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I have often wondered what the purpose of monopods is. It seems to me that they
remove only two degrees of freedom out of three possible degrees of freedom of
camera shake. And the situation is even worse for telephoto lenses, where one of the removed degrees of freedom isn't problematic, so that's one out of two removed. I usually don't need stabilization provided by a tripod unless it's for long exposures (where a monopod probably doesn't help) or tele photography (where I suspect monopod isn't that good either).

Camera shake can be, using aircraft terminology:

• Pitch shake: this is removed by a monopod


• Roll shake: this is removed by a monopod


• Yaw shake: this is completely unaffected by a monopod

When using a telephoto lens, my understanding is that the yaw and pitch shake become
more problematic and the problematicity of roll shake is not that big. So, when
taking a tele photograph, you are affected mainly by:

• Pitch shake: this is removed by a monopod


• Yaw shake: this is completely unaffected by a monopod

So, when taking a tele photograph, a monopod removes only one of the two
problematic degrees of freedom of camera shake.

Based on this, my intuition is that a monopod offers probably around one
additional stop in possible exposure time when taking tele photographs, if even that. Is this intuition correct? Are
monopods really useful in practice to be used with telephoto lenses? How much extra stops do they offer in
reality for tele photography?

A good image stabilizer can offer 3-4 stops of advertised improvement, and I genuinely believe it achieves most of that for intermediate exposures (not short or long), having tested the IS of Canon 55-250 mm lens.

Related: How much benefit can one expect from a monopod? ...although the existing question is for general purpose photography, not for tele photography. Here I'm interested only in answers related to tele photography, such as photographing birds or the moon with a monopod and without image stabilization (which I suspect won't be a good idea).

The monopod removes three degrees of liberty: distance from ground is fixed, and roll and pitch are linked to position in space.

But you are overlooking that you are no longer lifting the (potentially heavy) lens, so its shaking is no longer caused by your muscular control, itself affected by muscular fatigue(*).

Of course the 55-250mm is a rather light lens, so the monopod helps less with that than with a heftier 100-400...

(*) There is a poor man's monopod: attach a 6-foot piece of string under your camera. To take the picture, step on the string and pull the camera up. Fairly useful in museums where they don't allow flash or any kind of *-pods, but not as efficient as the monopod, since your muscles are tense (even if you rarely use heavy lenses in museums).

It's instructive to look where monopods are most often used: sporting events, and shooting wildlife. In all of these cases, it's not a matter of "how many stops" a monopod can provide. It's simply a matter of increasing the keeper rate of shots.

Competitive Sports (football, soccer, etc.)

Monopod are ubiquitous along the sidelines of professional football (international, American, etc.). For sports shooters, a fast shutter speed is necessary to capture the shot because the subject is moving fast. By-and-large, people like perfectly captured moments in time for these types of sports photos, with no motion blur. There are exceptions where motion blur in the background is desirable, but those shots are rare, and tend to be understood to be more artistic than the typical sports reportage.

Sports photographers have to be able to move, and move around each other, so tripods are unwieldy, impractical, and when near other sports photographers, inconsiderate. But when trying to capture events across the pitch, they need to use their longer lenses, and those lenses are heavy. So stability is required. The photographer could kneel down and use their knee as an elbow rest to provide some stability, but that limits them to shooting from a low position, and also reduces their mobility. A monopod allows comfortable shooting from a standing position, and the ability to move at a moment's notice. Even though a monopod is a compromise tool, it just so happens to be the best tool for this shooting situation.

Autosports

The distances are much greater in autosport, so there's more reliance on telephoto lenses than there is in football/stadium/arena sports. But there's an extra element not present in team/stadium/arena sports: controlled motion blur is actually desirable. Very fast shutter speeds in car racing yields boring shots, where the cars' entire motion is stopped — the wheels don't look like they're spinning. Other than perhaps a heavily loaded front corner suspension when the car is braking hard into a turn, with a fast shutter speed the cars look like they're static and parked on the asphalt, rather than dynamic.

So autosport photographers slow down the shutter speed, maybe as low as ¹⁄₃₀ s, depending on the cars' speed from the photographer's viewpoint, focal length, etc. But that's not enough, because the entire car will be blurred. The photographer also has to move the camera to follow the car's movement. This requires lots of practice, and even when done by seasoned professionals, results in a lot of subpar (unusable) shots. This shot is only really effective for a panning shot, which just so happens to be perfect for a monopod: the monopod doesn't restrict the panning axis (so-called yaw axis) at all, while removing or reducing the other axes of motion.

Incidentally, this is also why most lenses with a tripod mounting foot, and image stabilization, have 3 modes of IS: off; full on; and tripod-mode, meaning the IS ignores panning / yaw motion in its stabilization. Not specifically for motorsport, but for tracking laterally-moving subjects with a telephoto lens.

Wildlife

This is very similar to the competitive sports situation, but often at much further distances, similar to motorsports. In this case, the camera support is chosen depending on the particular subject intended to be tracked. Birders often sit in one spot, or move very infrequently and slowly, so a tripod is desirable for stability and to help carry the load of heavy supertelephoto lenses. Other game might require more mobility of the photographer, so the hassle of constantly moving and setting up a tripod would justify using a monopod instead.

This will be more than an actual answer, a complement of your question, which hopefully will help you answer your question.

Let's not use airplanes terminology. Let's use camera movements.

We have 6 camera movements related to 3-dimensional space and one additional for a total of 7.

We need to define our coordinate angles similar of those of a 3D program, with the XY plane parallel to the floor and the X-axis parallel to your feet.

Rotational movements (the ones you are describing)

1. Panning Rotating the camera on the Z angle. Rotating it left and right.




2. Tilting Rotating it upwards and downwards. (Using X axis)




3. Rolling Producing what is called a dutch angle (Y-axis)

Displacement movements

4. Dolly Moving back and forward (Y-axis)




5. Truck or travel (X-axis)




6. Pedestal Up and down. (Z-Axis)

Lens

7. Zoom

Now, the monopod does not only assist you on the rotating movements but also helps you with the pedestal one.

But let's explore more in depth. Human anatomy and the laws of physics. :o)

The force you are fighting mostly when holding a camera is the gravitational pull on the mass of the camera and lenses, using your skeleton and muscular systems... And the aim is to keep that steady using electrochemical pulsing signals firing the contraction of the Myocyte fibers...

Let's add some physiology (Yes, I am going nuts here)

Take your camera with the longest lens you have. The theory states that you should keep your arms close to your chest so it makes a steadier platform to your camera... Do you notice that the respiration produces a vertical movement and rotation?

Yes, you can hold respiration for a while before it turns into an issue. A purple photographer is not a good thing.

If a monopod helps you steading the vertical movement, and the vertical rotation, the vertical pull of the gravitational force... that is great!

The effort of steading the other movements are easier when you have the first under control.

How many stops? Depends on you, your anatomy, physiology, technique, practice and will to take that shot.

An additional point. You are probably using a monopod also because you want freedom. A monopod gives you more freedom of movement than a tripod, so all is not about fixing a camera, it is also about freeing yourself.

The most used camera movement is actually panning, and it is a good idea that this is not fixed by a monopod.

Quoting @Scottbb comment:

The monopod completely fixes/eliminates #6 (pedestal); it couples rolling and truck together (#3 & #5) tightly; and couples tilt and dolly (#2 & #4) together. That is to say, small truck or dolly vibrations are directly correlated with small rolling or tilting vibrations (within the vibrational flex limits of the monopod and mount). And for dolly especially, that motion is reduced/constrained by the photog's face, an important dampening factor/property

Here is a diagram on why only removes the pedestal, and do not removes the other, only deals with them:

So, besides your monopod, you need a photographer's face Do not forget to buy them in a bundle.

Thanks, @scottbb.

To be precise monopode do not remove completely pitch and roll. It just move the point of rotation to be not the camera itself but the point where monopod touch the ground. This mitigate a lot but do not remove completely.

About yaw shake - you should be not afraid of. You should name it freedom and I think many sport photographers will agree.

Longer monopod is better stabilization it provides
It's simple - while having a link with a ground, monopod converts all movements to circular and becomes a radius for camera movement in space, so bigger radius is - camera changes less angle while moving around for the same distance. And angle is what actually makes long-focus lens "shake" most of the time (rember external gyroscope stabilization systems - they fix angle shake) because when you move them along axis for 1cm, field of view moves also for 1cm (whatever lens focus distance is), but when you turn them for 1 degree, field of view changes depending on focus distance, and longer it is more change happens.
In practical usage, i don't like monopods (almost same weight as good true "manypod") but much less possibilities for long-exposure shots, but with modern cameras with axis stabs they work ok when you need to turn around more then long exposures.

How many extra stops do monopods offer for tele photographs?

In general I've found a monopod buys about three or four stops slower than the 1/(focal length X crop factor) rule for non-stabilized lenses. For stabilized lenses the monopod will help extend whatever benefit the Image Stabilization, Vibration Reduction, etc. provides by another stop or two.

A good tripod, on the other hand, will allow you to take much longer shots than a monopod.

So why use a monopod?

• 
  For me the primary benefit of a monopod is supporting the weight of a heavy telephoto lens during an extended shoot such as a sporting event or air show. It's not going to allow radically longer shutter speeds. For that you really do need a tripod.


• A monopod can significantly inhibit movements on the y-axis and z-axis while allowing the camera to pan along the x-axis.


• The other primary use case for a monopod is when you are in places that do not allow tripods but do allow monopods.

This image was shot while panning a plane over half a mile away with the pilot flying like his hair was on fire - at night. It was taken from a monopod at 1/60 sec, f/2.8, 200mm, ISO 6400 with a Canon EOS 7D (1.6X crop factor) + EF 70-200mm f/2.8L IS II. It is a fairly heavy crop of the full image. I was using the monopod to control movement on the y-axis and z-axis while panning with the plane's movement along the x-axis.

_{Shooting an airplane doing aerobatics while going about 300 mph from a position a half mile or more away - in the dark - is a challenge. I was panning with the plane using IS mode 2 to get anything resembling sharp. Some of the pans I shot matched the speed of the smoke more than the speed of the plane (the smoke slows ever so slightly as it is buffeted in the plane's turbulent wake before eventually slowing until it 'hangs' in the air several hundred feet behind the plane).}

This shot was taken from a monopod at 1/5 sec, f/3.5, 35mm focal length, ISO 1600. The lens used was a non-stabilized Canon EF 24-70mm f/2.8 L mounted on a Canon EOS 5D mark II. I was shooting a little over three stops below the 1/focal length rule. I had used my longer lens and a 7D on the monopod all afternoon at an air show. Not being aware there was going to be fireworks after dark, I had left my tripod in the car quite a distance from the venue. It is not razor sharp as there appears to have either been a little camera movement or my manual focusing left the foreground a little soft. Fortunately, the fireworks in the picture are very forgiving since they are very short bursts of bright light for any single spot in the photo.

This image, on the other hand, was exposed for 30 seconds and would not have been possible without a tripod or other solid camera support. There may be someone that could hold a monopod motionless for 30 seconds, but I've never met anyone that can.