Verify this charging circuit
Charge to AT LEAST 13.7V - maybe more. 

DO NOT discharge under 11.5V. 

IF the mains is off often and this drains often then you are in cycle mode and 14.4V should be used and still do not discharge under 11.5V. A LA battery floated at under 13.7V will die quickly. 

If floated at 13.7V the battery can be left connected indefinitely - the current will fall to zero in time.

If doing frequent discharges you should ideally charge to say 14.5 volts, hold it there "for a while" then set the voltage to 13.7V. "A while" above depends on how deep you discharged it. See Battery University - read all the lead-acid material.  

Also see these prior answers of mine some of which are relevant, and skim these for relevant material.



Modem: At 15V in to the 7809, if the modem draws 600 mA, then the 7809 will dissipate
Power = V x I = (15-9) x 0.6 = 3.6 Watts. It will need a substantial heatsink. You can reduce 7809 dissipation by placing a series resistor between supply and 7809 (NOT in battery circuit). Measure supply when supplying modem.
Rseries ~= (Vsupply -9V + 3V_headroom) / 0.6
If Vsupply = 15V then Rseries ~= (15-9+3)/.6 = 3/.6 = 5 Ohms.
Use say 4R7.  

Power in R = I^2 x R  ~= 1.8W.
Use a 5W resistor, air cooled. 

Measure Vin at 7809.
Probably needs 11 to 12V - see datasheet.
Adjust R at max modem draw to still have enough Vin to 7809.
Add capacitor at 7809 input - 100 uF at least - bigger is better.