To confirm, the solenoid being used is the 24V, not the 120V as originally stated. To be able to apply a power of 180 watt I would require 7.5 amp if Im not mistaken? This should then give the force profile as shown on the graph of the 10% duty cycle.
What do you mean by "the 24V"? On that page you linked to, the solenoids are specified by AWG number and resistance, along with a maximum voltage at various duty cycles (and 24V exactly doesn't appear in the table).
When using the arduino to switch a driver circuit on and off what component would be required in the switching circuit which acts as the "switch" which is controlled by the arduino?