ($M+m$) where $M$ is the primary.

You are right about the geosynchronous orbit...but the period is determined by the radius...

Apologies, I don't know how to do the math graphics so I'll try it in words

For a circular orbit orbital velocity = sqrt((G/r)(M+m))

so for orbital period: you know r, so calculate the length of the orbit (circumference, assuming it's a circle) and period is just length / velocity

Notice for a given primary the only thing that really matters is r

Mass of the satellite (assuming it's artificial) is kinda negligible (M+m) where M is the primary.

You are right about the geosyn orbit...but the period is determined by the radius...

Apologies, I don't know how to do the math graphics so I'll try it in words

You could just solve your first equation for T....

Simplistically:

For a circular orbit orbital velocity is constant at sqrt((G/r)(M+m))

so for orbital period: you know r (also constant), so calculate the length of the orbit (circumference, assuming it's a circle) and period is just length / velocity

Notice for a given primary the only thing that really matters is r

Mass of the satellite (assuming it's artificial) is kinda negligible (M+m) where M is the primary.

You are right about the geosyn orbit...but the period is determined by the radius...