The accretion luminosity is summed up for all cells surrounding the sink particle, and the total is given by 'E_acc'. This total accretion energy is then smoothed over a kernel of cells surrounding the sink, proportionally to a decreasing exponential, such that,

\Sigma ~dE_{acc} = E_{acc}

In other words, the exponential function is normalized.

We also want the exponential function to go to zero at the boundary of the kernel.

So given these 2 constraints, we have the following equation,

E_{acc} = \sum_{i=0}^{N} k(~e^{-dx*i} - e^{-Ndx} )~E_{acc}

which gives the normalization constant,

\boxed{k = \frac{1}{~\Sigma ~e^{-dx*i}-e^{-Ndx}}}

where N is the max number of cells in the kernel. Note that the sum runs over cells i=0 to 1, where the '0th' cell is the cell the sink is in, and the '4th' cell is the furthest cell from the sink in the kernel, that the exponential goes to zero at this boundary, and dx*i gives a position (which here assumes the sink is at the cell center — the actual function is described below).

Now, 2 things effect the shape of this smoothing function, 1. the number of cells in the kernel, and 2. dx.

Here is a plot showing, for dx=.1, a kernel of 4 cells (top), and a kernel of 40 cells (bottom):

As you can see, the smaller kernel looks like a triangle function, but the larger kernel begins to resemble a true exponential.

Now, keeping the kernel constant (let N=4), but changing dx from dx=.1 (top) to dx=1 (bottom).