Maximum Parallax Error

[JackHughs]

During the past few days, I participated in a parallax discussion on another thread in this forum. That discussion stimulated a bit of internet research which uncovered some very interesting results.

I've competed in various shooting disciplines over the years with a wide variety of riflescopes ranging from a 6 power Unertl Small Game scope up to Nightforce 15-55 Competition scopes. All this time, I've assumed that parallax error was a function of scope magnification. This seemed reasonable because the 6-8 power scopes had virtually no parallax error while the much higher power scopes required parallax adjustment for every change in distance.

It turns out that I was wrong. Maximum Parallax error is not a function of scope magnification. Rather it is a function of objective lens diameter. And, there is a formula to calculate maximum parallax error given the diameter or the objective lens, the distance at which parallax is zeroed, and the actual distance from muzzle to target.

Background

I wanted an inexpensive bullet-drop reticle scope for NRL-22. I'm old and slow so I chose a fixed 10 power SFP scope with a 40 mm objective. The low power provides for a wide depth of focus so I could set the focus at 70 yards and not have to take the time to refocus with each change in distance from 40 yards to 100 yards. This part works as predicted. I also assumed, incorrectly in retrospect, that the low power would also yield low parallax error from 40 yards to 100 yards. This assumption was incorrect. Zeroing parallax at 70 yards give me a 40 yard maximum parallax error of .806 MOA and a 100 yard maximum parallax error of .302 MOA. These are not insubstantial errors. Fortunately, I'm good at keeping my eye centered in the ocular.

The Maximum Parallax Error Formula

The parallax error formula, along with an excellent parallax error calculator can be found at https://www.lelandwest.com/parallax...onid=720BD8E8B3FC55A7B6D40669D1B3DFC0.cfusion

JackHughs

Note: I've edited this post to show that we are discussing "Maximum" parallax error. Based on the paper referenced in the following post, realizable parallax error can and will be less than maximum parallax error based on some additional variables. If I can, I 'll reduce the mathematics of realizable parallax error to plain English and post again.

JackHughs

 

[Advance]

This is an excellent paper that discusses this topic in detail. The key takeaway is that maximum possible parallax error does not equal the calculated parallax error for a given magnification and offset. Magnification does effect the amount of parallax error for a given offset because the exit pupil is inversely proportional to it.

Derivation of Riflescope Parallax Equation
By Michael G. Tappy

https://www.rimfirecentral.com/thre...0999&nested_view=1&sortby=oldest#post-2150999

 

[JackHughs]

I've read Mr. Tappy's paper. Here is the no-math summary.

1. Maximum Parallax Error as described in my first post is wholly dependent on objective lens diameter and wholly independent of magnification.
2. Visible (Actual) Parallax Error is dependent on three things: 1. the Maximum Parralax Error, 2. the deviation of the eye from the optical center of the scope and, 3. the scope magnification.
3. Because of magnification, Visible Parallax Error will always be less than Maximum Parallax Error
3. For any given deviation of the eye from the optical center, Visible Parallax Error will be greater at higher magnification.

Reference Mr. Tappy's Example No. 3:

A 3-9X40mm scope, where objective diameter, D = 40mm, parallax zeroed at 100 yards, and the target range at 25 yards.
Maximum Parallax Error at 25 yards = 15.00 mm (2.256 moa)
Visible Parallax Error at 25 yards with the eye offset 1 mm from optical center and magnification set to 3 = 2.25 mm (.085 moa)
Visible Parallax Error at 25 yards with the eye offset 1 mm from optical center and magnification set to 9 = 6.75 mm (.254 moa)

Conclusion

For any given scope, Visible (Actual) Parallax Error is always substantially less than Maximum Parallax Error. And, for any given scope, Visible (Actual) Parallax Error will be greater at higher levels of magnification.

Again, this is a qaulitative, no-math presentation. If you want to know more - and you enjoy math - Mr. Tappy's paper is a great read.

JackHughs

 

[JungleShooter]

Wow.

Now, there are a lot of fluff posts on this forum — I have contributed my fair share of those....


The above posts of @JackHughs and @Advance are not of that type.


Thank you for sharing this — I have immediately copied this to my Notes on Scopes document.

Matthias

 

[ChefSparrowRagazzi]

You're probably not good enough to account for the small dispersion.
Given all the other factors , on any given day.
Scope perfection cannot compensate for this. Zero mark will always move.
It's not the scope. It's ambient conditions, and, Air. And what air does when temperatures fluctuate
And when pressures change thanks to the atmosphere. Additionally , the rotation of the earth.
Not so much with airguns. But those algorithms.... persist.
Even with airguns. Or anything else considered , " accurate."
It happens and it happens often.

Factor in a 3 mph turbulence. Because that's what it is. It's not wind. Down range.
3 mph?????????????????? Now, you have to compensate for that. Additionally: Pellet weight. Additionally: efficiency consistency.
All of a sudden, it's not that simple. Regardless of glass.

In summary: You could be the best shooter the world has ever seen, or not.
If I write a book - they need pictures.

I hope you are all well, and , my message finds you, well........

 

[Airgun-hobbyist]

@JackHughs Having seen the formula for calculating parralax error and studied charted error at given ranges, I had the same thought as you regarding using a scope at a fixed setting. The concusion was the same as your's as well after use; there was an error but it "worked out" if I stayed behind the scope. Now though, what you "opened my eyes up to" is that parallax error was essentially compounded with more objective lense diameter. Thank you sir. And it brings me to this recent inadvertent discovery that I am happy to share with you...

Here's a nice clear little scope you may like to experiment with (based on my recent parallax discovery with it), I think you'll like the Discovery Optics HD 2-12X24SFIR FFP scope. This little 24mm objective scope can truly set parralax it at around 40 yards and have very minimal error up close at 20 yards and a clear image and virtually parallax free image at all magnification setting's beyond. With a good minimum and decent maximum power setting of 12X, I've found it to be a good all around scope hunting and useful at the shooting range at full magnification.

DISCOVERYOPT HD 2-12X24SFIR FFP-MIL Diameter:30MM Short Optics Scope

HD 2-12X24SFIR Compact First focal plane scopes 2-12x magnification range provides versatile zoom for hunting at various distances. A large 24mm objective lens ensures a bright and wide field of view. FFP (First Focal Plane) glass reticle design maintains identical aiming points at all...

us.discoveryopt.com

 

[JackHughs]

@JackHughs Having seen the formula for calculating parralax error and studied charted error at given ranges, I had the same thought as you regarding using a scope at a fixed setting. The concusion was the same as your's as well after use; there was an error but it "worked out" if I stayed behind the scope. Now though, what you "opened my eyes up to" is that parallax error was essentially compounded with more objective lense diameter. Thank you sir. And it brings me to this recent inadvertent discovery that I am happy to share with you...

Here's a nice clear little scope you may like to experiment with (based on my recent parallax discovery with it), I think you'll like the Discovery Optics HD 2-12X24SFIR FFP scope. This little 24mm objective scope can truly set parralax it at around 40 yards and have very minimal error up close at 20 yards and a clear image and virtually parallax free image at all magnification setting's beyond. With a good minimum and decent maximum power setting of 12X, I've found it to be a good all around scope hunting and useful at the shooting range at full magnification.

DISCOVERYOPT HD 2-12X24SFIR FFP-MIL Diameter:30MM Short Optics Scope

HD 2-12X24SFIR Compact First focal plane scopes 2-12x magnification range provides versatile zoom for hunting at various distances. A large 24mm objective lens ensures a bright and wide field of view. FFP (First Focal Plane) glass reticle design maintains identical aiming points at all...

us.discoveryopt.com

Hi,

Thank you for the scope recommendation. Among other things, it looks like a competent, low-cost scope for an AR and I would buy one in a heartbeat if I were more than a few years younger. Except for a once a month unregistered NRL22 match, I stay at the bench shooting fixed distances with RF rifles, tethered airguns, and my old, trusty Weaver T-36's.

JackHughs

 

[humbled.ag]

I can imagine the relation between the Ocular diameter, and objective tube diameter have an important part the play. With a large objective being necked down to say half it's diameter to meet the erector tube, this would be a cause for more pronounced error. My sightron with a 50mm objective tube, and 30mm erector tube seem to be more sentisitive to eye placement and proper focus to ensure proper parallax. Another sightron, with a 56mm objective, and 40mm erector tube seems* to be less sensitive, all other setting being equal. This is anecdotal to my experience.

What does the math say about this scenario?

 

[JackHughs]

I can imagine the relation between the Ocular diameter, and objective tube diameter have an important part the play. With a large objective being necked down to say half it's diameter to meet the erector tube, this would be a cause for more pronounced error. My sightron with a 50mm objective tube, and 30mm erector tube seem to be more sentisitive to eye placement and proper focus to ensure proper parallax. Another sightron, with a 56mm objective, and 40mm erector tube seems* to be less sensitive, all other setting being equal. This is anecdotal to my experience.

What does the math say about this scenario?

Hi.

The diameter of the scope body (the erector tube?) does not factor into either the Maximum Parallax Error calculation or the Actual Parallax Error calculation.

Sensitivity to eye placement is a function of the diameter of the scope's exit pupil. Most quality scopes show the diameter of the exit pupil in the specifications.

JackHughs