You see another car just ahead, cruising at a leisurely pace.

Hey it's your friend Bob!

**Hi Bob!**

So you step on the gas to approach his car.

To your horror, the harder you step on it, the further he drifts away.

But Bob is not changing his speed at all.

Eventually you start running out of fuel, but his car continues to recede in the distance.

How can this be!?!! What a nightmare!

On 3rd June 1965, two American astronauts faced this exact scenario.

James McDivitt and Edward White of the

**Gemini 4**space mission had just reached low Earth orbit at the height of about 200km.

They were instructed by mission control to attempt an orbital rendezvous with its spent second stage (a part of the rocket), which was circling the Earth at a slightly higher orbit.

This task was important to learn how to dock vehicles together in space, so that space stations and lunar missions can become reality.

Accustomed to the normal rules of flight, McDivitt aimed his spacecraft directly at the spent stage and fired his rear thrusters to catch up with it.

To his surprise, and the surprise of engineers on the ground as well, the target drifted further and further away.

It appeared as if he had accidentally thrown his own spacecraft into reverse. But there was nothing wrong with his vehicle - it was speeding up in the forward direction.

What happened? This article explains it:

This was where Newtonian mechanics kicked in.

*By increasing his craft’s speed, he had increased its distance from the earth. In this new, higher orbit, the craft’s linear velocity, measured in miles per hour, was greater than before.*

But its angular velocity—the rate at which it was traveling around the earth, measured in revolutions per hour—was lower. As Kepler had pointed out, objects in low orbits will complete an orbit around the earth faster than those in high orbits, even though their linear velocity is lower.

But its angular velocity—the rate at which it was traveling around the earth, measured in revolutions per hour—was lower. As Kepler had pointed out, objects in low orbits will complete an orbit around the earth faster than those in high orbits, even though their linear velocity is lower.

Click here to read more about the mathematics behind Kepler's equation.

At low Earth orbit, small differences in orbital height result in large differences in angular velocity. Johannes Kepler calculated this more than 300 years ago, but I guess NASA's Gemini 4 team was not prepared for such a striking effect during actual spaceflight.

Orbital mechanics is counterintuitive - unlike what you see in movies like Star Wars, you can't always approach a target simply by flying straight toward it.

Now, back to the story.

Eventually, despite many attempts, the Gemini 4 vehicle did not get any closer to the target.

McDivitt used up so much propellant that his thruster tanks drained to half-full (or half-empty?) and they decided to give up to avoid jeopardizing other mission objectives. Mission control agreed.

Although it seemed like a failure, this entire exercise was very important for the spaceflight endeavour because:

*NASA engineers and astronauts extracted a valuable lesson from this mission: It was difficult, if not impossible, to steer a spacecraft merely by eye. The orbital dynamics are so counterintuitive that—combined with the lack of references for judging distances—no human could do the job without help from electronic sensors.*

Another space mission would be needed to practice this manoeuvre.

Success finally came at end of the year. On 15th December 1965, Gemini 6A came within 30cm of its target, a passive Gemini 7 spacecraft.

These are the spacecraft shown in the photo above (Gemini 7 as seen from Gemini 6A).

How did they achieve this?

Firstly, instead of aiming straight for its target, the Gemini 6A spacecraft approached from a lower orbit so that it can catch with Gemini 7.

Next, the crew of Gemini 6A now had an onboard computer in command of the last 400km of the approach. The vehicle gradually traded angular velocity for height in a series of thruster burns until the two vehicles were just 40m apart.

Finally, at this distance, the more familiar rules of flight do apply, and the astronauts could close the gap by manually firing the thrusters.

This is fairly routine now, but imagine what went through the minds of all the people involved on that June day in 1965:

**Pressing forward makes you move backwards? WTF!!?!**

**Would you like to know more?**

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## 9 Comments:

Brilliant account of orbital motion, Lim!

I think I can tweak this to become an application question in my exams setting for my students! haha.. θδΈη ~~

Very interesting article, though I'm a bit shocked that NASA engineers wouldn't have been familiar with Kepler's work.

So that's why I can never catch up to my friend Bob... he's at a different ALTITUDE than me! If only I had read Kepler...

Keep it up, Lim!

Hi Lim,

After discussing with a colleague, we have some reservations about the reasoning employed here:

"By increasing his craft’s speed, he had increased its distance from the earth. In this new, higher orbit, the craft’s linear velocity, measured in miles per hour, was greater than before.

But its angular velocity—the rate at which it was traveling around the earth, measured in revolutions per hour—was lower. As Kepler had pointed out, objects in low orbits will complete an orbit around the earth faster than those in high orbits, even though their linear velocity is lower."

--------------------

"By increasing his craft’s speed, he had increased its distance from the earth."It was really by

directing the thrust away from earththat the distance from earth increased. One could direct the thrust towards earth and increase speed towards earth, thereby moving to a lower orbit.------------------------

"In this new, higher orbit, the craft’s linear velocity, measured in miles per hour, was greater than before."If the new higher orbit is a stable one, the total energy is greater than before. BUT, the new KE and hence linear speed should have been lower than the original KE in the lower orbit.

[ KE = GMm/(2r) ]

------------------------

angular velocity was lower.Yes. This is correct. In a higher orbit, the angular velocity is lower.

[ angular speed is

inversely proportionalto radius of orbit to the power of 3/2. - Kepler's law]------------------------

"objects in low orbits will complete an orbit around the earth faster than those in high orbits, even though their linear velocity is lower."The first part is correct, given Kepler's law. BUT, the linear speed is higher in a lower orbit.

[ v = (GM/r)^1/2 ]

------------------------

Hmmm... looks like some dubious reasoning employed here. Can anyone here care to check if I'm right and shed some more light on this?

To Kamel:

I'm pretty sure they were aware of Kepler's work, but I'm not certain why Gemini 4 was such a struggle for them. To read original mission reports and summaries, here is a useful resource -

http://www.geocities.com/bobandrepont/geminipdf.htm

To Teck:

Thanks for checking the quoted information!

Yes, the choice of words in the article is misleading, but I believe the concept is mainly correct.

Let's discuss this.

"By increasing his craft’s speed, he had increased its distance from the earth."

It was really by directing the thrust away from earth that the distance from earth increased. One could direct the thrust towards earth and increase speed towards earth, thereby moving to a lower orbit.

Not necessarily - because there are vectors involved (Oh no! Not vectors!!! :P )

The motion of a body in orbit can be resolved into two velocity vectors, the horizontal component (along a fixed altitude) and the radial component (towards/away from Earth).

If the astronaut fired engines such that the horizontal component of velocity increases, the vehicle will climb to a higher orbit.

Of course the radial component also matters, but resultant motion depends on both vectors - pointing downwards doesn't always lead to a lower orbit.

The easiest way (and much easier to calculate too) to drop to a lower orbit is reduce the horizontal component of velocity by flipping around and firing engines horizontally.

The space shuttle does this routinely to de-orbit for re-entry.

"In this new, higher orbit, the craft’s linear velocity, measured in miles per hour, was greater than before."

If the new higher orbit is a stable one, the total energy is greater than before. BUT, the new KE and hence linear speed should have been lower than the original KE in the lower orbit.

[ KE = GMm/(2r) ]

Again vectors are involved here. Let's say the target is ahead and above. The astronaut aims at the target, tilts his spacecraft to say, 45 degrees upward and fires engines.

As his vehicle increases in altitude, the horizontal component of velocity reduces (this is governed by Kepler's 2nd Law) but the radial component must necessarily increase, otherwise the vehicle can't climb to a higher orbit.

The resultant speed of his vehicle, measured in its direction of travel, does increase.

During orbital manoeuvers, the orbits of space vehicles are not circular, so the mathematics gets rather involved.

For more details about orbital mechanics, this is a superb resource:

http://aerospacescholars.jsc.nasa.gov/HAS/cirr/ss/3/2.cfm

Hope that this information will be helpful to you and your students!

Hi Lim,

Thanks for the explanation. Looks like u r quite a physics pro!

Will chew on the explanation and think over it..

Lim - The Biological Physicist

Excellent post Lim, thank you! I didn't know anything about orbital mechanics, but your explanation was clear. You really need to write books. Very engaging.

Hi John and Glendon, thanks for the encouragment. Truth be told my maths isn't that good, but I see the beauty of physics and try hard to make physics stories comprehensible to everyone.

I'm really busy now with work, but I'll put up a post about cancer biology soon, for the first Bayblab blog carnival.

Cheers!

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