Thanks, that'( i)s exactly what I needed.
Tkepler~ra1.5
Tkepler=k*ra1.5
Ok. Rooted gives
k=T/(r^1.5).
But, I do NOT see why (NOT).
Surely you mean ONLY the end result;
NOT the starting basis. ?
From Janus's post
Independent
of this thread's presentation syntax:
(as side_track)
I quickly (also) see**
(in her 1st line)
(that)
Newton's gravitational_force
Fg=G*(M/R)*(m/r)
is 2 mass_per_distance (linear_)ratios (M/R)*(m/r)
multiplied together,
& then multiplied by a proportionality constant G,
where their (common) center_to_center distances R=r
are (intended as) identical(ly the same).
(I.e. Caution (poor) syntax, (clash):
Distance, NOT just a radius.
It's a ruff approximation.
NOT to be confused with each (masses') radius separately
(with their own (different) radius size);
but instead the sum of both radii (distances + 1 height), (all) together.
E.g. r=rm+rM+h=R, h=separation_height, surface to surface.
Disclaimer: That'( i)s how I intuitively interpreted Fg, in a flash.
That example of (symbol) r has NOTHING to do with my orbit radius r, later.)
**
The Key (move=mano[e]uver): is
shift (=move)
the rooted_G*M (denominator)
1/((G*M)^0.5)
(to) under the 2*π (numerator)
(to obtain the proportionality constant k).
T=(2*Pi/((G*M)^(1/2)))*(r^(3/2)).
GM/(2pi)^2 = R^3 / T^2
(is (surely) enough info
to follow thru correctly,
& ignore the typo)
((2pi)^2)/GM = T^2 / R^3
Great inspiration.
Thanks.
---
So giving it another go, again.
A circular orbit_Period, is
Tcircle=2*Pi*((R/g)^0.5).
Equating, the Weight (Force)
Wt=m*g
of a mass m,
with Newton's gravity_Force
Fg=G*M*m/(R^2)
for the Earth's_mass M,
separated
by their total (radial) distance R
(center to center),
(& due to Newton's 3rd law
of opposite & equal reaction)
we have (equal & opposite forces, balancing (out))
Wt=Fg
m*g=G*M*m/(R^2), /m
& dividing both sides
by the (small(er)) mass m
we get
the free_fall (gravitational) acceleration
g=G*M/(R^2).
(Please Not(ic)e:
that g
is typically measured
near the Earth's surface;
but (g) gets smaller
as the separation ((e.g. orbit_)radius R)
gets large(r),
e.g. to a GSO
(geo_stationary orbit's) radius
RGSO~g*(T^2)/(2*(Pi^2))
where the weightless(ness)
gGSO=0
is zero.)
Inserting that (g, as inverse factor
1/g=(R^2)/(G*M))
into the circular orbit_Period
Tcircle=2*Pi*((R*(1/g))^0.5), gives
Tcircle=2*Pi*((R*(R^2)/(G*M))^0.5)
& we get R^3 under the root(_sign)
Tcircle=2*Pi*((R^3)/(G*M))^0.5).
The Key (move=mano[e]uver): is
shift (=move) the rooted_G*M (denominator)
1/((G*M)^0.5)
(to) under the 2*π (numerator)
(to obtain the proportionality constant k).
Tcircle=((2*Pi/(G*M))^0.5)*((R^3)^0.5), rooting the (R^3)
to (R^3)^0.5=R^(3/2)=R^1.5
we get
Tcircle=(2*Pi/((G*M)^0.5)*(R^3/2), 3/2=1.5
Tcircle=(2*Pi/((G*M)^0.5)*(R^1.5)
or ingesting(=incorporating)
the 2*π
into under the root_sign,
as rooted 4*(Pi^2);
we have
Tcircle=((4*(Pi^2)/(G*M))^0.5)*(R^1.5).
Janus's (wonderful method) tells me:
Newtonian Physics says the period of an orbiting object is
T=2*Pi*(R^(3/2))/((G*M)^0.5), *1/(T*2*Pi)
Thus
1/(2*Pi)=(R^(3/2))/(T*((G*M)^0.5)), ^2=Square both sides
1/((2*Pi)^2)=(R^3)/((T^2)*G*M), *G*M
Move G*M to the left side of the equation
(by multiplying both sides by G*M,
gives):
(G*M)/((2*Pi)^2)=(R^3)/(T^2), invert both sides
((2*Pi)^2)/(G*M)=(T^2)/(R^3), swap sides
(T^2)/(R^3)=((2*Pi)^2)/(G*M), multiply by R^3
T^2=(((2*Pi)^2)/(G*M))*(R^3), ^0.5=root both sides
T=(2*Pi/((G*M)^0.5)*(R^1.5)
So Swansont's (searched (for (proportionality)))
constant, is
k=2*Pi/((G*M)^0.5)
for the circular orbit_period
T=k*(R^1.5)
with orbit_radius R.