▁a ▁hyper bol a . ▁ ▁In ▁particular , ▁ ▁for ▁perfectly ▁circular ▁or bits ▁( the ▁central ▁force ▁exactly ▁equals ▁the ▁cent rip etal ▁force ▁requirement , ▁which ▁determ ines ▁the ▁required ▁angular ▁velocity ▁for ▁a ▁given ▁circular ▁radius ). ▁ ▁For ▁a ▁rep uls ive ▁force ▁( k   >   0 ) ▁only ▁e   >   1 ▁applies . ▁ ▁Solution ▁in ▁ped al ▁coordinates ▁If ▁we ▁restrict ▁ourselves ▁to ▁the ▁orbit ing ▁plane , ▁there ▁is ▁an ▁easy ▁way ▁how ▁to ▁obtain ▁a ▁rough ▁shape ▁of ▁the ▁orbit ▁( without ▁the ▁information ▁about ▁paramet riz ation ) ▁in ▁ped al ▁coordinates . ▁Remember ▁that ▁a ▁given ▁point ▁ ▁on ▁a ▁curve ▁in ▁ped al ▁coordinates ▁is ▁given ▁by ▁two ▁numbers ▁, ▁where ▁ ▁is ▁the ▁distance ▁from ▁the ▁origin ▁and ▁ ▁is ▁the ▁distance ▁of ▁the ▁origin ▁to ▁the ▁tang ent ▁line ▁at ▁ ▁( the ▁symbol ▁ ▁stands ▁for ▁a ▁vector ▁per pend icular ▁to ▁— ex act ▁orientation ▁is ▁un important ▁here ). ▁ ▁The ▁Ke pler ▁problem ▁in ▁a ▁plane ▁asks ▁for ▁a ▁solution ▁of ▁the ▁system ▁of ▁differential ▁equations : ▁▁▁▁ ▁where ▁ ▁is ▁the ▁product ▁of ▁the ▁gravit ational ▁body ' s ▁mass ▁and ▁gravit ational ▁constant . ▁M aking ▁the ▁scalar ▁product ▁of ▁the ▁equation ▁with ▁ ▁we ▁obtain ▁▁▁▁▁ ▁Integr ating ▁we ▁get ▁the ▁first ▁conser ved ▁quantity ▁: ▁▁▁▁ ▁which ▁corresponds ▁to ▁the ▁energy ▁of ▁the ▁orbit ing ▁object . ▁Similarly , ▁making ▁the ▁scalar ▁product ▁with ▁ ▁we ▁get