Practical Astronomy Focus: A Deep Dive Into Kepler's 2nd Law Of Planetary Motion

 Kepler's 2nd law of planetary motion, i.e.  a planet completes equal areas of its orbit in equal intervals of time, is perhaps the most important of the 3 laws.  In assessing it the diagram shown below is instructive. 

 In the first instance we have for the change in area, D A :

D A  =    ½  [r   x (r +  D  r )] =   r^  x   Dr  

Now, the areal velocity at P  is by definition:

lim Dt _®0  (D A /D t)  

Then :  dA/ dt =  ½   r   x  r'  =     ½  (  r ²   dq  /dt )

Or   dA/dt =  h/ 2

(Since:  h  =  r ²   dq  /dt )

We can also use a "torque" model for radial motion,  in concert with the polar coordinates of a point mass, to further examine an orbit of radius r under gravitational attraction to the Sun,  viz:

For the radial acceleration:

a _r   =   dr ²/ dt ²   -  r (dq  /dt ) ²

Similarly:

a _q  =  r (d² q  /dt ²)   + 2 dr/dt ((dq  /dt )  

In polar coordinates we are also interested in the angular momentum, e.g.

L = mr ²   dq  /dt   =   r p _q  =  r p sin q 

Note that the angular momentum for a planet orbiting the Sun is constant! The force components in polar coordinates are:

F _r    =  m[dr ²/ dt ²   -  r (dq  /dt ) ² ]

F q    =  m  [r (d² q  /dt ²)   + 2 dr/dt ((dq  /dt ) ]

If we take:  d/dt (r p _q )  we get: 

r  d p _q /dt   =   r F q    =   dL/ dt

The rate of change of angular momentum which is also called the torque of the force F about the point 0, is.

 

t   =   dL/ dt

The change in angular momentum is then:

dL/ dt = r F q    =

 m  [r (d² q  /dt ²)   + 2 dr/dt ((dq /dt ) ]

=   d/ dt  m r ² (dq  /dt)

Note the angular momentum per unit mass m is just:

h  =  L/ m  =  r ²   dq  /dt   =  r p _q   /  m

This  quantity is exactly the h used in the Kepler 2^nd law, and a constant of the motion.  In effect, with a bit of further working: 

r ²   dq  /dt   =  2 dA/ dt  =   h  or dA/dt =  h/ 2

The preceding can be integrated to obtain:

A  =  ½  h t +   c

Then the area swept out  by the radius vector in time  t   -  t ₀ ,  according to Kepler’s 2^nd law would be:

A  =   ½  h (t   -  t ₀)

Given a period of revolution P,   i.e. for elliptic motion about a fixed origin, when t = t + P, and P > 0, the area swept out is:

A  +  p ab  =    ½  h (t  + P   -  t ₀)

b  =  a (1   -  e ² ) ^1/2

 Subtracting the first equation above from the second, i.e.

½  h (t  + P   -  t ₀)  -   A  +  p  ab 

And solving for P, one arrives at:

P  =   2 p  ab h ^-1

An alternative approach uses:

dq =    h dt/  r ²

Then integrate to obtain:

q =  q ₀    +    ò ^t  ₀      h dt/  r ² (t) 

Application Problem:

An asteroid moves in an elliptic orbit around the Sun. The lengths of the major and minor axes are 2a and 2b, respectively. If the asteroid’s velocity at the point of closest approach (where it crosses the major axis) is   v _o   then how much time is needed for the object to make one complete orbit?  (Hint:  Take the area of an ellipse to be:   p ab  )

Degenerate Leaders Who Gut Foreign Aid - Inciting Violence Among The Most Destitute - Do Not Merit A Nobel Peace Prize

 

                Trump would rather let poorest kill each other than allow food aid

What the ongoing Trump tyranny shows is a yen for cruelty and no regard for the poorest and most destitute on our planet. This was confirmed in the recent WSJ piece: 'African Refugees Scrape, Scrap As Aid Ebbs', Aug. 12, p. A16).  Noting:  

"With the Trump Administration's gutting of humanitarian aid to Africa, refugees from war-torn countries have been reduced to literally fighting each other over the scraps of food that remain."

In other words, these monsters who've replaced a once functional government with a criminal extortion enterprise opposed to the rule of law - have deliberately created and stoked an ultimate version of the hunger games in the poorest region of the world. What once distinguished America as a land of compassion and generosity has now mutated - since January 20th - into a despot-ruled rogue nation.  One dominated by a rogue criminal (convicted felon) and traitor intent on spreading misery and degradation to the most vulnerable.

Well, we should not be surprised if we possess any sentience. After all, as Chris Hayes recently showed on ALL In, e.g.

Hayes blasts Trump: ‘Most pro-criminal president of my lifetime’

Trump is the most unabashedly pro-criminal president in our lifetimes. I mean, frankly, he makes Richard Nixon look like a veritable choir boy. Indeed, one recent Facebook post best summed up the warp and woof of the demented orange fungal turd who calls himself a president:

It expresses exactly the nature of this deranged imp who - according to the WSJ lead story last Thursday - is now "unfettered,  no controls". So believes he can do whatever he wants, to anyone. In the words of Rice University historian Douglas Brinkley (WSJ, p. A4, 8/28):

"Trump is motivated by having control over all American institutions. He seems to want to grab everyone by the neck and say 'I'm in charge'."

He also seems to believe he's worthy of a Nobel Peace Prize for having a few delusions about settling some recent dustups - like between India and Pakistan over Kashmir. But any so-called leader who cuts off critical food assistance to the neediest of our world - thereby spawning internecine refugee violence -  merits no peace prize.  

He more merits a lump of coal atop his Mickey D burger and gator piss in his diet coke. Especially after the recent outbreaks of refugee upon refugee mayhem as recently occurred in South Sudan and Uganda. As one Norwegian citizen aptly summed up Dotard's aspirations in a Facebook entry:

As the WSJ piece about African Refugees reported: 

"At Uganda's sprawling Kiryandongo refugee settlement, residents who had fled South Sudan attacked the shelters of new arrivals from Sudan, stealing food, killing one and injuring almost 100."

Meanwhile:

"During a four day rampage in July, hundreds of South Sudanese refugees, armed with machetes and sticks, stormed a large, separate compound housing Sudanese newcomers."

Making Trump directly responsible for the refugee -on- refugee mayhem, given the food scarcity and desperation spawned by his gutting of aid. The WSJ piece again:

"Until President Trump took a chain saw to U.S. foreign assistance programs this year, American funding accounted for 60% of U.N. World Food program relief efforts in Uganda...and 40% of the global humanitarian aid budget."

In other words, Trump's taking a chain saw to vital food aid directly resulted in bestial, inhuman behavior and violence. Predictable when scarcity faces the destitute driven to desperation by starvation and disease.

 Nobel Peace Prize?  This Turd doesn't deserve a wad of warm doggie lickspittle.

See Also:

Defying Congress, Trump Moves to Cut $4.9 Billion in Foreign Aid - The New York Times

Excerpt:

The move largely targets accounts funding the United States’ contributions to the United Nations and soft power programs run by the State Department and the U.S. Agency for International Development, which has already largely been dismantled by the Trump administration.

The single biggest clawback would be a $445 million cut to U.S. funding of peacekeeping operations abroad, including through the United Nations. The request also proposes a $132-million rescission of the $140 million approved by Congress for the Democracy Fund at the State Department

And:

Trump Declares He's Now Beyond All Laws Because Of "Divine Orders" - He's Flat Wrong

And:

Yale Professor Gives Darker Reasons For Why Trump Unconditionally Pardoned Over 600 Violent Insurrectionists

And:

Other Voices Weigh In On Trump's Fascist LA Power Grab

And:

'Blatantly unconstitutional': Law professor says key Trump executive orders won't survive

And:

Trump's Outbursts & Castigation Of Zelensky Reveals The Orange Turd As The Hard Russian Asset We Always Suspected

And:

Donald Trump

Bully-in-Chief Trump Delights in Cruelty

by Andrea Mazzarino | August 28, 2025 - 4:55am | permalink

— from TomDispatch

US President Donald Trump, his cabinet, and those who have profited from his rise seem to revel in public displays of cruelty.

Take former Department of Government Efficiency (DOGE) head Elon Musk, holding a chainsaw at a televised event to celebrate the firing of civil servants. Or Trump’s White House sharing a video featuring Immigration and Customs Enforcement (ICE) officers marching handcuffed immigrants onto a deportation flight, with Jess Glynne’s musical hit “Hold My Hand” playing in the background. Or how about ICE allowing right-wing TV host Dr. Phil to film its sweeping immigration raids for public consumption? And don’t forget those federal agents tackling Democratic California Sen. Alex Padilla to the floor (and handcuffing him!) when he asked a question at a Department of Homeland Security press conference. Or what about during the first Trump presidential campaign, when the then-candidate boasted that he could shoot someone on Fifth Avenue in New York City and he wouldn’t lose a voter?

» article continues...

Solving More Difficult Partial Differential Equations (Part 5, Sec. iii)

 Look at the condition u(x,y,0) =

 T _o  xy (x - a) (y - b)

Then:

 å^¥ _n=1  å^¥ _m= 1 a _nm sin (np/ a) x sin  (mp/ b)

Suppose a function f(x, y) which is periodic of period 2p 

Consider y fixed and suppose the function has the Fourier Series expansion;

f(x, y) =   å^¥ _n=o [a (y) cos nx + b(y) sin nx]

Suppose in addition that a(y) and b(y) have Fourier series expansions, e.g.

a(y) = å^¥ _n=o [a _nm cos ny + b _nm sin ny]

And:

b(y) = å^¥ _m=o  (g _nm cos my + d _nm sin ny]

Substitution of a _n (y),  b _n (y) into series for f(x,y) results in terms of the form:

cos nx cos my, cos nx sin my, sin nx cos my, sin nx sin my

Hence:

f(x, y) = å^¥ _n=o å^¥ _m=o  a _nm (cos nx cos my) + b _nm (cos nx sin my)  +

c _nm (sin nx cos my) +   d _nm (sin nx sin my)

Note that the preceding is a double Fourier series.

This brief diversion enables us to continue our treatment as follows, so writing:

ò ^p _-p  f(x,y) sin ax dx = p å^¥ _m=1   c _am cos my + p å^¥ _m=1  d _am sin my

Next, multiply both sides by sin by and integrate, i.e.

Which tells us

ò ^p _{-p   [}ò ^p _-p  f(x,y) sin ax dx] sin by dy

Þ

å _m  ò ^p _-p  d _am sin by  sin my dy   =  p d _ab

Which tells us:

 d _ab   =   1/ p ²  ò ^p _{-p   [}ò ^p _-p  f(x,y) sin ax sin by dx dy

Similarly:  

a _ab   = 1/ p ò ^p _{-p   [}ò ^p _-p  f(x,y) cos ax cos by dx dy

b _ab   = 1/ p ò ^p _{-p   [}ò ^p _-p  f(x,y) cos ax sin by dx dy

c _ab   = 1/ p ò ^p _{-p   [}ò ^p _-p  f(x,y) sin ax cos by dx dy

And further:  

d _n0 =  d _0m   =    c _0m  =    b _n0

_{Now take:}

_{ò ^p -p   [ò ^p -p  f(x,y) sin ax dx dy =}  p (2 p c _a0)

c _a0   = 1/ 2p ²  ò ^p _{-p   [}ò ^p _-p  f(x,y) sin ax dx dy

And:

b _0b   =   1/ 2p ²  ò ^p _{-p   [}ò ^p _-p  f(x,y) sin by dx dy

Using the preceding we may write:

 1/ 2p ²  ò ^p _{-p   [}ò ^p _-p  f(x,y) cos ax dx dy =  a _a0

Similarly:

a _ab   = 1/ 2p ²  ò ^p _{-p   [}ò ^p _-p  f(x,y) cos by dx dy 

To get  a _a0:    Take:  ò ^p _-p  f(x,y)  dx = 

2p å^¥ _m=0   a _0m cos my  + 2p å^¥ _m=0   b _0m sin my  etc.

If f(x,y) is an odd function in x and y then:

a _oo =  a _0m   =  a _nm  =    b _0m  =    b _nm=    c _no  =  c _nm  = 0

The series then reduces to:

å^¥ _n=1  å^¥ _{m= 1   d nm} sin nx sin my

And:

d _nm   = 4/p ²  ò ^p _{-p   [}ò ^p _-p  f(x,y) sin nx sin my dx dy   {if f(x,y) odd