Airplane flight and the rotation of the Earth

[deZordine]

If a plane departs from the North Pole (where the Earth's rotation speed around its axis is 0 Km / h) on the median line to Romania, around the 45th parallel (where the Earth's rotation speed around its axis is 1178, 80 km / h) would the plane reach its destination only by flying to *South* ( the initial-virtual centerline), if it would use star navigation?
The distance between the North Pole and a landmark in Romania is 6000 km. If the airplane speed is on average 500 km/h, then it would reach its destination in 12 hours.
However, during the twelve hours, Romania deviated from the initial meridian line from the moment of take-off, by 12 hours x 15 degrees, which means that it would be diametrically opposed to the initial line by 180 degrees.
If I flew to Romania on the meridian line from the time of departure (meaning only to the South), after 12 hours would I arrive in Romania? Wouldn't it be diametrically opposed to the Globe? What forces should be considered?

 

[hutchphd]

If I flew to Romania on the meridian line from the time of departure (meaning only to the South), after 12 hours would I arrive in Romania?

Absent atmospheric currents (jet stream etc), if the "speed" you quote is airspeed then the answer is yes. Note that your velocity relative to the air is not the same as your velocity relative to the "fixed stars"

 

[HallsofIvy]

Your airplane is supported by the atmosphere. In addition to its motion through the atmosphere, it moves as the atmosphere moves. And the atmosphere rotates with the earth.

 

[jbriggs444]

What forces should be considered?

The usual choice is to adopt the rotating coordinate system in which the Earth and its atmosphere are at rest. In this coordinate system, there is a Coriolis force acting to deflect the aircraft rightward from its southward path. This force is small enough that the pilot will automatically counter it with some minor control inputs.

If uncompensated, the result would be a bit of uncoordinated side-slip. The remediation for a rightward side-slip would be to command a bit of left bank.

The other remediation for a right side-slip is some right rudder, but the pilot is steering a southward course, which means that any rightward course deviation has to be corrected with left rudder. And left rudder needs to be associated with left bank if the turn is to remain coordinated.

[Not a pilot here. Real pilots may control course with bank and slip with rudder similar to the way they control speed with pitch and altitude with throttle].

 

[sophiecentaur]

In this coordinate system, there is a Coriolis force acting to deflect the aircraft rightward from its southward path. This force is small enough that the pilot will automatically counter it with some minor control inputs.

There must be a similar thing when sailing in tidal waters. If you have any sense then you will 'do the triangle' on the chart and steer a course that does not point at your destination but in the direction that produces the wanted resultant Course Over Ground. It usually works OK if you do this in hourly steps because the tidal stream is pretty constant over that length of time.

If you are going further or sailing for longer (crossing the English Channel in a sailing boat is a good example) you steer a course directly towards your destination and ignore the up channel and down channel flows over 12 out 24 hours. Your COG will be wildly sinusoidal but the only work your sails have done will be in the direction of the destination. The tide (over a day) will have had little or no effect.

I can't make up my mind which of the above applies when Coriolis is significant but the heading will not need constant adjustment to eliminate Coriolis - you just predict the necessary heading and use that.

A bigger and more significant navigational procedure is to make sure that you follow a Great Circle path (constant heading takes you along a Rhumb line which is not the shortest distance and is a corkscrew path and cost more fuel) a GC course requires frequent adjustment and correction of the required heading.

 

[deZordine]

Your airplane is supported by the atmosphere. In addition to its motion through the atmosphere, it moves as the atmosphere moves. And the atmosphere rotates with the earth.

Inertia, lost of inertia, atmospheric motion (resistance to movement, it is a force that opposes the movement of anybody that moves in a fluid, so in the air)

The airplane, passes from a point with speed 0 to a point with speed 1667km / h. (the case from North pole to the Equator)
Depending on the forward speed, this may mean a lateral acceleration of 6mm / s / s.
A bus that accelerates from 0 to 60km / h in one minute has an acceleration of 23cm / s / s.
If you jump as the bus accelerates, the air inside it keeps you above an X point? Of course not!

The problem is when you are in an accelerated system.
https://youtu.be/2-UzBitLmf8
The helium balloon is not pushed, but looks for its lower pressure area, in front of the car, in its accelerated motion.
The air balloon stays in the back.
https://youtu.be/jkKEMF38FIQ

The atmosphere of the Earth, it's behave in this way?

Angular momentum (Angular momentum 15 degrees rotation every hour, at the North Pole is transposed in 0Km/h and at the Equator in 1670Km/h) plus atmospheric motion (resistance to movement, it is a force that opposes the movement of anybody that moves in a fluid, so in the air) could not maintain on the flight line (flight without deviation) a plane from the North Pole to a fixed destination from the Equator.
If it uses stellar navigation and flies a fixed heading relative to the stars while keeping its altitude constant relative to the Earth's surface (in other words in polar ECI coordinates) the plane will fly several time zones west of destination.

https://www.quora.com/If-a-plane-departs-from-the-North-Pole-speed-of-the-Earth-around-its-axis-is-0-km-h-on-the-median-line-to-Romania-where-the-Earths-rotation-speed-around-its-axis-is-1-178-80-km-h-would-the-plane-reach-its

 

[russ_watters]

I'm really not clear what point you are trying to make.

The airplane, passes from a point with speed 0 to a point with speed 1667km / h. (the case from North pole to the Equator)
Depending on the forward speed, this may mean a lateral acceleration of 6mm / s / s.
A bus that accelerates from 0 to 60km / h in one minute has an acceleration of 23cm / s / s.
If you jump as the bus accelerates, the air inside it keeps you above an X point? Of course not!

So...you agree it is insignificant for the plane?

The helium balloon is not pushed, but looks for its lower pressure area, in front of the car, in its accelerated motion.
The air balloon stays in the back.

I don't see the relevance of this.

If it uses stellar navigation and flies a fixed heading relative to the stars while keeping its altitude constant relative to the Earth's surface (in other words in polar ECI coordinates) the plane will fly several time zones west of destination.

Why would anyone do that/so what?

BTW, your links aren't working - there is something wrong with how you cut and paste them.

 

[jbriggs444]

If it uses stellar navigation and flies a fixed heading relative to the stars while keeping its altitude constant relative to the Earth's surface (in other words in polar ECI coordinates) the plane will fly several time zones west of destination.

As @sophiecentaur points out, there is a difference between heading and course over ground.

A plane on a due south heading in a 600+ mph (relative to the non-rotating frame) eastward cross-wind over a 600+ mph (relative to the non-rotating frame) eastward-moving ground will be deflected eastward at approximately 600+ mph and arrive correctly at the moving destination.

 

[sophiecentaur]

Why would anyone do that/so what?

That's a very good point but it's a valid 'what if?' question, however impractical, I think.

Any navigator would normally go for a great circle course but not if another factor came into it - such as avoiding an area of contrary high wind and (possibly?) a situation in which Coriolis was relevant - all in the interest of minimising fuel cost. I don't know how to calculate the total fuel cost when sticking to a GC course, compared with pre-compensating for Coriolis with a constant westward heading correction or a variable one over the flight. That would be regarded as 'first year work' for a long haul pilot, I'm sure.

 

[jbriggs444]

pre-compensating for Coriolis with a constant westward heading correction or a variable one over the flight

No pre-compensation for Coriolis is called for. The pilot will maintain zero left/right drift relative to the wind. His wind-relative velocity will match his heading exactly.

To say it differently, the required correction is achieved with roll, not yaw.

 

[sophiecentaur]

Yaw probably right @jbriggs444 But he can only steer a constant heading if he’s on a line of longitude.
Also the wind will not remain the same as he goes south. Coriolis will definitely have affected that, according to every weather map you can find.
It’s a combination of simple and complicated.

 

[GlenLaRocca]

First, Navigation is determining you position, velocity, and attitude. Guidance determines where the aircraft goes based on the navigation data. Star navigation would have no effect on guidance.

It is all determined by the guidance commands that fly the aircraft. Typically a roll is commanded to fly some flightpath (guidance mode)--like a great circle to get from one point (latitude, longitude) to another. The great circle calculation does not need to account for the rotating Earth and that calculation will result in a changing commanded heading (here heading means direction of the velocity vector, not attitude of the aircraft) as you progress. The roll command would be continuously calculated based on the difference between the current aircraft heading and the commanded heading, Winds and Coriolis are basically disturbances that are offset by the roll command. Because the calculations are done continuously, the great circle calculations can ignore Earth's flatness and assume a sphere.

But you can change your guidance mode from great circle to anything. If your guidance law was to fly a constant commanded heading (rhumb line steering) , then you have a few special cases that result in great circles but mostly get a strange flightpath. Starting at the equator and flying east or west will fly the equator. If you due north or south you will stay on whatever meridian you are on. Anything else results in a weird spiral tour of the planet always ending at one of the poles. The trivial guidance law would be to always command zero roll--just keep the aircraft level. This would be almost like commanding a constant heading in the direction the aircraft was initially heading except now winds and Coriolis would push the aircraft around--it would be a knuckleball. The effect of these forces would depend on the speed of the aircraft. If your guidance mode did the calculations to fly in an inertial strait line, and compensate for disturbances from that, you would go somewhere else. That somewhere else might want to leave the atmosphere or burrow through the Earth if you let your flight control system control altitude.

 

[zanick]

I wonder if flight times from Dallas to Chicago (a north east destination) are measurably faster than going north to south. it seems the coriolis force would be an aid.

 

[A.T.]

I wonder if flight times from Dallas to Chicago (a north east destination) are measurably faster than going north to south. it seems the coriolis force would be an aid.

The vertical component of the Coriolis force reduces or increases the required lift, and thus affects fuel consumption slightly, when flying east vs. west.

 

[zanick]

The vertical component of the Coriolis force reduces or increases the required lift, and thus affects fuel consumption slightly, when flying east vs. west.

you are speaking of Eotvos. that effectively lowers the weight of the plane flying east . I was specifically speaking of flying to a location north east where you could take advantage of the conservation of angular momentum. (Coriolis) with a 500mph cruise speed, the deflection might be something like 1.5mph per minute. normally going north, a pilot would fight against this unknowingly (small corrections).

 

[A.T.]

you are speaking of Eotvos.

Yes, that's what Earth Science calls the vertical Coriolis force. It has a small effect on fuel.

normally going north, a pilot would fight against this unknowingly (small corrections).

Directional corrections of the horizontal Coriolis force, would have an even smaller effect.

 

[jbriggs444]

you are speaking of Eotvos. that effectively lowers the weight of the plane flying east . I was specifically speaking of flying to a location north east where you could take advantage of the conservation of angular momentum. (Coriolis) with a 500mph cruise speed, the deflection might be something like 1.5mph per minute. normally going north, a pilot would fight against this unknowingly (small corrections).

If the plane is traveling north-east and aiming north-east (or, indeed at any angle at all) then the Coriolis force is at right angles to its path. No improvement in flight time is available in that manner (except for the component of lift mentioned already).

 

[zanick]

Your airplane is supported by the atmosphere. In addition to its motion through the atmosphere, it moves as the atmosphere moves. And the atmosphere rotates with the earth.

Yes, and all objects that would move south would have to follow the conservation of momentum laws...… so, the plane would have to have a force added to it to travel directly to Romania from the north pole. this "force" would be caused by the increase in momentum needed to reach the higher tangential velocity of that destination latitude . it would slowly be accelerated to the east via the airplane to match the new tangential velocity (and therefore, angular momentum) it was missing before it started its flight

 

[zanick]

If the plane is traveling north-east and aiming north-east (or, indeed at any angle at all) then the Coriolis force is at right angles to its path. No improvement in flight time is available in that manner (except for the component of lift mentioned already).

If there was a plane flying directly north from the equator, we all agree that there would be an eastward deflection. if the plane wanted to fly due north, it would have to use the air and the planes control surfaces to scrub off tangential velocity by slightly turning west the entire way. if you didn't have to control the plane to counteract that natural deflection, then that would imply less drag forces acting against the plane giving it a quicker flight time, or more fuel efficiency. another way to think about it, is in this thought experiment: think of a straight train track leading North. if a high speed rail car was traveling on it north, there would be force exerted on the east side of the tracks, scrubbing off tangential velocity as it traveled north.

 

[jbriggs444]

If there was a plane flying directly north from the equator, we all agree that there would be an eastward deflection. if the plane wanted to fly due north, it would have to use the air and the planes control surfaces to scrub off tangential velocity by slightly turning west the entire way. if you didn't have to control the plane to counteract that natural deflection, then that would imply less drag forces acting against the plane giving it a quicker flight time, or more fuel efficiency. another way to think about it, is in this thought experiment: think of a straight train track leading North. if a high speed rail car was traveling on it north, there would be force exerted on the east side of the tracks, scrubbing off tangential velocity as it traveled north.

Do I understand correctly that you have abandoned the claim that Coriolis assists in a northeast trajectory and are now promoting the claim that the horizontal component of Coriolis slightly impedes all trajectories where it is non-zero?

Yes, I agree with that. There is a trivial component of induced drag associated with horizontal Coriolis.

 

[GlenLaRocca]

I wonder if flight times from Dallas to Chicago (a north east destination) are measurably faster than going north to south. it seems the coriolis force would be an aid.

Coriolis, acting on the plane, is so small compared to the force of the winds and Coriolis is also driving the winds.

 

[zanick]

Coriolis, acting on the plane, is so small compared to the force of the winds and Coriolis is also driving the winds.

We could ignore the winds, and as you said, they too are assisted east via Coriolis. The conservation of momentum will point to an aid to an object traveling north east, and create a need for more energy going south east to gain that momentum back.

 

[jbriggs444]

We could ignor the winds, and as you said, they too are assisted east via Coriolis. The conservation of momentum will point to an aid to an object traveling north east, and create a need for more energy going south east to gain that momentum back.

Stuff and nonsense.

 

[zanick]

Stuff and nonsense.

Why wouldn't the higher tangential velocity at take off and therefore , greater linear momentum vs that plane a destination on the same longitude, aid in getting to the destination ?

if the rotation was a cylinder, going north, parallel its axis of rotation, eastward linear/angular momentum would be conserved and the airplane would follow a straight line north. however on a sphere, traveling north from the equator, the airplane would accelerate ahead of the surface below in the non inertial reference frame.. if it left the equator at 1030 mph tangential velocity, it would end up traveling via a north initial orientation, to a region at 45 degrees north, at a tangential velocity of 730mph. over that trip, it would end up having a eastward tangential velocity component of 300mph by the time it reached its destination, (providing some average tangential velocity over the time of the trip) conversely, in a south west path, it would take more energy to gain the momentum required to reach an eastward, lower latitude destination.

 

[jbriggs444]

Why wouldn't the higher tangential velocity at take off and therefore , greater linear momentum vs that plane a destination on the same longitude, aid in getting to the destination ?

The surface of the Earth is an equipotential surface in the rotating frame.

Edit to add...

An equipotential surface is [obviously] a surface along which potential energy is a constant. Conservation of energy allows you to conclude that an object freely following the surface retains constant kinetic energy and, accordingly, constant speed.

In the case of the surface of a rotating Earth, there are two contributions to the potential: Gravitational force and centrifugal force. Both have associated potential fields. We are all familiar with the gravitational potential field. It scales as the inverse of the radius [from the center of a spherically symmetric gravitating object as long as one is in the region outside said object].

We are not often taught about the potential field associated with the centrifugal force. However, it has one. It is, after all, a central force whose magnitude depends only on the distance from the center. It follows that it has a potential field. The force scales as the square of distance from the center. The potential (the integral of the force), accordingly, scales with the cube of the distance from the center.

If you take the sum of the gravitational potential and the centrifugal potential and find the set of points where that sum takes on a particular value, you have an equipotential surface. Along this surface, the vector sum of centrifugal force and gravitational force always points tangent to the surface.

An object that is rolling, flying or otherwise moving along this equipotential surface never experiences any forward, rearward or side-ward acceleration due to the vector sum of centrifugal and gravitational force. The vector sum of the two forces (i.e. a plumb line) is always normal to the surface.

What about Coriolis force, you may ask. The answer is that Coriolis can produce a sideways force. But a sideways force never alters an object's velocity. As judged from the rotating frame, the Coriolis force can never do work. It can never make an object move faster or more slowly. It can only change an object's direction of motion.

Billiard balls do not roll northeast more easily than southwest.

Edit again...

The above is an analysis in the rotating frame. If you are going to speak of the Coriolis force at all, you need to jump in with both feet and adopt the rotating frame. It is no fair cheating and mixing in references to effects that hold in the inertial frame. That'll just get you labelled as a "frame jumper" and all the kids on the playground will start laughing behind your back.

If you try to analyze the situation from the inertial frame you have a problem: You are not flying north-east any more. Nor are you trying to reach a point north-east of where you started. You are jumping off from a moving point and trying to intercept another moving point while swimming in a moving fluid. [The complexity of this problem is what motivates the shift to the rotating frame of reference]

Attaining a high eastward velocity is no longer an important metric for success. Instead, you'll need a westward acceleration to match speeds with your destination. That "Coriolis" boost that you were expecting to help you is actually a velocity delta that you eventually need to fight.

 

[zanick]

"Frame Jumper" That's a good one! :) yes, I agree. we don't want any frame jumpers.
Lets look at only the non-inertial reference frame from a point of reference on the ground where a plane begins it flight due north. we can also look at warm air moving north. In this reference frame, the plane is apparently deflected to the east with a measurable west to east velocity... so is the air.. so much so that the air now is moving at a faster rate east than it moved from the lower latitudes.

From the inertial reference frame, a point of reference in space, the plane /projectile, takes off with a tangential velocity of 1030mph and travels due north. as the Earth spins, it appears that the object speeds ahead of the Earth's surface as it conserves momentum with respect to the surface below. this apparent deflection is in the rotating frame , and an acceleration with respect to the Earth's surface in the inertial frame. I am thinking it's for the same reasons a skater pulls in her arms and increases angular velocity. on other side of the coin, going southward with a given amount of angular/linear momentum, the southbound flight would appear to lag behind the surface moving underneath its path, thus requiring a force to increase momentum to reach a point of higher tangential velocity.
the change in velocity (acceleration), IS due to a change of direction and not magnitude, and any change of velocity will require a force 90 degrees opposed. it is considered a pseudo force because the "force" has no origin.
let me know what you think.

The surface of the Earth is an equipotential surface in the rotating frame.

An object that is rolling, flying or otherwise moving along this equipotential surface never experiences any forward, rearward or side-ward acceleration due to the vector sum of centrifugal and gravitational force. The vector sum of the two forces (i.e. a plumb line) is always normal to the surface.

What about Coriolis force, you may ask. The answer is that Coriolis can produce a sideways force. But a sideways force never alters an object's velocity. As judged from the rotating frame, the Coriolis force can never do work. It can never make an object move faster or more slowly. It can only change an object's direction of motion.

Billiard balls do not roll northeast more easily than southwest.

Edit again...

The above is an analysis in the rotating frame. If you are going to speak of the Coriolis force at all, you need to jump in with both feet and adopt the rotating frame. It is no fair cheating and mixing in references to effects that hold in the inertial frame. That'll just get you labelled as a "frame jumper" and all the kids on the playground will start laughing behind your back.

If you try to analyze the situation from the inertial frame you have a problem: You are not flying north-east any more. Nor are you trying to reach a point north-east of where you started. You are jumping off from a moving point and trying to intercept another moving point while swimming in a moving fluid. [The complexity of this problem is what motivates the shift to the rotating frame of reference]

Attaining a high eastward velocity is no longer an important metric for success. Instead, you'll need a westward acceleration to match speeds with your destination. That "Coriolis" boost that you were expecting to help you is actually a velocity delta that you eventually need to fight.