One of the demands and difficult tasks for indoor bonsai growing is that tree needs enough light in order to develop. When having the tree inside the word “enough” usually refers to “provide as much as you can”. In the most if not all cases it is really true. It is very difficult, if even possible, to provide too much light and yet live in the same rooms together with the trees 😀
Nevertheless, I questioned myself often what is “enough” and how “good” light conditions my trees have inside.
Before jumping into conclusions and start just assuming 100k lx all around the tree would fulfill the requirements I tried to calculate how much light a tree receives from the sun during one day in nature. The calculation and derivation of the equations can be found here. As mentioned in that page if the reader notices a mistake in derivations/approximations/assumptions please do comment the page. Actually, comment the page in any case if you like 🙂 Any feedback is better than no feedback. It was not intended to overengineer the subject, which perhaps I managed exactly doing that, but to think a little bit how some values are related around light in general.
I wanted to have an equation which shows me approximate light energy the tree receives during the day as a function of the tree height. With that approach it would be easy to estimate the quality of light conditions a specific tree grows in.
Results
The table 1 shows the derived equation dependent on tree height (H) and all other parameters already given under conditions mentioned under the link above. The table also contains the equations derived for 2 special cases. All explained in the link above.
The last column (fictive constant illuminance) contains lux values if the sun light would be constant the whole length of the day (12h) and perpendicular at each bit of the surface. The values help to compare an already existing indoor lighting or location for indoor bonsai growing by measuring it with a lightmeter.
Light type 
Luminous energy,
f(E,H)
[lumen * h] 
Luminous energy,
f(H)
[lumen * h] 
Luminous
power [lumen]

Fictive constant illuminance
[lx]

Direct

9.82∗E_{v}∗H^{2}

785600*H^{2}

65466*H^{2}

41898

Diffuse

18.84∗E_{0}∗H^{2}

376800*H^{2}

31400*H^{2}

20096

Total

(9.82*E_{v}+18.84*E_{0})*H^{2}

1162400*H^{2}

96866*H^{2}

61994

Cloudy day

18.84∗Ec∗H^{2}

37680*H^{2}

3140*H^{2}

2009

Allaroundsun

18.84∗E_{v}∗H^{2}

1884000*H^{2}

157000*H^{2}

100000 = E_{v}

Trees upper
half only 
6.33∗E_{v}∗H^{2}

633000*H^{2}

52750*H^{2}

33760

The table 2 is focused in estimating how many LED spot lights are enough to provide the same amount of luminous energy as received by the sun during the day under conditions mentioned in the link above, a perfect sunny day.
It is considered that if the table shows X LED spot lights that means that the light power is distributed over the tree uniformly. In practice it is very difficult to succeed in that. Some light will overlap and produce the peak on that spot, some light will not hit the tree foliage which is therefore lost. All of that would mean that careful consideration has to be done when distributing the light to be applied to the tree.
H=10cm

H=20cm

H=30cm


Luminous
power [lumen] 
968

3874

8717

~
Nr of LED spot lights (~400lm
each) 
23

910

2122

Table 2.
Conclusion
So, what’s the big deal in these equations/results?
Below are listed several of conclusions I see out of these numbers:
 30% of light energy the tree receives on a sunny day comes from the diffuse light.
 The upper half of the trees foliage receives half of the light energy…sounds logical?! Keep in mind that the surface of the upper part of the tree under conditions mentioned in the above link is 4 times smaller than the lower part!
 100k lx applied to the whole tree all around its foliage gives almost 2x the maximum light energy the tree gets in the nature from the sun. If the tree is being under lights for longer than 12h then the received light energy gets even higher.
The values in the tables are maximum values of light energy the tree can receive in nature! The conditions listed in the link above consider the tree standing alone on the field with no object throwing a shadow and the sky is perfectly clear. More than this the tree can not receive in the nature!
If one starts replicating the values of luminous energy found in the table above consider that those values are going to apply daily in your bonsai shelf/set or wherever the tree is located. The tree in nature does have some days without the maximum luminous energy. The tree under artificial lights does not.
Some trees might be able to extract more out of more light they receive, speeding up their development, some perhaps not. I do not have enough competence to validate more on this issue but it should be logical that some species perhaps even suffer under the max light conditions, 12h a day, every single day.
The tables allow one to compare the light levels their trees receive under the present conditions and evaluate if that is enough and how “enough” that is. Providing light for a longer period of time might in some cases be enough and these equations could help in figuring out is it so.
Example
 Tree = 20cm, ficus microcarpa
 5x LED spot lights (400lm) = 2000 lm
 lights are ON for 15h a day
 Luminous energy it receives is: 2000lm * 15h = 30k lumen hours
 or 2500 lumen during 12h of a day
 or 6 LED spot lights minimum
2500 is less than 3874 lumens from table 1 but 3874 is the maximum number a tree can receive in nature on a perfectly sunny day. The ficus is such to be able to survive with very low levels of light. That would perhaps give me a hint the tree is accustomed to being in a group or inside the darker forest, or under bigger trees shading it. If that all is true I consider 64% (100* 2500/3874 ), or 50% with some losses in light distribution, of max luminous energy to be received every day to be enough.
Jani
Nice! You ended up pretty much equal spot amounts, than I did. With my first test with 10 spots light levels were perhaps too high, now tree has 6 and is happier – but humidity is higher, so its bad science.
tato
Thanks! It is very good that there are also empirical proves of this “theory” thus I guess it makes both the empirical value of yours and theoretical value of this page seem correct 🙂
BSnaga
Ludo, burazere, ludo!
tato
Hvala 🙂